Multifractal analysis for non-conformal and non-compact ...reeveh//HenryWJReevePhDThesis.pdf · Multifractal analysis for non-conformal and non-compact systems Henry W. J. Reeve A

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Multifractal analysis for

non-conformal and

non-compact systems

Henry W. J. Reeve


Multifractal analysis for

non-conformal and

non-compact systems

Henry W. J. Reeve

A dissertation submitted to

the University of Bristol in

accordance with the requirements

for award of the degree of

Doctor of Philosophy in

the Faculty of Science

School of Mathematics

2012


Abstract

We shall study the multifractal analysis of Birkhoff averages in a variety of non-conformal and non-compact

settings. We begin with a brief overview of the relevant background in the dimension theory of dynamical

systems.

In Chapter 3 we investigate the packing dimension of the level sets for Birkhoff averages of continuous

potentials on a self-affine Sierpinski sponge. We also settle a question of Olsen and show that the the packing

spectrum for local dimension need not attain the full packing dimension of the repeller.

In Chapters 4 and 5 we consider the dimension theory of infinitely generated non-conformal limit sets. The

central objective is to establish conditional variational principles for the Hausdorff dimension of level sets for

Birkhoff averages. In Chapter 4 we focus on a special class of infinite non-conformal iterated function systems

which generalise the self-affine systems considered by Gatzouras and Lalley. In Chapter 5 we study typical

infinitely generated self-affine sets in the plane. The results of Chapter 5 are part of joint project with Antti

Kaenmaki.

In Chapter 6 we consider a diagonal flow on the space of unimodular lattices. We prove a conditional

variational principle for the Hausdorff dimension of the level sets for uniformly continuous potentials which

go to infinity in the cusp. We also show that both the set of points for which the Birkhoff averages tend to

infinity, and the set of points for which the Birkhoff averages do not exist, have full Hausdorff dimension.

5


Acknowledgments

I would like to begin by thanking my supervisor Dr. Thomas Jordan for his constant support and encourage-

ment throughout my time at the University of Bristol.

I would also like to thank my undergraduate tutor Charles Batty for devoloping my interest in pure math-

ematics.

I would like to thank the Engineering and Physical Sciences Research Council for their financial support.

It has been a great pleasure to collaborate with Tomas Persson from the University of Lund and Antti

Kaenmaki from the University of Jyvaskyla. I would like to thank both of them for sharing their insight and

passion for dimension theory.

I would also like to thank everyone in the Ergodic Theory and Dynamical Systems research group for creat-

ing a fantastic research atmosphere. A particular debt of gratitude goes to Alex Gorodnik and Corinna Ulcigrai

for introducing me to homogeneous and Teichmuller dynamics, respectively, and to Andrew Ferguson, Shirali

Kadyrov and Felipe Ramirez for their helpful conversations and guidance.

This PhD contains research completed during my stay at the Instytut Matematyczny PAN in Warsaw and

I would like to thank everyone at the Institute, especially Michał Rams and Feliks Przytycki, for their kind

hospitality. I would also like to thank the Marie Curie training network ”Conformal Structures and Dynamics”

for funding my time abroad.

I would like to thank all of my friends and colleagues in Bristol for making my time here so entertaining.

Finally I would like to thank my family for being an unrelenting source of strength and inspiration.

7


Author’s declaration

I declare that the work in this dissertation was carried out in accordance with the requirements of the Univer-

sity’s Regulations and Code of Practice for Research Degree Programmes and that it has not been submitted for

any other academic award. Except where indicated by specific reference in the text, the work is the candidate’s

own work. Work done in collaboration with, or with the assistance of, others is indicated as such. Any views

expressed in the dissertation are those of the author.

Signed:

Date:

9


Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Author’s declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1 Introduction 15

1.1 The dimension theory of dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Iterated function systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.1 Conformal iterated function systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.2 Non-conformal iterated function systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 Ergodic theory and the thermodynamic formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.1 The dimension of an ergodic measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.2 Ergodic measures of maximal Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . 19

1.4 Multifractal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.5 A summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Background 23

2.1 Geometric measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.1 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.2 Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.3 Packing dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1.4 Mass distribution principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.5 Frostman’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.6 Marstrand’s fibre lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.7 Box counting dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11

12 Table of contents

2.2.1 Topological dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.2 Measure preserving systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.3 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.4 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.5 Birkhoff’s ergodic theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Entropy theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.2 Information and the entropy of a partition . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.3 The entropy of a transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.4 Calculating entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.5 The entropy map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.6 Conditional information and conditional entropy . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.7 The Abramov Rokhlin entropy formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3.8 The Shannon-McMillan-Breimann theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4 Symbolic spaces and the left shift operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.1 The symbolic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.2 Metric structure of the symbolic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.3 Invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.4 Entropy for the left shift map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 The thermodynamic formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5.1 The variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5.2 Gibbs measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5.3 The submultiplicative pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6 Diagonal actions on quotients of the special linear group . . . . . . . . . . . . . . . . . . . . . . . 40

2.6.1 The special linear group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6.2 Quotient spaces of discrete subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6.3 Quotient spaces of lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.7 Level sets for Birkhoff averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.7.1 A measure-theoretic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.7.2 Multifractal analysis of Birkhoff averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.7.3 Multifractal analysis of local dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.8 Iterated function systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.8.1 Conformal iterated function systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.8.2 Infinite conformal iterated function systems . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.8.3 Self-affine Sierpinski sponges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.8.4 Lalley-Gatzouras repellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.8.5 Typical self-affine sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Table of contents 13

2.9 Multifractal analysis for dynamical systems without a Markov partition . . . . . . . . . . . . . . 58

3 The packing spectrum for Birkhoff averages on a self-affine repeller 61

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2 Notation and statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Symbolic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4 Proof of the lower estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5 Proof of the upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.6 The shape of the spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.8 Generalisations and open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4 Infinite non-conformal iterated function systems 85

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


4.3 The upper bound for locally constant potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3.1 Building a cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3.2 Constructing a measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4 Approximation arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.5 Preliminary lemmas for the lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.5.1 Dimension lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.6 Convergence lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.7 Proof of the lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5 Multifractal analysis for typical infinitely generated self-affine sets 117

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117


5.2.1 Thermodynamic formalism for sub-multiplicative potentials . . . . . . . . . . . . . . . . . 117

5.2.2 Infinitely generated self-affine sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2.3 Multifractal analysis of Birkhoff averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.3 Thermodynamic formalism for quasi-multiplicative potentials . . . . . . . . . . . . . . . . . . . . 123

5.3.1 Existence of Gibbs measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.3.2 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3.3 Differentiation of pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.4 Dimension of infinitely generated self-affine sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.5 Multifractal analysis of Birkhoff averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.5.1 Proof of the upper bound in Theorem 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.5.2 Symbolic tree structure in level sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

14 Table of contents

5.6 Conditional variational principle for bounded potentials . . . . . . . . . . . . . . . . . . . . . . . 143

5.6.1 Space of integrals with respect to invariant measures . . . . . . . . . . . . . . . . . . . . . 143

5.6.2 Upper bound for interior points of the spectrum . . . . . . . . . . . . . . . . . . . . . . . . 145

5.6.3 Quasi upper-semicontinuity lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.6.4 Finitely many potentials lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.6.5 Proof of the upper bound in Theorem 5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.6.6 Proof of the lower bound in Theorem 5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.7 Self-affine sets of a prescribed dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6 Multifractal analysis on the space of lattices 157

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.2 Multifractal analysis for flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.3 Multifractal analysis for the time one map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.4 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.4.1 Metric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.4.2 The chain lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.4.3 A uniform law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.5 Proof of the upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.5.1 Returns to compact regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.5.2 Growth of separated sets on the unstable manifold . . . . . . . . . . . . . . . . . . . . . . 169

6.5.3 Constructing a measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.5.4 The entropy function is concave and continuous . . . . . . . . . . . . . . . . . . . . . . . . 176

6.6 Proof of the lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.6.1 Compactly supported measures with high entropy . . . . . . . . . . . . . . . . . . . . . . 178

6.6.2 Central lemma in the proof of the lower bound . . . . . . . . . . . . . . . . . . . . . . . . . 180

6.6.3 Reduction to unstable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.6.4 Constructing a partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.6.5 Lower bound construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

1

Introduction

In this thesis we shall consider the dimension theory of dynamical systems. Whilst the dimension theory of

conformal hyperbolic maps on a compact phase space is well understood, much less is known in the setting of

non-conformal maps or maps with a non-compact phase space. Our aim is to extend the theory of multifractal

analysis into such settings.

In this chapter we shall give a brief overview of the current state of play within this research area before

summarising the main results of the thesis.

1.1 The dimension theory of dynamical systems

One of the most fascinating features of dynamical systems is their ability to give rise to intritcately structured

invariant sets. Such invariant sets often display fractal-like features including fine structure at arbitrarily small

scales and smaller parts of the set resembling the set itself. This level of complexity means that such sets

cannot be described with the concepts of classical geometry. Instead we classify such sets according to their

scaling behaviour through some form of fractal dimension. In this thesis we shall be primarily concerned with

the Hausdorff dimension and the packing dimension. We shall also make some use of the coarser concept of

box counting dimension as a means of obtaining upper bounds on a set’s packing dimension. Each of these

concepts, along with the relationships between them, will be formally introduced in the next chapter.

The focus upon dimension as a geometric characteristic underlines one of the central tenets of the modern

theory of dynamical systems. Whilst hyperbolic dynamical systems might appear to be so complex as to be

beyond the reaches of understanding, this intricacy conceals a multitude of simple relationships which only

become visible when one takes the appropriate non-classical standpoint. Take for example the issue of pre-

dictability. Hyperbolic dynamical systems display a level of sensitivity upon initial conditions which makes

any attempt to extract information about an individual orbits behaviour from its initial locality essentially

pointless. One would be forgiven for infering that such systems are entriely unpredictable. However, hyper-

bolic dynamical systems are predictable when one focuses not upon the behaviour of individual orbits, but on

the aggregate behaviour of the system. In fact such systems obey a variety of statistical laws through which

one may extract a great deal of information concerning the system’s distributional behaviour (see [BoEq]).

15

16 Chapter 1. Introduction

When one turns to the geometry of dynamical systems the contrast becomes even more striking. The

various invariant sets and measures which one might seek to study in hyperbolic dynamical systems often

display a level of intricacy which make many of the concepts of classical geometry non-applicable. However,

when we turn to studying the scaling behaviour of these dynamically defined objects we uncover a variety of

elegant relationships between geometric structure and dynamical invariants such as entropy and Lyapunov

exponents. It is this tradition which we wish to contribute to in this thesis.

1.2 Iterated function systems

The protypical example of a fractal is the limit set of an iterated function systems. Here the fractal property

of smaller parts resembling the whole takes the strong form in which the limit set is precisely the union of

contractions of the whole set. Iterated function systems often arise from the process of looking at the back-

ward trajectories of a suitable repelling dynamical systems. As such the limit sets of iterated function systems

provides us with a tractable family of model dynamical systems with repellers displaying fractal features.

1.2.1 Conformal iterated function systems

The first rigorous treatment of iterated function systems appears in the work of Hutchinson [Hu]. Inspired by

related work of Moran [Mo] Hutchinson proved the existence of the attractor and showed that for finite col-

lections of contracting similitudes satisfying suitable seperation conditions the Hausdorff dimension is given

by the unique root of a simple equation involving only the contraction ratios of the similitudes.

The work of Hutchinson was subsequently generalised by Ruelle to include iterated function systems con-

sisting of finitely many conformal contractions [Ru]. Ruelle’s analysis demonstrates that for such systems the

Hausdorff, packing and box counting dimensions coincide and are given by the unique root of an equation

involving the thermodynamic pressure of the log-Jacobian (see [F2, Chapter 4]). This result is often referred to

as Bowen’s equation in light of its close resemblance to an earlier result of Bowen [Bo].

Mauldin and Urbanski extended the work of Ruelle to include iterated function system consisting of a

countable infinity of conformal contractions [MU1]. For such systems the pressure will be infinite for all suf-

ficiently small values of the parameter and may drop below zero as soon as it becomes finite. Consequently,

the pressure function need not have any root. Instead the Hausdorff dimension is given by the infimal value

for which the associated pressure function drops below zero. The packing dimension for attractors of infinite

iterated function systems is much less well understood.

1.2.2 Non-conformal iterated function systems

A major challenge in the study of iterated function systems is to understand the dimension of non-conformal

limit sets. In an ideal world we would possess a general theory containing formulae for the Hausdorff and

packing dimensions of limit sets for any iterated function system consisting of finitely many, well seperated,

affine contractions. However, such a goal is currently well out of reach. Nonetheless significant progress on

1.2. Iterated function systems 17

the dimension theory of self-affine sets has been made and what we do know reveals that any general theory

of non-conformal iterated function systems will differ markedly from the theory of conformal iterated function

systems.

There are essentially two different strands of research in the dimension theory of non-conformal limit sets.

The first approach is to focus on particular families of examples for which the contractions align in such a way

as to make the dynamics along one direction appear as a factor, thus lending themselves to the techniques of

dimension theory. The first example of this kind was studied independently by Bedford [Be] and McMullen

[Mc], who considered the self-affine limit set obtained by dividing the unit square into some collection of

equally sized rectangles, deleting some configuration from amongst them, before taking an affine copy of the

configuration inside each of the remaining rectangles, and iterating the process. One interesting feature of these

limit sets is that, unlike their conformal counterparts, the Hausdorff and packing dimensions of these limit

sets do not usually coincide. Later Kenyon and Peres extended Bedford and McMullen’s example to higher

dimensions [KP]. Gatzouras and Lalley have generalized the self-affine limit sets of Bedford and McMullen

to include a range of planar iterated function systems with a variable Lyapunov exponent [LG]. Baranski has

studied an alternative generalization of the self-affine limit sets of Bedford and McMullen, also with a variable

Lyapunov exponent, and given an elegant proof which also works in the setting of Gatzouras and Lalley [Bar].

The second approach to the theory of self-affine sets stems from the work of Falconer [F1]. Suppose we

have a collection of linear mappings of norm no more than a third. By composing each of these maps with

a translation we form an iterated function system of affine maps. Falconer gave a formula known as the

singular value dimension which gives the Hausdorff and packing dimensions of the corresponding limit set for

almost every collection of translation vectors. Later Solomyak showed that Falconer’s proof may be adjusted

to include collections of linear maps of norm up to a half [S]. The proof develops a general philosophy in

dimension theory which dates back to Kaufman’s potential theoretic proof of Marstrand’s projection theorem

[Ka]. The Hausdorff dimension of a set may be related to the finiteness of a certain integral. Even when we are

not able to evaluate this integral for a particular set we are sometimes able to show that it is finite for typical

members of a parameterizable family of such sets by integrating the integral to be bounded over the paramater

space, using Fubini’s theorem to change the order of integration and showing that the combined integral is in

fact finite. A related approach was later deployed by Solomyak to solve a long standing conjecture of Erdos on

the absolute continuity of Bernoulli convolutions [SB].

Previously to Solomyak’s result, Przytycki and Urbanski had shown that the dimension of any given

Bernoulli convolution has implications for the dimension of an associated self-affine graph resembling the

Weirstrass function [PU]. By combining Przytycki and Urbanski’s work [PU] with the result of Solomyak [SB]

we obtain almost sure results for another family of self-affine sets where the parameter is not the translation

vector but the vertical contraction ratio. For typical values of the parameter the Hausdorff dimension is given

by the value suggested by Falconer’s formula. However, Przytycki and Urbanski also proved the existence of

a countable family of countraction ratios arbitrarily close to a half for which the Hausdorff dimension drops

below the value suggested by Falconer’s formula. Edgar has shown that in fact this dimension drop does not


depend upon the particular choice of translation vectors and as such Solomyak’s constraint upon the norms in

Falconer’s theorem is optimal [Ed3]. Note that such sensitive dependence upon the precise choice of contrac-

tion ratios can also occur in the self-similar setting [SiSo] but only when the seperation conditions of Hutchin-

son’s theorem fail to hold. Another surprising feature of self-affine sets is that the Hausdorff dimension of

the self-affine sets considered by Bedford and McMullen is typically lower than the singular value dimension,

suggested by Falconer’s theorem. In fact most particular examples of self-affine sets for which we can actually

compute the Hausdorff dimension behave non-typically with respect to Falconer’s theorem. We infer that the

dimension of self-affine sets depends sensitively upon the particular choice of translation vector. In certain cir-

cumstances the box counting and packing dimensions can also drop below the singular value dimension. Note

that the central role played by Fubini’s theorem in Falconer’s proof means that we currently have no general

critereon for deciding when a particular self-affine set behaves typically with respect to Falconer’s theorem.

1.3 Ergodic theory and the thermodynamic formalism

As we have already seen, the thermodynamic pressure is required to state the formulae for the dimension of

limit sets for both conformal and self-affine iterated function systems. In fact the thermodynamic formalism

and related notions such as entropy play a key role in the dimension theory of dynamically defined sets.

1.3.1 The dimension of an ergodic measure

Manning showed that for any continuously differentiable Axiom A diffeomorphism on a surface the Hausdorff

dimension of the set of generic points for an ergodic invariant measure intersected with the local unstable

manifold is equal to the entropy of the measure divided by the expected Lyapunov exponent [Ma]. It is clear

from the proof that the same is true for an ergodic invariant measure supported on the limit set of a conformal

finitely generated iterated function system satisfying Hutchinson’s separation conditions. Later Ledrappier

and Young obtained an expression for the Hausdorff dimension of ergodic measures which are invariant for

an arbitrary twice continuously differentiable diffeomorphism on a Riemannian manifold [LY]. The formula

is given in terms of a sum of ratios of conditional entropies and expected Lyapunov exponents in different

directions. A similar formula holds for an ergodic measure supported upon Bedford-McMullen carpets, or

more generally the self-affine limit sets of Kenyon and Peres, Lalley and Gatzouras, and Baranski [KP, LG, Bar].

One may also show that for typical self affine sets the dimension of an ergodic measure is equal to the unique

value for which the Lyapunov exponent of corresponding singular value function is equal in magnitude to the

entropy [JPS].

In the setting of conformal iterated functions systems we may apply the classical variational principle for

pressure on a compact sets to show that the zero of the associated pressure function, and hence the dimension

of the attractor, is equal to the supremum of the dimensions of invariant measures supported on the limit set.

When the dimension of a limit set is equal the supremum of the dimensions of invariant measures supported

on the limit set we shall say that the limit set satisfies the variational principle for Hausdorff dimension.

1.4. Multifractal analysis 19

The variational principle for Hausdorff dimension holds for both infinitely generated conformal iterated

function systems and also typical self-affine iterated function systems. In both cases this may be proven using

the pressure-theoretic formulae for the dimension combined with an appropriate version of the thermody-

namic formalism. For infinitely generated conformal iterated function systems Mauldin, Urbanski [MU1] and

Sarig [S1, S2] developed a thermodynamic formalism for symbolic spaces defined over a countable alphabet.

For typical self-affine iterated function systems Kaenmaki developed a thermodynamic formalism for subad-

ditive potentials [K].

Whilst the Hausdorff dimension of the self-affine limit sets of Bedford and McMullen [Be, Mc], Kenyon

and Peres [KP], Gatzouras and Lalley [LG] and Baranski [Bar] are not naturally related to the thermodynamic

pressure, it turns out that they do satisfy the variational principle for Hausdorff dimension.

1.3.2 Ergodic measures of maximal Hausdorff dimension

A key component in the theory of hyperbolic dynamical systems are the Sinai-Ruelle-Bowen measures; ergodic

invariant measures which are absolutely continuous with respect to Lebesgue measure. These measures give

us a characterisation of the statistical behaviour of Lebesgue typical orbits. For dynamical systems restricted

to invariant sets with a Hausdorff dimension below that of the ambient space it is clear that no such measure

can exist. Instead we seek out the natural analogue of an ergodic measure of maximal Hausdorff dimension.

For the attractors of finitely generated conformal iterated function systems the existence of an ergodic

measure of full Hausdorff dimension follows straightforwardly from the variational principle for Hausdorff

dimension. One may use the compactness of the space of invariant measures combined with the upper-

semicontinuity of entropy and the ergodic decomposition theorem. This approach also works for the self-affine

Sierpinski sponges of Kenyon and Peres as well as for typical finitely generated self-affine sets.

For infinitely generated iterated function systems this argument breaks down. The space of invariant mea-

sures is no longer compact and the entropy is no longer upper semi-continuous. Nonetheless Mauldin and

Urbanski have used the existence of Gibbs measures to show that whenever the pressure function has a root

there exists an ergodic measure of full Hausdorff dimension.

1.4 Multifractal analysis

A central aim in the theory of dynamical systems is to classify points according to their long-run asymptotic

behaviour. To this effect we study sets of points with a given Birkhoff average. The aim of the multifractal

analysis of Birkhoff averages is to study the fractal dimension of these level sets. More generally, multifractal

analysis is concerned with using global quantities, such as dimension and entropy, to study the complexity

of level sets for invariant local quantities. This problem is motivated by the seminal paper of Halsey et al.

[HJKPS] which emphasized the importance of identifying the variety of scaling behaviours which appear in

non-linear physics.


Another motivation comes from questions in number theory regarding the representation of real numbers

by strings of digits. Indeed, the first excursions into multifractal analysis were undertaken by Besicovitch and

Eggleston who studied asymptotic frequencies of digits for expansions of real numbers [Bes, Eg]. The modern

theory of multifractal analysis of Birkhoff averages began with the work of Pesin and Weiss who generalised

the work of Eggleston by studying Holder continuous potentials on a symbolic space with a finite alphabet

[PW1]. We now have a well-developed theory encompassing arbitrary continuous potentials on the attractor

of a finite conformal iterated function system [FFW, BS, FLW, Ol2, Ol4].

In recent years this theory has been extended to deal with a wide range infinite conformal iterated function

systems [JK, KMS, KS, IJ, FLM, FLWWJ, FLMW, FJLR]. For such system there is a well known conditional

variational principle; the Hausdorff dimension of the set of points for which the Birkhoff average convereges to

a given value is equal to the supremum of the dimensions of ergodic invariant measures which are supported

on the level set. For finite conformal iterated function systems, the packing dimension is also given by this

value. Moreover, for Holder continuous potentials the multifractal spectrum is an analytic.

The multifractal analysis of non-conformal systems is far less well understood. Nonetheless, the condi-

tional variational principle for Hausdorff dimension has been established for a few select non-conformal sys-

tems. Barral, Mensi and Feng have proved a conditional variational principle for the self-affine Sierpinski

sponges of Kenyon and Peres [BM1, BF]. This result has been generalised to include the self-affine limit sets of

Lalley and Gatzouras by the present author [R2]. The multifractal spectrum has also been studied for typical

self-affine sets with a diagonal linear part by Jordan and Simon [JS].

A closely related area of research is the multifractal analysis of local dimension. Here one takes a dynami-

cally defined measure such as a Gibbs measure and considers sets of points with a given local dimension. The

development in this research area in many ways runs parallel to the progress made in the multifractal analysis

of Birkhoff averages. The problem is well understood in the conformal setting. The Hausdorff dimension is

given by a conditional variational principle [I] and for finite conformal iterated function systems the packing

and Hausdorff dimensions coincide [Ol1]. Again, far less is known for non-conformal limit sets. The Hausdorff

spectrum for Bernoulli measures on Bedford-McMullen carpets with strong separation conditions was estab-

lished by King [Ki]. Olsen extended King’s result to Bernoulli measures on d dimensional self-affine Sierpinski

sponge [Ol5]. The Hausdorff spectrum for Gibbs measures was determined by Barral and Mensi [BM2] for

Bedford-McMullen carpets, and by Barral and Feng [BF] for a d dimensional self-affine Sierpinski sponge. Re-

cently Jordan and Rams gave the Hausdorff spectrum for Bernoulli measures on a Bedford-McMullen carpet

without the strong separation conditions required by King [JR]. In contrast very little is known about the

packing spectrum for pointwise dimension on a self-affine Sierpinski sponge.

1.5 A summary of results

In Chapter 3 we consider the level sets for Birkhoff averages on a self-affine Sierpinski sponge. We give a

formula for the packing dimension of the level sets. This formula is expressed in terms of a weighted sum

1.5. A summary of results 21

of different supremas of entropies of the various factors. Recall that for conformal iterated function systems

the packing and Hausdorff spectrums coincide and for Holder continuous potentials the spectrum is analytic.

Our formula shows that in the non-conformal setting the spectrums need not coincide and that the spectrum

for Holder continuous potentials need not be analytic. We also consider the packing spectrum for pointwise

dimension in the self-affine setting. We give a formula for the packing dimension of the level sets for a very

limited class of Bernoulli measures on self-affine Sierpinski sponges with strong separation conditions. The

theorem also allows us to refute a conjecture of Olsen on a related question regarding the packing dimension

of sets of a given local dimension [Ol5, Conjecture 4.1.7] by showing that the packing spectrum need not attain

the full packing dimension of the repeller.

In Chapter 4 we consider a class of infinite non-conformal iterated function systems which generalise the

self-affine iterated function systems of Lalley and Gatzouras [LG]. Motivating examples include the planar

limit set obtained by taking the continued fraction expansion along the horizontal axis and taking the binary

expansion along the vertical axis and considering the collection of points that remain if certain digit pairs are

prohibited. We show that the Hausdorff dimension is equal to the supremum of the dimensions of invariant

measures supported on the limit set, which is in turn equal to the the supremum of the Hausdorff dimen-

sions of its finitely generated subsystems. We also prove a conditional variational principle for the Hausdorff

dimension of the level sets for Birkhoff averages of countable infinities of uniformly continuous potentials.

In Chapter 5 we consider typical infinitely generated self-affine limit sets in the plane. The results of this

chapter are part of a joint project with Antti Kaenmaki. We consider collections of affine maps for which

the linear parts satisfy a suitable generic irreducibility condition. Given the irreducibility condition we extend

Falconer’s formula for the dimension of a typical self-affine set to include infinite iterated function systems and

show that the dimension of the attractor is equal to the supremum of the dimensions of the finitely generated

subsystems. We also provide a sufficient condition for the existence of an ergodic Gibbs measure of maximal

dimension. Finally we give a conditional variational principle for the dimension of the level sets for Birkhoff

averages of countable infinities of uniformly continuous potentials and derive a more informative formula for

the special case in which each of the potentials is bounded. All of the results in this section rest upon the

development of a thermodynamic formalism for quasi-multiplicative potentials on a symbolic space with a

countable alphabet.

In Chapter 6 we consider a one-parameter diagonal flow on the space of unimodular lattices. These systems

are interesting as they provide a natural family of dynamical systems which are neither compact nor do they

have any Markov-type structure. We prove a conditional variational principle for the Hausdorff dimension

of the level sets for uniformly continuous potentials which go to infinity in the cusp. We also show that both

the set of points for which the Birkhoff averages tend to infinity and the set of points for which the Birkhoff

averages do not exist have full Hausdorff dimension.


2

Background

2.1 Geometric measure Theory

2.1.1 Fractal dimension

The concept of fractal dimension generalizes the notion of the dimension of a real manifold to arbitrary metric

spaces. A set is ascribed a given dimension in accordance with its scaling behaviour. Just as the k-dimensional

Lebesgue measure of a small ball in a k-dimensional embedded manifold will be about 2k times as much as a

ball of half the radius a set of fractal dimension s is one for which a small ball will contain about 2s times as

much “stuff” as a similarly centered ball of half the radius. To make this notion precise we need to introduce

a family of geometric measures which scale in an s-dimensional way. There are two natural approaches to

constructing such a measure, and as we shall see, they are not equivalent.

2.1.2 Hausdorff dimension

Fix s ∈ R and take a subset A ⊂ X of a metric space (X, d). We define the s-dimensional Hausdorff measure by

Hs(A) := limε→0

inf

∑j∈N

diam(Bj)s : A ⊆

⋃j∈N

Bj and diam(Bj) < ε

.

Note that if we take a doubling similarity transformation Φ : X → Y mapping into some metric space

(Y, ρ) with ρ(Φ(x1), Φ(x2)) = 2 · d(x1, x2) for x1, x2 ∈ X then Hs (Φ(A)) = 2s · Hs(A). This corresponds to

the intuition that for an s-dimensional measure doubling the distances between points should produce a set

containing 2s times as much stuff.

Given a set A there will be a critical exponent sh such that for all s < sh Hs(A) = ∞ and for all s > sh

Hs(A) = 0. We may think of sh as being the correct exponent with respect to which we may measure the

associated Hausdorff measure of A, since this is the only exponents for which there is a possibility of obtaining

a quantity which is both positive and finite. This motivates the following definition,

dimH(A) := sup s : Hs(A) = ∞ = inf s : Hs(A) = 0 .

We refer to dimH(A) as the Hausdorff dimension of A (see [F3, Chapter 2] for details).

23

24 Chapter 2. Background

One can easily show that the Hausdorff dimension is monotone, in the sense that if A1 ⊆ A2 then dimH A1 ≤

dimH A2 and countably stable, meaning that it does not increase under countable unions ie.

dimH

( ⋃n∈N

An

)≤ sup

n∈N

dimH An .

2.1.3 Packing dimension

The packing measure is a natural dual to the notion of Hausdorff dimension. Whilst the Hausdorff measure

is defined in terms of minimal coverings, the packing measure is defined via maximal packings. However, to

define the s-dimensional packing measure requires more care. Fix s ∈ R and take a subset A ⊂ X of a metric

space (X, d). We first define the s-dimensional packing pre-measure by

P s(A) := limε→0

sup

∑

j∈N

diam(2rj)s : ∃(xj)j∈N ⊂ A, d(xi, xj) > ri + rj for i 6= j

.

Unfortunately the packing pre-measure is not countably subadditive on the Borel sigma-algebra. We define

the s-dimensional packing measure by

P s(A) := inf

∑j∈N

P s(Aj) : A ⊆⋃

j∈N

Aj

.

Again, if we take a doubling similarity transformation Φ we will have P s (Φ(A)) = 2s · P s(A). Moreover,

given a set A there will also be a critical exponent sp such that for all s < sp P s(A) = ∞ and for all s > sp

P s(A) = 0 and so we define,

dimP(A) := sup s : P s(A) = ∞ = inf s : P s(A) = 0 ,

and refer to dimP(A) as the packing dimension of A (see [F3, Chapter 3] for details).

Note that packing dimension, like Hausdorff dimension, is monotone and countably stable. Note also the

Hausdorff dimension cannot exceed the packing dimension.

Proposition 2.1. dimH A ≤ dimP A.

Proof. See [F3, Chapter 3].

We shall see examples for which the inequality is strict.

By extension we can also define the Hausdorff and packing dimension of a measure.

Definition 2.2. Let µ be a Borel probability measure on a metric space (X, d). Let B be the sigma algebra of all Borel

sets. The (upper) Hausdorff and packing dimensions of µ are defined respectively by

dimH(µ) := inf dimH(A) : A ∈ B, µ(A) = 1 ,

dimP(µ) := inf dimP(A) : A ∈ B, µ(A) = 1 .

2.1. Geometric measure Theory 25

2.1.4 Mass distribution principles

In order to provide an upper bound for the Hausdorff dimension of a set it suffices to find a suitable cover.

However, to find a lower bound we must find some way of dealing with all possible coverings at once. It turns

out that it suffices to find a measure with the correct local scaling behaviour. Given a measure µ on a metric

space X together with a point x ∈ X we define the lower local dimension at x by

dimH(µ, x) := lim infr→0

log µ(B(x, r))log r

.

Proposition 2.3. Let A ⊂ Rn be a Borel set and µ a finite Borel measure on Rn with µ(A) > 0. Suppose that

dimH(µ, x) ≥ s for all x ∈ A. Then dimH(A) ≥ s.

Proof. See [F2, Proposition 2.3].

A stronger result holds for packing dimension. Given a measure µ on a metric space X together with a

point x ∈ X we define the upper local dimension at x by

dim(µ, x) := lim supr→0


.

Proposition 2.4. Let A ⊂ Rn be a Borel set and µ a finite Borel measure on Rn with µ(A) > 0. Suppose that

dim(x) ≥ s for all x ∈ A. Then dimP(A) ≥ s.


When constructing measures with a view to applying propositions 2.3 and 2.4 we shall make frequent use

of the following instance of the Daniell-Kolmogorov consistency theorem.

Theorem 2.5 (Existence of product measures). Suppose that for each n ∈N we have a probability space (Xn,Bn, µn)

in which Xn is a locally compact, sigma-compact metric space and Bn is the sigma-algebra of Borel sets. Let X∞ :=

∏n∈N Xn be the product space, under the product topology and B∞ the sigma-algebra of Borel sets. There exists a unique

Borel probability measure µ∞ on X∞ with the property that for each sequence of sets (An)n∈N with each An ∈ Bn and

An = Xn for all but finitely many n, we have

µ∞

(∏

n∈N

An

)= ∏

n∈N

µn (An) .

Proof. See [T, Theorem 2.4.4].

2.1.5 Frostman’s lemma

As an aside we note that Propositions 2.3 and 2.4 have the following partial converses.

Proposition 2.6. Suppose that A ⊂ Rn is a non-empty Borel set with dimH A > s. Then there exists a Borel probability

measure µ supported on A such that for µ-almost every x ∈ A we have dimH(µ, x) ≥ s.



Proposition 2.7. Suppose that A ⊂ Rn is a non-empty Borel set with dimP A > s. Then there exists a Borel probability

measure µ supported on A such that for µ-almost every x ∈ A we have dim(µ, x) ≥ s.


The first of these is known as Frostman’s lemma. By combining Propositions 2.6 and 2.7 with Proposi-

tions 2.3 and 2.4 we have the following alternative characterisations of the concepts of Hausdorff and packing

dimension.

Given a Borel set A ⊂ Rn we let P(A) denote the set of all Borel probability measures µ on Rn with

µ(A) = 1.

Proposition 2.8. Let A ⊂ Rn be a Borel set. Then

dimH A = sup

s ∈ R : ∃µ ∈ P(A) with dimH(µ, x) ≥ s for µ almost every x ∈ A

.

Proof. Combine Proposition 2.3 with Proposition 2.6.

Proposition 2.9. Let A ⊂ Rn be a Borel set. Then

dimP A = sup

s ∈ R : ∃µ ∈ P(A) with dim(µ, x) ≥ s for µ almost every x ∈ A

.

Proof. Combine Proposition 2.4 with Proposition 2.7.

2.1.6 Marstrand’s fibre lemma

Choose n, k ∈ N. For each x = (xi)ni=1 ∈ Rn we define the corresponding k-dimensional hyperplane L(x) ⊂

Rn+k by

L(x) :=(x, y) : y ∈ Rk

.

The following proposition due to Marstrand allows us to relate the dimension of set in Rn+k to the dimension

of its fibres.

Proposition 2.10. Let A be any subset of Rn+k and B a subset of Rn. The following formula holds

dimH A ≥ dimH B + inf dimH (A ∩ L(x)) : x ∈ B .

Proof. See [F3, Corollary 7.12].

2.1.7 Box counting dimension

We define the lower and upper box counting dimension as follows. Given a set A ⊂ X within a metric space

we define for each r > 0,

Nr(A) := min

n ∈N : ∃(xi)

ni=1 ⊂ A, A ⊂

n⋃i=1

B(xi, r)

.

2.2. Dynamical systems 27

We define the lower box counting dimension by

dimB(A) := lim infr→0

log Nr(A)

log r.

and the upper box counting dimension by

dimB(A) := lim supr→0

log Nr(A)

log r.

When the lower and upper box dimensions of a set A coincide dimB(A) = dimB(A) we refer to the common

value as the box counting dimension dimB(A).

From a purely mathematical point of view the box counting dimension is less satisfactory than the Haus-

dorff and packing dimensions. In particular, it lacks the property of being countably stable.

Example 2.11. dimB(0 ∪

n−1 : n ∈N

)= 1

2 .

Nonetheless the box dimension is often much easier to calculate than the Hausdorff and packing dimen-

sions. Moreover, the following theorem allows us to convert an estimate of the upper box counting dimension

into an upper bound on the packing dimension.

Proposition 2.12. Let A ⊆ Rn. Then

dimP A = inf

supj∈N

dimBBj

: A ⊂

⋃j∈N

Bj

.


2.2 Dynamical systems

A dynamical system consists of an action by a semi-group G, usually N or R+, upon a set X. We think of G as

designating time, X as a phase space of possible states of our system, and the action as a rule for the evolution

of the phase space. We shall focus on discrete time dynamical systems, where the acting semigroup is N. An

action is specified by the action of the generating set, so for discrete time dynamical systems it suffices to study

the action by one, which is given by a single map T : X → X. We shall also give some attention to continuous

time dynamical systems or flows, which are actions by R. Equivalently, a continuous time dynamical system

consists of a set X together with a family of maps ( ft)t∈R satisfying the composition rule fs+t(x) = fs( ft(x))

for all x ∈ X and all s, t ∈ R.

The concepts introduced in this section will be exemplified in sections 2.4 and 2.6 where we shall introduce

two classes of dynamical system which are of central importance to our enquiry.

2.2.1 Topological dynamical systems

In order to say something meaningful about the geometry of a dynamical system we must first impose some

additional structure.


Definition 2.13 (Topological Dynamical Systems). Suppose we have a metric space (X, d) consisting of set X and

a metric d on X. Given a continuous map T : X → X we shall say that (X, d, T) is a topological dynamical system.

Similarly, a continuous time topological dynamical system is a metric space (X, d) combined with a family of continuous

maps ( ft)t∈R+ satisfying the composition rule.

Topogical dynamical systems may be related to one another via the concept of topological semi-conjugacy.

Definition 2.14 (Semi-conjugacy). Suppose we have a pair of topological dynamical systems (X, d, T) and (Y, ρ, S).

Suppose further that there exists a continuous surjective map Φ : X → Y such that for all x ∈ X we have Φ(T(x)) =

S(Φ(x)), so the following diagram commutes.

X

Φ

T// X

Φ

YS// Y.

Then we shall say that (X, d, T) and (Y, ρ, S) are (topologically) semi-conjugate. If Φ is a homeomorphism we say that

(X, d, T) and (Y, ρ, S) are (topologically) conjugate.

Definition 2.15 (Topological mixing). A topological dynamical system (X, d, T) is said to be topologically mixing if for

every pair of non-empty open sets U, V ⊂ X there exists a natural number N(U, V) ∈N such that for all n ≥ N(U, V)

we have U ∩ Tn(V) 6= ∅, or equivalently T−n(U) ∩V 6= ∅.

2.2.2 Measure preserving systems

Whilst the primary object of study will be certain families of topological dynamical systems measure preserv-

ing systems shall play a key role.

Definition 2.16 (Measure Preserving Systems). Suppose we have a probability space (X,B, µ) consisting of a set X, a

σ-algebra B of subsets of X and a probability measure µ defined on B. Suppose T : X → X is measureable ie. T−1B ⊆ B.

We shall say that T preserves the measure µ or is µ preserving if µ(T−1B

)= µ(B) for all B ∈ B. Alternatively, we

can view T as fixed and say that µ is a T-invariant measure. In either case, we shall say that (X,B, µ, T) is a measure

preserving system. We can extend this concept to continuous time dynamical systems by requiring that the family of

maps ( ft)t∈R+ are all measureable and for each t ∈ R+, µ f−1t = µ.

We give a useful way to check that a given measure is invariant.

Lemma 2.17. Suppose that (X,B, µ) is a probability space, T : X → X is measureable. Let S be a semi-algebra which

generates the sigma-algebra B. Suppose that µ(T−1 A) = µ(A) for all A ∈ S . Then µ is a T-invariant measure.

Proof. See [W, Theorem 1.1].

We shall be interested in relationships between a given topological dynamical system and the collection

of measure preserving systems which are compatible with that system’s topological structure. Let (X, d) be a

metric space. We letM(X) denote the collection of all Borel probability measures on X. Given a continuous


map T : X → X we letMT(X) denote the set of all µ ∈ M(X) which are T-invariant. Similarly given an R+

action by f = ( ft)t∈R+ on X we defineMf(X) to the collection of f-invariant Borel probability measures on X.

Equivalently,Mf(X) :=⋂

t∈R+M ft(X).

Theorem 2.18. Suppose that (X, d, T) is a topological dynamical system consisting of a continuous transformation T

acting upon a locally compact metric space (X, d). ThenMT(X) is a compact in the weak star topology and convex in

the sense that if ρ is a probability measure onMT(X) and we construct a measure ν by

∫f (x)dν(x) =

∫ (∫f (x)dµ(x)

)dρ(µ),

for continuous compactly supported functions f : X → R, then ν is also a member ofMT(X).

Proof. Given any continuous compactly supported function f : X → R we have

∫( f T)(x)dν(x) =

∫ (∫( f T)(x)dµ(x)

)dρ(µ)

=∫ (∫

f (x)dµ(x))

dρ(µ) =∫( f T)(x)dν(x).

It follows that ν = ν T−1, so ν ∈ MT(X).

A common way of constructing invariant measures is as limit points of certain averaged measures. Suppose

that T is a continuous map of a metric space (X, d). Given a Borel probability measure µ ∈ M(X) we define

for each n ∈N, An(µ) ∈ M(X) by

An(µ) :=1n

n−1

∑j=0

µ T−j.

Lemma 2.19. Suppose that we have a sequence of probability measures (µn)n∈N ⊂ M(X). Suppose that there is some

subsequence (nk)k∈N and a measure ν ∈ M(X) for which limk→∞ Ank (µnk ) = ν, in the weak star topology. Then

ν ∈ MT(X) is invariant.

2.2.3 Ergodicity

Ergodic measures are the indecomposable atoms of measure preserving systems.

Definition 2.20 (Ergodic measures). Let (X,B, µ, T) be a measure preserving system. The invariant measure µ is said

to be ergodic if there is no set B ∈ B with T−1B = B and 0 < µ(B) < 1.

Given a map T : X → X, where (X, d) is a metric space, we let ET(X) denote the set of all members of

MT(X) which are ergodic. Recall that an extreme point in a convex set is any point which cannot be written

as a non-trivial convex combination of other members of the convex set.

Theorem 2.21. Suppose that (X, d, T) is a topological dynamical system consisting of a continuous transformation T

acting upon a compact metric space (X, d). Then the set of extreme points of the compact convex setMT(X) is precisely

the set ET(X). Moreover, if µ, ν ∈ ET(X) and µ 6= ν then µ and ν are mutually singular.



Theorem 2.22 (Ergodic Decomposition Theorem). Suppose that (X, d, T) is a topological dynamical system con-

sisting of a continuous transformation T acting upon a compact metric space (X, d). Then for each invariant measure

µ ∈ MT(X) there is a unique Borel probability measure τ on MT(X) with τ (ET(X)) = 1 such that for all continuous

functions f : X → X we have ∫f (x)dµ(x) =

∫ (∫f (x)dν(x)

)dτ(ν).

Proof. This follows from Theorem 2.21 combined with the Choquet representation theorem [Ph].

2.2.4 Mixing

Consider the following theorem.

Theorem 2.23. Suppose that (X,B, µ, T) be a measure preserving system. The measure µ is ergodic if and only if for

every pair A, B ∈ B we have

limn→∞

1n

n−1

∑j=0

µ(

A ∩ T−jB)= µ(A) · µ(B).

In fact, it suffices for this limit to hold for all members A, B ∈ S where S ⊂ B is some semi-algebra which generates the

sigma-algebra B.


Theorem 2.23 states that if we consider a pair of states A and B and consider the proportion of points in

the initial state A that are in the state B after n time steps then, on average, this will be roughly equal to the

probability of B. A system is mixing if this happens not only on average, but as a precise limit.

Definition 2.24 (Mixing). Let (X,B, µ, T) be a measure preserving system. The invariant measure µ is said to be

mixing if for every pair A, B ∈ B we have

limn→∞

µ(A ∩ T−nB) = µ(A) · µ(B).

Corollary 2.25. Let (X,B, µ, T) be a measure preserving system. If µ is mixing then µ is also ergodic.

Proof. This is immediate from Theorem 2.23.

We note that it suffices to check mixing on a generating semi-algebra.

Theorem 2.26. Let (X,B, µ, T) be a measure preserving system and let S ⊂ B be a semi-algebra which generates the

full sigma algebra B. The invariant measure µ is mixing if for every pair A, B ∈ S we have

limn→∞

µ(A ∩ T−nB) = µ(A) · µ(B).


Mixing for measure preserving systems is the analogue of topological mixing for topological dynamical

systems. In certain cases mixing implies topological mixing.


Definition 2.27. Let (X, d) be a metric space. A Borel measure µ on X is said to be fully supported if for each non-empty

open set U ⊂ X we have µ(U) > 0.

Proposition 2.28. Let (X, d, T) be a topological dynamical system and suppose that µ ∈ MT(X) is a fully supported

invariant measure. Then, if the measure preserving system (X,B, µ, T) is mixing then the topological dynamical system

(X, d, T) is topologically mixing.

Proof. Given any pair of non-empty open sets U, V ⊂ X we have µ(U), µ(V) > 0, so by mixing

limn→∞

µ(U ∩ T−nV

)= µ(U) · µ(V) > 0.

So for all sufficiently large n, µ (U ∩ T−nB) > 0 so U ∩ T−nV 6= ∅.

2.2.5 Birkhoff’s ergodic theorem

At the core of ergodic theory is a limit law obeyed by all measure preserving systems.

Given a function ϕ : X → V from a space X to a vector space V. For each n ∈ N we define the n-th level

Birkhoff sum

Sn(ϕ) :=n−1

∑j=0

ϕ T j.

For our purposes V will always be either Rn for some n ∈N or RN.

Theorem 2.29 (Birkhoff’s ergodic theorem). Let (X,B, µ, T) be a measure preserving system. Then for any real-

valued function ϕ : X → R with∫|ϕ|dµ < ∞ there exists an integrable, T-invariant function, ϕ∗ : X → R with the

property that∫

ϕ∗dµ =∫

ϕdµ and for µ almost every x ∈ X,

limn→∞

n−1Sn(ϕ)(x) = ϕ∗(x).


Corollary 2.30. Let (X,B, µ, T) be an ergodic measure preserving system and ϕ : X → R a real valued function with∫|ϕ|dµ < ∞. Then for µ almost every x ∈ X we have

limn→∞

n−1Sn(ϕ)(x) =∫

ϕdµ.

In applications we shall often require uniform convergence. This is obtained through Egorov’s theorem.

Theorem 2.31 (Egorov’s Theorem). Let (X,B, µ) be a probability space and suppose we have a sequence of real-valued

functions ( fn)n∈N with fn : X → R together with a function f : X → R such that for µ almost every x ∈ X we have

limn→∞ fn(x) = f (x). Then for each ε > 0 there exists a set A ⊂ X with µ(A) > 1− ε such that ( fn)n∈N converges

uniformly to f on A.

Proof. See [T, Theorem 1.3.26].


Corollary 2.32. Let (X,B, µ, T) be an ergodic measure preserving system and ϕ : X → R a real valued function with∫|ϕ|dµ < ∞. For each ε > 0 there exists a set A(ε) > 0 with µ (A(ε)) > 1− ε and a number N(ε) ∈ N such that

for all x ∈ A(ε) and all n ≥ N(ε) we have

∣∣n−1Sn(ϕ)(x)−∫

ϕdµ∣∣ < ε.

Proof. Apply Theorem 2.31 to Corollary 2.30.

2.3 Entropy theory

It is hard to overstate the importance of entropy to the modern theory of dynamical systems. Shortly after its

inception it was shown to be an isomorphism invariant which enabled the settling of a long standing open

question regarding the isomorphism of Bernoulli schemes [Ko]. Entropy has also played a pivotal role in

partial results towards the measure rigidity conjectures of Furstenburg and Margulis with applications to Dio-

phantine approximation (see [Fu1, Rud, Mg1, EKL]). For our purposes, the importance of entropy is its close

ties to both packing and Hausdorff dimension.

2.3.1 Partitions

By a partition of a probability space (X,B, µ) we mean a disjoint collection of sets from the sigma-algebra B

whose union is X.

Given two partitions ξ and η we write ξ ≤ η to mean that every element of ξ is a union of elements of η.

Given two partitions ξ = Aii∈D(ξ) and η = Cjj∈D(η), where D(ξ) and D(η) are indexing sets for ξ and

η, respectively, the join of the two partitions is the partition

ξ ∨ η :=

Ai ∩ Cj : i ∈ D(ξ), j ∈ D(η)

.

Given a transformation T : X → X which preserves the measure µ we define a new partition for each

q ∈N by

T−qξ = T−q Aii∈D(ξ).

We note the following,

T−n(ξ ∨ η) = T−nξ ∨ T−nη

ξ ≤ η ⇒ T−nξ ≤ T−nη.

2.3.2 Information and the entropy of a partition

Suppose we have a probability space (X,B, µ) along with a finite or countable partition ξ = Aii∈D(ξ). We

shall think of the sets Ai as corresponding to the possible outcomes of an experiment, where each outcome Ai

has a probability µ(Ai). We wish to associate to the partition a function a value which measures our amount of

uncertainty about the outcome of the experiment, or equivalently, the average amount of information gained

by performing the experiment.

2.3. Entropy theory 33

We begin introducing an information function I(ξ) : X → R>0, for each partition ξ = Aii∈D(ξ), which

specifies the amount of information gained by learning which element of ξ a point x ∈ X is in. We assume

no previous knowledge of the state of x and that the information gained from a particular event A is fully

reflected in the likelihood that partcular event µ(A) ie. I(ξ)(x) = f (µ(A)) where x ∈ A ∈ ξ for some function

f : [0, 1] → [0, ∞). Given a pair of independent sets A ∈ ξ and B ∈ η, their independence would suggest that

the information gained by learning that x ∈ A ∩ B is the sum of the information gained by learning that x ∈ A

and learning that x ∈ B, so

f (µ(A)) + f (µ(B)) = I(ξ)(x) + I(η)(x)

= I(ξ ∨ η)(x)

= f (µ(A ∩ B)) = f (µ(A) · µ(B)) .

It follows that f is a non-increasing positive valued function with the property that f (xy) = f (x) + f (y) for

x, y ∈ [0, 1). The only functions of this form are x 7→ −λ log x for some λ ≥ 0. This justifies the following

definition.

Definition 2.33 (Information). Let (X,B, µ) be a probability space and ξ a finite or countable partition. We define

I(ξ) : X → R>0 by

Iµ(ξ)(x) := − ∑A∈ξ

1A(x) log µ(A).

The entropy is defined to be the average amount of information gained by learning which element of a

partition a point is in.

Definition 2.34 (Entropy). Let (X,B, µ) be a probability space and ξ a finite or countable partition. We define

Hµ(ξ) :=∫

Iµ(ξ)(x)dµ(x) = − ∑A∈ξ

µ(A) log µ(A).

2.3.3 The entropy of a transformation

Suppose we have a measure preserving system (X, µ,B, T). Given a partition ξ = Aii∈D(ξ). For each n we

let

n−1∨i=0

T−iξ := ξ ∨ T−1ξ ∨ · · · ∨ T−(n−1)ξ

=

Aj0 ∩ T−1 Aj1 ∩ · · · ∩ T−(n−1)Ajn−1 : ji ∈ D(ξ)

.

One may show that the following sequence is subadditive (see [W, Corollary 4.9.1]).1n

Hµ

(n−1∨i=0

T−iξ

)n∈N

.

It follows that the following definition makes sense (see [W, Theorem 4.9]).

Definition 2.35. Let (X, µ,B, T) be a measure preserving system. For each finite or countable partition ξ ⊂ B we define

hµ(T, ξ) = limn→∞

1n

Hµ

(n−1∨i=0

T−iξ

)= inf

n∈N

1n

Hµ

(n−1∨i=0

T−iξ

).


The quantity hµ(T, ξ) gives the average amount of information gathered from determining which partition

element a point is in after successive iterations of the map.

Definition 2.36 (Entropy of a transformation). Let (X, µ,B, T) be a measure preserving system. We define,

hµ(T) := sup

hµ(T, ξ) : ξ ⊂ B is a finite or countable partition and hµ(T, ξ) < ∞

.

Definition 2.37 (Entropy of a flow). Suppose we have a flow f = ( ft)t∈R on a space X together with an f invariant

probability measure µ, so µ = µ f−1t for all t ∈ R. Then (X, µ,B, f1) is a measure preserving system. We define,

hµ( f ) := hµ( f1).

2.3.4 Calculating entropy

We have defined entropy as a supremum taken over all finite or countable partitions. Since the collection of

such partitions will be at least as large as the set itself, this leaves us with a problem as to how to estimate the

entropy, given that we cannot hope to rifle through all possible partitions.

Our first approach gives us a way of selecting partitions for which the entropy is arbitrarily close to the

entropy of the system.

Theorem 2.38. Let (X,B, µ, T) be a measure preserving system. Suppose that we have a sequence Ann∈N such that

each ξn ⊂ B is a finite or countable partition,

ξ1 ≤ ξ2 ≤ · · · ≤ ξn ≤ · · ·

and such that B is the unique sigma algebra containing ξn for all n ∈N. Then

hµ(T) = limn→∞

h(T, ξn).


Our second approach gives us a way of selecting partitions for which the entropy is maximized.

Theorem 2.39 (The Kolmogorov Sinai Theorem). Let (X,B, µ, T) be a measure preserving system. Suppose that

ξ ⊂ B is a finite or countable partition such that B is the unique sigma algebra A ⊆ B with∨n−1

i=0 T−iξ ⊂ A for all

n ∈N. Then hµ(T) = h(T, ξ).


We shall apply this theorem in subsection 2.4.4.

2.3.5 The entropy map

Suppose we have a topological dynamical system T : X → X. The entropy map is the function fromMT(X)

to R≥0 by µ 7→ hµ(T).

2.3. Entropy theory 35

Theorem 2.40. Given a topological dynamical system T : X → X the entropy map µ 7→ hµ(T) is affine. That is, given

a probability measure ρ on the space of invariant measuresMT(X), the measure µ ∈ M(X) defined by∫f (x)dµ(x) =

∫ ∫f (x)dν(x)dρ(ν)

for continuous functions f : X → R satisfies

hµ(σ) =∫

hνdρ.

Proof. See [ELW, Theorem 5.27].

For the following theorem we require T : X → X to be an expansive map on a compact metric space.

Definition 2.41. A map T : X → X on a metric space (X, d) is said to be expansive if there exists some δ > 0 such that

for all x 6= y ∈ X there exists some n ∈N∪ 0 with d(Tn(x), Tn(y)) > δ.

Theorem 2.42. Given a topological dynamical system T : X → X, where X is a compact metric space and T is expansive,

the entropy map µ 7→ hµ(T) upper semi-continuous.

Proof. See the proof of [W, Theorem 8.2]. Theorem 8.2 [W] requires that T be a homeomorphism and uses the

anologue of expansiveness for Z-actions. However, by applying Theorem [W, Theorem 4.17] in place of [W,

Theorem 4.17] the proof may be adapted to deal with continuous N-actions.

The assumption of compactness here is key. Indeed on may construct one can readily construct sequences

of Bernoulli measures on a countable shift space for which the entropy of the limit falls below the lower limit

of the entropies.

2.3.6 Conditional information and conditional entropy

We shall also require the notion of conditional information and conditional entropy. We may think of the

information conditioned on a sigma-algebra C as being the information gained, given we already know which

element of C a point belongs to.

To define these concepts precisely requires the notion of a conditional measure.

Theorem 2.43 (The existence of conditional measures). Suppose that (X,B, µ) is a Borel probability space and that

C ⊂ B is a countably generated sub-sigma-algebra. Then there exists an C measureable set X′ ⊂ X with µ(X′) = 1 and

a family of Borel probability measures

µCx : x ∈ X′

, known as conditional measures, with the following properties:

(i) For each x ∈ X′, µCx (⋂

x∈A∈C A) = 1,

(ii) Given f ∈ L1(X,B, µ) the function x 7→∫

f dµCx depends C measurably upon x,

(iii) For each f ∈ L1(X,B, µ) and A ∈ C we have∫A

∫f (y)dµCx (y)dµ(x) =

∫A

f dµ.

Conditional information and conditional entropy are defined in terms of conditional measures as follows.


Definition 2.44 (Conditional information and entropy). Let (X,B, µ) be a probability space, let C ⊂ B be a count-

ably generated sub-sigma-algebra and ξ a finite or countable partition. We define I(ξ|C) : X → R>0 by

Iµ(ξ|C)(x) := − ∑A∈ξ

1A(x) log µCx (A).

We define the conditional entropy H(ξ|C) by

Hµ(ξ|C) :=∫

Iµ(ξ|C)(x)dµ(x).

Definition 2.45. Let (X, µ,B, T) be a measure preserving system and C ⊂ B a sub-sigma-algebra. For each finite or

countable partition ξ ⊂ B we define

hµ(T, ξ|C) = limn→∞

1n

Hµ

(n−1∨i=0

T−iξ|C)

= infn∈N

1n

Hµ

(n−1∨i=0

T−iξ|C)

hµ(T|C) := sup

hµ(T, C|ξ) : C ⊂ B is a finite or countable partition and hµ(T, C|ξ) < ∞

.

2.3.7 The Abramov Rokhlin entropy formula

The Abramov Rokhlin entropy formula states that given any subsystem or factor of a measure preserving

system, the entropy of the whole system is equal to the sum of the entropy in the factor added to the conditional

entropy remaining in the fibres of the projection.

Definition 2.46. Let (X,B, µ, T) be a measure preserving system. A factor is a measure preserving system (Y, C, ν, S)

for which there exists a map φ : X → Y such that for all C ∈ C, µ(φ−1C) = ν(C) and such that for all x ∈ X,

φ(T(x)) = S(φ(x)).

Theorem 2.47 (The Abramov Rokhlin formula). Let (X,B, µ, T) be a measure preserving system and (Y, C, ν, S) a

factor. Then,

hµ(T) = hν(S) + hµ(T|φ−1C).

Proof. See [ELW, Corollary 5.18].

2.3.8 The Shannon-McMillan-Breimann theorem

Theorem 2.48 (Shannon-McMillan-Breimann). Suppose T is an ergodic measure-preserving transformation of a Borel

probability space (X,B, µ). Let ξ be a countable partition for which B is the smallest sigma algebra containing∨n−1

i=0 T−iξ

for all n ∈N and let C ⊂ B be a sub sigma algebra with T−1C ⊂ C. Then for µ almost every x ∈ X we have

limn→∞

Iµ

(n−1∨i=0

T−iξ|C)(x) = hµ(T|C).

Proof. The Lemma is a mild generalisation of [Pa, Theorem 7, Chapter 2] and may proven in the same way.

2.4. Symbolic spaces and the left shift operator 37

2.4 Symbolic spaces and the left shift operator

2.4.1 The symbolic space

The first example of a dynamical system we shall consider is known as the left shift on a symbolic space.

Definition 2.49 (Left shift map on a symbolic space). Suppose we have a finite or countable set D. The set Σ := DN

consisting of all infinite words from the alphabet D is referred to as the symbolic space. We define the left shift map

σ : Σ→ Σ by σ : (ωi)i∈N 7→ (ωi+1)i∈N.

The semigroup N acts on Σ by (n, x) 7→ σn(x), so the map σ defines a discrete time dynamical system on

Σ.

2.4.2 Metric structure of the symbolic space

We let Σ∗ :=⋃

n∈NDn denote the collection of all finite words from the alphabet D. Given a finite word

ω = (ωi)ni=1 ∈ Σ∗ we let |ω| = n denote the length of ω. If ω ∈ Σ then |ω| := ∞. Given a pair of finite or

infinite words ω, τ ∈ Σ∗ ∪ Σ we let ω ∧ τ denote the maximal common initial segment, which is a member of

Σ∗ unless ω = τ ∈ Σ, in which case ω ∧ τ = ω ∈ Σ. We define a metric d defined by

d(ω, τ) := exp (−|ω ∧ τ|) ,

for ω, τ ∈ Σ, where we adopt the convention that exp(−∞) := 0. We refer to this metric as the symbolic

metric. It is compatible with the natural topology on Σ, which is the Tychonoff product topology taken over

the discrete topology on D.

The left shift map σ : Σ → Σ is continuous with respect to the symbolic metric, so (Σ, d, σ) is a topological

dynamical system. (Σ, d) is compact if and only if D is finite.

2.4.3 Invariant measures

Take the dynamical system consisting of the space Σ acted on by the map σ. For each ω = (ωi)ni=1 ∈ Σ∗ we

define the associated cylinder set [ω] ⊂ Σ by

[ω] := τ = (τi)i∈N ∈ Σ : τi = ωi for all i ≤ n .

Choose (pd)d∈D so that each pd ≥ 0 and ∑d∈D pd = 1. We may define a probability vector on the finite or

countable digit set D by giving each digit d the weight pd. By Theorem 2.5 we may define a Borel probability

measure µ on Σ with the property that µ ([ω]) = pω1 · · · pωn for ω = (ωi)ni=1 ∈ Σ∗. Measures formed in

this way are known as Bernoulli measures. By Lemma 2.17 we see that Bernoulli measures are invariant. By

Lemma 2.26 we see that Bernoulli measures are mixing and hence ergodic. Moreover, if we choose (pd)d∈D so

that each pd > 0 then the corresponding Bernoulli measure µ will be fully supported and so by Lemma 2.28

(Σ, d, σ) is topologically mixing.


It is clear that thare are uncountably many distinct Bernoulli measures hence Eσ(Σ) and Mσ(Σ) are un-

countably infinite. For all of the dynamical systems (T, X, d) which we studyMT(X) is uncountably infinite.

We should note however that this by no means always the case. Indeed if Rα : S1 → S1 is an irrational rotation

of the circle then ERα(S1) =MRα(S

1) is a singleton and if T+ : R → R is the map x 7→ x + 1, and R is given

the Euclidean topology, thenMT+(R) is empty.

2.4.4 Entropy for the left shift map

For the left-shift on a symbolic space we have the following simple formula for the entropy of an invariant

measure. This is a corollary to the Kolmogorov Sinai entropy theorem (Theorem 2.39)

Proposition 2.50. Consider the topological dynamical system consisting of the left shift map σ : Σ → Σ acting upon

the symbolic space Σ. Given µ ∈ Mσ(Σ) the entropy is

hµ(σ) = − limn→∞

1n ∑

ω∈Dnµ([ω]) log µ([ω]).

Proof. We shall apply Theorem 2.39 to the finite or countable partition ξ = [d]d∈D . For each n ∈N we have

n−1∨i=0

T−iξ = [ω] : ω ∈ Dn .

The family⋃

n∈N [ω] : ω ∈ Dn is a basis for the topology on Σ, so the only sigma algebra of Borel sets

containing∨n−1

i=0 T−iξ for all n ∈N is the Borel sigma algebra. Thus,

hµ(σ) = hµ(σ, ξ)

= limn→∞

Hµ

(n−1∨i=0

T−iξ

)

= − limn→∞

1n ∑

ω∈Dnµ([ω]) log µ([ω]).

Example. Take the topological dynamical system consisting of the left shift map σ : Σ → Σ acting upon the symbolic

space Σ. Choose pd ≥ 0 so that ∑d∈D pd = 1 and let µ ∈ Mσ(Σ) denote the invariant measure satisfying µ([ω]) =

pω1 · · · pωn for all ω = (ωi)ni=1 ∈ Dn. Proposition 2.50 implies that

hµ(σ) = − ∑d∈D

pd log pd.

2.5 The thermodynamic formalism

We shall introduce the theormodynamic formalism on a countable symbolic space. Recall that D is a finite or

countable set of digits, Σ := DN with a metric d compatible with the product topology, and σ : Σ → Σ is the

left shift operator.

Given a real valued function φ : Σ→ R we define the variation

varn(φ) := sup |φ(ω)− φ(τ)| : ωi = τi for all i ≤ n .

2.5. The thermodynamic formalism 39

Note that φ is uniformly continuous if and only if limn→∞ varn(φ) = 0. We say that φ has summable variations

if ∑n∈N varn(φ) < ∞.

In defining the pressure we shall follow the approach of Mauldin and Uranski [MU1].

Definition 2.51 (Pressure). Suppose that φ : Σ→ R is uniformly continuous. We define the pressure by

P(φ) := limn→∞

1n

log

(∑

ω∈Dnsup exp (Sn(φ)(x)) : x ∈ [ω]

).

2.5.1 The variational principle

The pressure satisfies the following properties (see Mauldin and Uranski [MU1, MU2] and Sarig [S1, S2] for

details).

Theorem 2.52 (The variational principle). Let φ : Σ→ R be a uniformly continuous potential. The pressure satisfies

P(φ) = sup

hµ(σ) +∫

φdµ : µ ∈ Mσ(Σ),∫

φdµ > −∞

.

The variational principle was originally proved in the setting of compact spaces by Walters [W1].

Definition 2.53 (Equilibrium measures). An equilibrium measure is a measure µ ∈ Mσ(Σ) which satisfies

P(φ) = hµ(σ) +∫

φdµ.

Note that when D is finite Σ is compact and so the existence of equilibrium measures follows straightfor-

wardly from the variational principle combined with the upper semi-continuity of entropy. However, when

D is countably infinite this line of argument breaks down. Instead one must construct the measure via a fixed

point for the Ruelle transfer operator [S1]. When φ has summable variations the resultant equilibrium state

will be a Gibbs measure.

2.5.2 Gibbs measures

Definition 2.54 (Gibbs measure). A probability measure µ ∈ Mσ(Σ) is said to be a Gibbs measure if there exists some

C > 1 such that for all n ∈N, all ω ∈ Dn and all x ∈ [ω] we have

C−1 <µ ([ω])

exp(−nP(φ) + Sn(φ)(x))< C.

Theorem 2.55. Let φ : Σ → R be a potential with summable variations. Suppose that P(φ) < ∞. Then there exists a

unique ergodic equilibrium measure, which is also a Gibbs measure.

The existence of Gibbs equilibrium measures was first proved by Bowen in the setting of symbolic spaces

over a finite alphabet [BoEq].

2.5.3 The submultiplicative pressure

When dealing with self-affine sets it will be necessary to adopt a more general framework. We shall now deal

with real-valued functions defined on the set of all finite words.


Definition 2.56. A function ϕ : Σ∗ → R is said to be submultiplicative if for all pairs ω, τ ∈ Σ∗ we have ϕ(ωτ) ≤

ϕ(ω) · ϕ(τ). We say that ϕ : Σ∗ → R is almost-multiplicative if there also exists some c > 0 such that for all pairs

ω, τ ∈ Σ∗ we have ϕ(ωτ) ≥ c · ϕ(ω) · ϕ(τ).

It follows from the submultiplicativity of ϕ that the pressure is well-defined.

Definition 2.57 (Submultiplicative pressure). Given a submultiplicative potential ϕ : Σ∗ → R we define the corre-

spondng pressure by

P(ϕ) := limn→∞

1n

log

(∑

ω∈Dnϕ(ω)

).

Note that this generalises the usual notion of pressure by taking the submultiplicative potential

ω 7→ sup exp (Sn(φ)(x)) : x ∈ [ω] .

Given an invariant measure µ ∈ Mσ(Σ) we define the Lyapunov exponent Λ(ϕ, µ) by

Λ(ϕ, µ) := limn→∞ ∑

ω∈Dnµ([ω]) log ϕ(ω).

Again, it follows from the submultiplicativity of ϕ that this limit exists.

Kaenmaki showed that when Σ is compact the submultiplicative pressure satisfies the following variational

principle [K].

Theorem 2.58 (The variational principle). Suppose that #D < ∞ and let ϕ : Σ∗ → R be a subadditive potential. It

follows that the pressure satisfies

P(ϕ) = sup

hµ(σ) + Λ(ϕ, µ) : µ ∈ Mσ(Σ)

.

This theorem was subsequently generalised to the setting of toplogical dynamical systems on a compact

metric space by Cao, Feng and Huang [CFH]. Recently Iommi and Yayama have given a thermodynamic

formalism for almost-additive potentials on a symbolic space with a countable alphabet [IY].

2.6 Diagonal actions on quotients of the special linear group

Most of the dynamical systems we shall consider will be either conjugated or semi-conjugated to the left shift

map on an appropriate symbolic space. However, in the final chaper we shall consider an interesting family of

flows on homogeneous spaces.

Take d ∈ N. We let Md(R) denote the set of all d × d real matrices. We endow Md(R) with the usual

topology, obtained through its natural identification with the Euclidean space Rd2. The general linear group

GLd(R) of degree d is the collection of all invertible members of Md(R),

GLd(R) = m ∈ Md(R) : det(m) 6= 0 .

Recall that any group acts upon itself by both left multiplication Lg and right multiplication Rg,

Lg : G → G by h 7→ g · h,

Rg : G → G by h 7→ h · g−1,

2.6. Diagonal actions on quotients of the special linear group 41

for g ∈ G.

A topological group is a group together with a topology under which both of these actions are continuous.

The general linear group GLd(R) ⊂ Md(R) forms a topological group under the subspace topology induced

from Md(R). We let e ∈ GLd(R) denote the identity element.

Definition 2.59 (Closed linear groups). A closed linear group is a closed subgroup of GLd(R).

2.6.1 The special linear group

The special linear group G = SLd(R) of degree d is the collection of all d× d real matrices of determinant one,

SLd(R) := h ∈ Md(R) : det(h) = 1 .

The determinant map h 7→ det(h) is a continuous multiplicative map on GLd(R). It follows that the special

linear group SLd(R) ⊂ GLd(R) forms a closed linear group.

As well as being a closed linear group SLd(R) is also connected [EW, Example 9.8]. This allows us to apply

the following theorem.

Recall that a left-invariant metric dH on a group H is a metric for which the left-muliplication map Lg is an

isometry for all g ∈ G, so given any triple g, h1, h2 ∈ G we have

dG (g · h1, g · h2) = dG (h1, h2) .

Theorem 2.60. Suppose that H ⊆ GLd(R) is a closed linear group. Suppose further that H is connected. Then H

carries a left-invariant Riemannian metric dH compatible with the subspace topology on C. Moreover, given any norm

|| · || on Md(R) there exists some η0 > 0 and C0 > 1 such that for all g1, g2 ∈ H with either dH(h1, e), dH(h2, e) < η0

or ||h1 − e||∞, ||h2 − e||∞ < η0 we have

C−10 ||g1 − g2|| ≤ dH (g1, g2) ≤ C0||g1 − g2||.

Proof. See [EW] Corollary 9.11 and Lemma 9.12.

Consequently, the special linear group G = SLd(R) is a connected Riemannian manifold with a left-

invariant Riemannian metric dG.

Take d1, d2 ∈ N so that d1 + d2 = d. We let Md2×d1(R) denote the collection of all d2 by d1 matrices. We

define ud1,d2 : Md2×d1(R)→ SLd(R) by

ud1,d2(t) :=

Id1×d1 0d1×d2

t Id2×d2

,

for t = (tij)ij ∈ Md2×d1(R). One can easily check that ud1,d2 : Md2×d1(R)→ SLd(R) is a group monomorphism

with Md2×d1(R) viewed as an additive group.

The following elementary observation is due to Dani [D, Remark 2.11].


Proposition 2.61. Given d1, d2 ∈ N with d1 + d2 = d there exists a subset P(d1, d2) ⊂ SLd(R) such that for every

ε > 0 the set p ∈ P(d1, d2) : ||p− e||∞ < ε is of Hausdorff and packing dimension (d2 − d1d2 − 1) and there exists

some η1 > 1 so that every g ∈ G = SLd(R) with dG(g, e) < η1 may be written uniquely as g = p · ud1,d2(t) for

some p ∈ P(d1, d2) and t ∈ Md2×d1(R). Moreover, there exists C1 > 1 so that given g1, g2 ∈ G with dG(g1, e) < η1,

dG(g2, e) < η1 and g1 = p1 · u(t1), g2 = p2 · u(t2) with pi ∈ P(d1, d2) and ti ∈ Md2×d1(R), we have

C−11 dG(g1, g2) ≤ max

||p1 − p2||d×d, ||t1 − t2||d2×d1

≤ C1dG(g1, g2).

Proof. We begin by defining u, c ⊂ Md(R) by

p :=

(mij)i,j : mij = 0 for all pairs (i, j) with i > d1 and j ≤ d1 and

d

∑i=1

mii = 0

,

u :=(mij)i,j : mij = 0 for all pairs (i, j) with i ≤ d1 or j > d1 or both

.

Note that p, u are linearly independent subspaces of dimension d2− d1d2− 1 and d1d2, respectively. We define

a map ψ : p× u→ SLd(R) by (p, u) 7→ exp(p) exp(u).

First note that ψ is well-defined since tr(p) = tr(u) = 0 and det(exp(h)) = exp(tr(h)) for all h ∈ Md(R),

so det(p) = det(u) = 1. Moreover, the derivative of ψ at (0, 0) is the linear map D(ψ) : p× u → Md(R)

by (p, u) 7→ p + u, which is invertible, since p, u are linearly independent. Thus, ψ is locally injective and

bi-Lipschitz at (0, 0). Since SLd(R) is a d2 − 1 dimensional manifold ψ must also be locally surjective onto a

neighbourhood of e = exp(0) · exp(0). We let P(d1, d2) := exp(p) and note that every member u ∈ exp(u)

may be written in the form u = ud1,d2(t) for some t ∈ Md2×d1(R). It follows that their is a neighbourhood

U0 surrounding the identity for which every point g ∈ SLd(R) may be written uniquely as g = p · ud1,d2(t)

for some p ∈ P(d1, d2) and t ∈ Md2×d1(R). Since the exponential map is differentiable and locally invertible

at (0, 0), P(d1, d2) intersected with any neighbourhood of the identity has Hausdorff and packing dimension

d2 − d1d2 − 1.

In order to show the local equivalence of the metrics we begin by observing that ||t1 − t2||d2×d1 = ||u(t1)−

u(t2)||d×d. Moreover, the exponential map is locally bi-Lipschitz about 0 ∈ Md(R). Also recall that all norms

on Euclidean space are Lipschitz equivalent. Thus, the proposition follows from the fact that the map ψ :

p× u→ SLd(R) is locally bi-Lipschitz around (0, 0) with respect to the Euclidean norm.

An important feature of SLd(R) is that bounded sets are precompact.

Theorem 2.62 (Hopf-Rinow Theorem). Let M be a Riemannian manifold. Suppose that M is complete as a metric

space. Then closed and bounded subsets K ⊂ M are compact.

Proof. See [DC, Theorem 2.8].

Corollary 2.63. Closed and bounded subsets of SLd(R) are compact.

Proof. By the Hopf-Rinow theorem it suffices to show that G = SLd(R) is complete as a metric space, with

respect to the Riemannian metric dG. Now take a Cauchy sequence (gn)n∈N ⊂ G. Take η0 > 0, C0 > 1 as


in Theorem 2.60. Since (gn)n∈N is a Cauchy sequence there exists some N0 such that for all n ≥ N0 we have

dG(gN0 , gn) < η0 and so by left-invariance d(e, g−1N0

gn) < η0/2. Hence, for all n, m ≥ N0 we have

C−10 ||g

−1N0

gn − g−1N0

gm|| ≤ dH

(g−1

N0gn, g−1

N0gm

),

where || · || denotes the Euclidean norm on Md(R).

Thus, (g−1N0

gn)n∈N is a Cauchy sequence with respect to the norm || · || and so converges to some point

g∞ ∈ Md(R). Let g∞ := gN0 g∞. Since g 7→ det(g) is continuous we have det(g∞) = 1 and hence g∞ ∈ SLd(R)

and d(e, g−1N0

g∞) < η0 so we have

dG(gn, g∞) = dG(g−1N0

gn, g−1N0

g∞) ≤ C0||g−1N0

gn − g−1N0

g∞||,

so (gn)n∈N converges to g∞ ∈ SLd(R) with respect to dG. Hence, (G, dG) is complete and so by the Hopf-

Rinow theorem a subset K ⊂ G is compact if and only if it is closed and bounded.

The special linear group possesses an invariant measure.

Theorem 2.64. There exists a fully supported measure mG supported on G = SLd(R) such that for all g ∈ G, mg is both

left and right-invariant, so for any Borel set B ⊂ G and any g ∈ G we have mG(B) = mG(L−1g (B)) = mG(R−1

g (B)).

Proof. We begin by defining a measure mGL on GLd(R) by letting

∫GLd(R)

ϕ(g)dmGL(g) =∫

GLd(R)ϕ(g) · det(g)−d

d

∏i,j=1

dgij

for all compactly supported continuous ϕ : GLd(R)→ R.

Note that given any g ∈ GLd(R) both the left-multiplication map Lg : h 7→ g · h and the right-multiplication

Rg : h 7→ h · g−1 are linear maps with Jacobian det(g)d and det(g)−d. Consequently, a simple calculation shows

that mL is invariant under both left and right multiplication. That is, for all g ∈ GLd(R) we have

∫L

ϕ(h)dmL(h) =∫

Lϕ(Lg(h))dmL(h) =

∫L

ϕ(Rg(h))dmL(h).

Moreover, the Lebesgue measure is absolutely continuous with respect to mL and so mL is positive on open

sets.

To obtain an invariant measure mG on G = SLd(R) we use the continuous surjection π : GLd(R)→ SLd(R)

by π : g 7→ (det(g))−1/d · g. Given a Borel set B ⊂ G we let mG(B) = mGL(π−1B

). Since π is continuous mG

is clearly positive on open sets, and hence fully supported. The fact that mG is left and right-invariant follows

straightforwardly from the left and right-invariance of mGL.

We refer to mG as the Haar measure on G.

2.6.2 Quotient spaces of discrete subgroups

Let G be the closed linear group SLd(R). Given a discrete subgroup Γ ⊂ G we may use the left-invariant

metric dG on G to define a natural metric dX on the quotient space X = Γ\G of left cosets of Γ in G.


Proposition 2.65. Let G be a closed linear group and Γ ⊂ G a discrete subgroup. We may define a metric dX upon

X = Γ\G by

dX(Γg1, Γg2) = infγ1,γ2∈Γ

dG(γ1g1, γ2g2) = infγ∈Γ

dG(γg1, g2),

for all pairs x1 = Γg1, x2 = Γg2 ∈ X.

Proof. See page 297 in [EW].

Given r > 0, g ∈ G and x ∈ X we define

BG(g, r) := h ∈ G : dG(g, h) < r ,

BX(x, r) := y ∈ X : dX(x, y) < r

The metric space (X, dX) is locally isometric to (G, dG).

Proposition 2.66. Let G be a closed linear group and Γ ⊂ G a discrete subgroup. Then for any x = Γh ∈ X = Γ\G

there exists some η(x) > 0 such that the map φx : G → X by g 7→ xg = Γ(hg) is an isometry which maps BG (g, η(x))

onto BX (x, η(x)). Moreover given any compact set K ⊂ X we may take η(K) > 0 so that η(x) > η(K) for all x ∈ K.

Proof. See [EW, Proposition 9.14].

In addition X = Γ\G has the following global property.

Proposition 2.67. Let X = Γ\G be a quotient of G by some discrete subgroup Γ and consider the metric space (X, dX).

The closed and bounded subsets of X are compact.

Proof. Let K ⊂ X be closed and bounded. Fix some point x0 ∈ X. Since K is bounded there exists some

r ∈ (0, ∞) such that d(k, x0) < r for all k ∈ K. Take some sequence (xn)n∈N ⊂ K, so dX(xn, x0) < r for all

n ∈ N. By the definition of dX there must exist (gn)n∈N∪0 so that xn = Γgn for each n ∈ N ∪ 0 and

dG(gn, g0) < r. Thus, (gn)n∈N is contained in the bounded set g ∈ G : dG(g, g0) < r, and so by Corollary

2.63 there exists a convergent subsequence (gnj)j∈N converging to some point g∞ ∈ G. Take x∞ = Γg∞. Then

for j ∈N, dX(xnj , x∞) ≤ dG(gnj , g∞) so limj→∞ dX(xnj , x∞) = 0. Since K is closed x∞ ∈ K.

2.6.3 Quotient spaces of lattices

Definition 2.68 (Fundamental domains). Let Γ be a subgroup of G. A fundamental domain is a subset F ⊂ G such

that G =⋃

γ∈Γ γF and for all γ ∈ Γ\e we have γF ∩ F = ∅. A subroup Γ ⊂ G with a fundamental domain F of

finite Haar measure mG(F) < ∞ is said to be of finite covolume.

Definition 2.69 (Lattice). A lattice Γ ⊂ G is a discrete subgroup of finite covolume.

Example 2.70. The set SLd(Z) ⊂ SLd(R) consisting of all determinant one matrices with integer entries is a lattice

[BM, Theorem 2.7].


Given a lattice Γ ⊂ G contained within G = SLd(R) we are interested in the quotient space X = Γ\G.

As we have seen, X carries a natural metric dX . Whilst we cannot, in general extend the left-action Lg to the

quotient space we may extend the right action to a map Rg : G → G by

Rg(x) = x · g = (Γh)g = Γ(hg) = Γ(Lg(h)),

for g ∈ G and x = Γh ∈ X. Note that (h1g−1) · (h2g−1)−1 = h1 · h−12 and so if Γh1 = Γh2 then Γh1g = Γh2g, so

the right-action Rg is well-defined on X.

The property of Γ being of finite covolume allows us to define a natural right-invariant measure on X.

Proposition 2.71. Let G be the special linear group and let Γ ⊂ G be a lattice with fundamental domain Γ. Then we

may define a non-trivial finite measure mX on the quotient space X = Γ\G by

mX(B) := mG (g ∈ G : Γg ∈ B ∩ F) ,

for each Borel set B ⊂ X. This definition is independent of the particular choice of fundamental domain F. Moreover,

given g ∈ G mX is invariant under the right multiplication map Rg.

Proof. See [EW, Proposition 9.20].

We let mX denote the Borel probability measure B 7→ mX(B)/mX(X). Clearly mX is right-invariant.

Proposition 2.72. The right-invariant probability measure mX on X = Γ\F is fully supported.

Proof. It is clear from the definition of dX that the projection G → Γ\G by g 7→ Γg is Lipschitz. Thus,

given an non-empty open subset U ⊂ X the set g ∈ G : Γg ∈ U is open in G. Since mG is fully supported

mG (g ∈ G : Γg ∈ U) > 0. Since Γ is discrete and G is seperable Γ must be countable and hence,

mG (g ∈ G : Γg ∈ U) = ∑γ∈Γ

mG (g ∈ G : Γg ∈ U ∩ γF) ,

where F is a fundamental domain for Γ. In particular, mG (g ∈ G : Γg ∈ U ∩ γF) > 0 for some γ ∈ Γ.

Moreover, by left-invariance

mG (g ∈ G : Γg ∈ U ∩ γF) = mG

(γ−1 g ∈ G : Γg ∈ U ∩ F

)= mG (g ∈ G : Γγg ∈ U ∩ F)

= mG (g ∈ G : Γg ∈ U ∩ F)

= mX(U) = mX(X) ·mX(U).

Hence, mX(U) > 0 for all open subsets U ⊂ X. This completes the proof of the proposition.

We let D ⊂ G denote the subgroup of all diagonal matrices with determinant one. Given some element

g ∈ D\e we shall be interested in the discrete time dynamical system obtained by repeated iteration of the

map Rg : X → X. We shall also be interested in the flow (Rgt)t∈R. As we have seen mX is a fully supported


Borel probability measure which is invariant with respect to both actions. In particular, (X,B, mX , Rg) is a

measure preserving system. It turns out that (X,B, mX , Rg) is also mixing and hence ergodic.

Recall that given a sequence (yn)n∈N in a locally compact metric space Y we say that limn→∞ yn = ∞ if

given any compact set K ⊂ Y there exists some N(K) such that yn /∈ K for all n ≥ N(K). Given a Hilbert space

H equiped with some inner product 〈·, ·〉 we let U (H) denote the collection of all linear operators T : H → H

with 〈T(x), T(y)〉 = 〈x, y〉 for all x, y ∈ H.

Theorem 2.73 (Howe-Moore). Let G = SLd(R) be the special linear group. Suppose that H is a Hilbert space with a

continuous unitary representation π : G → U (H). Suppose further that for each u ∈ H\0 there exists some g ∈ G

with π(g)(u) 6= u. Then for any pair u, v ∈ H and any sequence (gn)n∈N with limn→∞ gn = ∞ we have

limn→∞

〈π(gn)u, v〉 = 0.

Proof. See [BM, Chapter 3, Theorem 1.1].

The Howe-Moore theorem has the following well-known consequence.

Corollary 2.74 (Mixing). Let Γ ⊂ G be a lattice and take X = Γ\G. Suppose that g ∈ G is a diagonal matrix which

is not equal to the identity. Then the measure preserving system (X,B, mX , Rg) is mixing and hence ergodic. Moreover,

the topological dynamical system (X, dX , Rg) is topologically mixing.

Proof. Clearly to show that (X,B, mX , Rg) is mixing it suffices to show that for every pair ϕ1, ϕ2 ∈ L2(mX) we

have

limn→∞

(∫ϕ1 (Rg)

n · ϕ2dmX

)=∫

ϕ1dmX ·∫

ϕ2dmX .

Since the collection C0(X) ⊂ L2(mX) of all compactly supported continuous real-valued square integrable

functions is L2 dense in L2(mX) it suffices to show that this limit for all pairs ϕ1, ϕ2 ∈ C(X). Now C(X) is a

real Hilbert space under the inner product 〈ϕ1, ϕ2〉 =∫

ϕ1 ϕ2dmX . By the bilinearity of the inner product, it

suffices to prove that for all ϕ1, ϕ2 ∈ C(X) with∫

ϕ1dmX =∫

ϕ1dmX = 0 we have

limn→∞

(∫ϕ1 (Rg)

n · ϕ2dmX

)= 0.

We define a Hilbert spaceH by

H :=

ϕ : X → R : ϕ is continuous,∫|ϕ|2dmX < ∞ and

∫ϕdmX = 0

,

with the inner product 〈 f , g〉 =∫

f · gdµ.

Given g ∈ G we consider the map Rg : H → H by ϕ 7→ ϕ Rg. The map Rg is clearly linear and maps

continuous compactly supported functions to continuous compactly supported functions. Moreover, mX is

invariant under the map Tg and so the map Tg is a well-defined member of U (H). The map π : G → U (H) by

π : g 7→ Rg is clearly continuous since members of H are uniformly continuous. Also Re is the identity and

given g, h ∈ G we have

π(g) π(h)(ϕ) = ϕ Rg Rh = ϕ Rgh = π(gh)(ϕ).

2.7. Level sets for Birkhoff averages 47

Thus, π : G → U (H) is a continuous representation.

Finally for each ϕ ∈ H\0 we have some x ∈ X with ϕ(x) 6= 0, so since∫

ϕdmX = 0 there exists

x−, x+ ∈ G with ϕ(x−) < 0 < ϕ(x+). Take g−, g+ ∈ G so that x− = Γg− and x+ = Γg+. Then π(g−1+ g−)(ϕ) =

Rg−1+ g−

(ϕ) 6= ϕ since

Rg−1+ g−

(ϕ)(x−) = ϕ(x−(g−1+ g−)−1) = ϕ(Γg−g−1

− g+) = ϕ(x+).

Thus, π : g 7→ Rg satisfies the conditions of the Howe-Moore theorem. Conesequently, for any sequence

(gn)n∈N with gn → ∞ as n→ ∞ we have

limn→∞

(∫ϕ1 (Rgn) · ϕ2dmX

)= 0.

Moreover, for any g ∈ D\e we have limn→∞ gn = ∞ and (Rg)n = Rgn , so

limn→∞

(∫ϕ1 (Rg)

n · ϕ2dmX

)= 0.

This completes the proof of the corollary.

2.7 Level sets for Birkhoff averages

Let T : X → X be a topological dynamical system. One of the primary objectives of the theory of dynamical

systems is understand and classify points according to their long-run asymptotic behaviour. A natural way

to make this question precise is to take a continuous real-valued observable ϕ : X → R and classify points

according to the limiting behaviour of the Birkhoff averages n−1Sn(ϕ).

For each α ∈ R we consider the level set

Jϕ(α) :=

x ∈ X : limn→∞

n−1Sn(ϕ)(x) = α

.

We would like to understand the size and complexity of the level sets Jϕ(α).

2.7.1 A measure-theoretic approach

The measure theoretic approach to this question proceeds from Birkhoff’s ergodic theorem. Suppose we have

an ergodic invariant measure µ ∈ ET(X). By Birkhoff’s ergodic theorem we know that µ almost every x ∈ X

will satisfy

limn→∞

n−1Sn(ϕ)(x) =∫

ϕdµ.

Equivalently,

µ(

Jϕ(α))

:=

1 if α =

∫ϕdµ

0 if α 6=∫

ϕdµ.

However, this approach is only useful if µ has some natural geometric flavour; if µ were supported on a single

periodic orbit we would have learnt very little.


Suppose we have an ergodic invariant measure µ ∈ ET(X) which is equivalent to the Lebesgue measure L.

Then

L(

Jϕ(α))

:=

1 if α =

∫ϕdµ

0 if α 6=∫

ϕdµ.

However, many of the dynamical systems we shall be interested in will be restricted to fractal sets, often

with Hausdorff dimension strictly below that of the ambient space. In such circumstances there will be no

physical measure. Instead we search for an ergodic invariant measure of full Hausdorff dimension.

Definition 2.75 (Invariant measures of full Hausdorff dimension). Let T : X → X be a topological dynamical

system. An invariant measure of full Hausdorff dimension µ is an invariant Borel probability measure µ ∈ MT(X) with

dimH(µ) = dimH(X). That is, inf dimH(A) : µ(A) = 1 = dimH(X).

The importance of ergodic measures of full Hausdorff dimension is that they characterise the statistical

behaviour for a subset of the phase space of full Hausdorff dimension. Note however that there may be

multiple measures of full Hausdorff dimension and the support of a measure of full Hausdorff dimension

need not have full Hausdorff measure in the corresponding exponent [Ra]. That said, in the absence of any

physical measure, ergodic invariant measures of full Hausdorff dimension seem the most natural choice of

invariant measure from the geometric viewpoint.

2.7.2 Multifractal analysis of Birkhoff averages

We would like to obtain a finer analysis of the level sets Jϕ(α). The reason for this is twofold. Firstly, there are

many systems for which their is no single natural geometric invariant measure. Secondly, even when their is a

natural geometric invariant measure the remaining measure zero level sets are often still large in the sense of

both topology and cardinality; they are dense invariant subsets with the cardinality of the continuum. Thus,

we seek to classify these level sets according to their scaling behaviour by studying their fractal dimensions.

We shall consider the following spectra.

α 7→ dimH(

Jϕ(α))

and α 7→ dimP(

Jϕ(α))

.

We refer to these maps as the Hausdorff and packing spectra, respectively.

The primary aim in the multifractal analysis of Birkhoff averages is to provide formulae for the Hausdorff

and packing specta relating these geometric characteristics to dynamical quantities such as the entropy and

Lyapunov exponents of certain measures. We are also interested in the shape of the spectra; are these functions

concave, continuous, analytic?

2.7.3 Multifractal analysis of local dimension

A closely related field of study is the multifractal analysis of local dimension. Here we fix a measure µ, usually

an invariant measure with some particular dynamical significance, such as a Bernoulli measure or a Gibbs


measure. Given α ∈ R we consider the level sets

Dµ(α) :=

x ∈ X : limr→0

log (µ(B(x, r)))log r

= α

.

Again we are interested in the corresponding Hausdorff and packing spectra,

α 7→ dimH(

Dµ(α))

and α 7→ dimP(

Dµ(α))

.

Whilst distinct, the spectra for Birkhoff averages and pointwise dimension are closely linked and it is some-

times possible to pull across results regarding one class of spectra to another.

2.8 Iterated function systems

Suppose we have a compact metric space (X, ρ). An iterated function system consists of a finite or countable

family Sdd∈D of invertible maps Sd : X → X from the space to itself. Given ω = (ωi)ni=1 ∈ Σ∗ we let

Sω := Sω1 · · · Sωn . Suppose that our iterated function system Sdd∈D is uniformly contracting meaning that

there exists some constant c < 1 such that

ρ (Sd(x), Sd(y)) ≤ c · ρ (x, y)

for all d ∈ D and all x, y ∈ X. Let Σ be the symbolic space corresponding to the alphabet D. Since each of the

maps Sd : X → X is uniformly contracting we know that each ω ∈ Σ there is a unique point Π(ω) ∈ X for

which

Π(ω) =⋂

n∈N

Sω|n(X).

In this way we define a projection map Π : Σ → X. We define the limit set or attractor of the iterated function

system Sdd∈D by Λ := Π (Σ). We note that Λ satisfies

Λ =⋃

d∈DSd (Λ) .

We are particularly interested in cases where Si (Λ) ∩ Sj (Λ) = ∅ for i 6= j. In such cases we may define an

expanding map T : Λ→ Λ by

T(x) = (Si)−1(x) for all x ∈ Si(Λ).

Moreover, the projection Π : Σ → X will be an injective conjugacy map. That is, Π satisfies Π(σ(ω)) =

T(Π(ω)) for all ω ∈ Σ.

Σ

Π

σ// Σ

Π

ΛT// Λ.

The limit sets of iterated function systems provide us with a useful class of models for the dimension theory

of dynamical systems.


2.8.1 Conformal iterated function systems

We shall begin with a brief overview of conformal iterated function systems. An iterated function system

Sdd∈D is said to be conformal if X is a subset of a Riemannian manifold and each map Sd is a continuously

differentiable map whose derivative is a scalar multiple of an isometry. For simplicity we shall study the

one dimensional case where this assumption holds by necessity. However, the results below may be readily

extended to conformal iterated function systems in higher dimensional Euclidean space (see [MU1, B]).

Definition 2.76 (Interval Iterated Function Systems). By an interval iterated function system we shall mean a family

Sd : d ∈ D of C1 maps Sd : I → I, indexed over some finite or countable digit set D, which satisfies the following

assumtions.

(UCC) Uniform Contraction Condition. There exists a contraction ratio ξ ∈ (0, 1) and N ∈ N such that for all n ≥ N

and all ω ∈ Dn we have

supx∈I|S′ω(x)| ≤ ξn.

(OSC) Open Set Condition. There exists a non-emty open subset U ⊂ [0, 1] such that⋃

d∈D Sd(U) ⊂ U and for each

pair d1, d2 ∈ B with d1 6= d2, we have

Sd1(U) ∩ Sd2(U) = ∅.

(TDP) Tempered Distortion Property. There exists some sequence ρn with limn→∞ ρn = 0 such that for all n ∈ N and

for all ω ∈ Dn and all x, y ∈ I we have

e−nρn ≤ |S′ω(x)||S′ω(y)|

≤ enρn .

If D is finite then Sd : d ∈ D is said to be a finite interval iterated function system.

2.8.1.1 The dimension of conformal attractors

The dimension of a finite conformal iterated function system is given by the unique zero of an associated

pressure function.

We define a potential φ : Σ → R by ω 7→ − log |S′ω1(Π(σ(ω)))| and given µ ∈ Mσ(Σ) we let λµ(φ) :=∫

φdµ.

The following result is due to Ruelle [Ru] who referred to it as “Bowen’s equation”; Bowen had previously

proved a similar result for quasi-circles [Bo].

Theorem 2.77 (Ruelle). Let Sd : d ∈ D be a finite interval iterated function system. Then there is a unique s ≥ 0

for which P(−sφ) = 0 and this value gives both the Hausdorff and packing dimensions of the limit set

dimHΛ = dimPΛ = s.

Moreover,

dimHΛ = sup

hµ(σ)

λµ(φ): µ ∈ Mσ(Σ)

.


Proof. For a proof of dimHΛ = dimPΛ = s see [F2, Theorem 3.2]. The second part follows via the variational

principle for pressure 2.52.

2.8.1.2 Ergodic measures of full dimension

The following result is due to Manning [Ma].

Proposition 2.78 (Manning). Let Sd : d ∈ D be an interval iterated function system. Let µ ∈ Eσ(Σ) be an ergodic

measure. Then,

dimH (µ) =hµ(σ)

λµ(φ).

In particular dimH (Λ) = sup dimH(µ) : µ ∈ Mσ(Σ). Moreover, there exists an ergodic measure µ ∈ Eσ(Σ) with

dimH(µ) = dimH (Λ).

Proof. See [B, Chapter 5]. This essentially follows from the Shannon-McMillan-Breimann theorem (Theorem

2.48) combined with Birkhoff’s ergodic theorem (Theorem 2.29).

2.8.1.3 The multifractal analysis of Birkhoff averages

The earliest results on the multifractal analysis of Birkhoff averages date back to Besicovitch [Bes] and Eggle-

ston [Eg].

Fix a ∈N with a ≥ 2 and for each p := (pi)a−1i=0 ∈ [0, 1]a with ∑a−1

i=0 pi = 1 we define

Λ(p) :=

x ∈ [0, 1] : ∃ω ∈ 0, · · · , a− 1N x =

∞

∑j=1

ωja−j limn→∞

n−1 · #

j ≤ n : ωj = i for each i < a= pi

.

Theorem 2.79 (Besicovitch, Eggleston). For all p := (pi)a−1i=0 ∈ [0, 1]a with ∑a−1

i=0 pi = 1 we have

dimH (Λ(p)) =−∑a−1

i=0 pi log pi

log a.

The multifractal analysis of Birkhoff averages for more general classes of potentials and conformal iterated

function systems was developed by Pesin and Weiss [PW1], Fan, Feng and Wu [FFW], Barreira and Saussol

[BS], Feng, Lau and Wu [FLW] and Olsen [Ol2, Ol4]. The dimension of the level sets may be described via a

conditional variational principle resembling the variational characterisation of the dimension in Theorem 2.77.

In order to study the multifractal analysis of Birkhoff averages for iterated function systems we use the

projection map Π to rephrase the problem in terms of the symbolic space. Given a continuous potential ϕ :

Σ→ RN we define

Eϕ(α) :=

ω ∈ Σ : limn→∞

n−1Sn(ϕ)(ω) = α

,

and let Jϕ(α) := Π(Eϕ(α)

). In cases where Si (Λ) ∩ Sj (Λ) = ∅ for i 6= j this corresponds to a level set

expanding map who’s local preimages are the maps Sd.

Theorem 2.80 (Feng, Lau, Wu). Let Sd : d ∈ D be a finite interval iterated function system and let ϕ : Σ→ RN be

a continuous potential. Then for all α ∈ RN we have

dimH(

Jϕ(α))= dimP

(Jϕ(α)

)= sup

hµ(σ)

λµ(φ): µ ∈ Mσ(Σ),

∫ϕdµ = α

.


A similar variational formula also holds for the pointwise dimension spectrum for Gibbs measures (see [B,

Chapter 6]).

2.8.2 Infinite conformal iterated function systems

In this section we shall consider conformal iterated function systems Sdd∈D consisting of a countable infinity

of maps. The theory of infinite iterated function systems begins with the work of Mauldin and Urbanski [MU1].

For simplicity we describe the relevant results in the contexts of interval iterated function systems, but all of

the following may be generalised to higher dimensions with appropriate additional constraints (see [MU1] for

details).

We shall make the following assumptions not required for finite iterated function systems.

(OIC) Open Interval Condition. For all d1, d2 ∈ D with d1 6= d2, we have

Sd1((0, 1)) ∩ Sd2((0, 1)) = ∅.

(BDP) Bounded Distortion Property. There exists a constant C > 1 such that for all ω ∈ ⋃n∈NDn and all x, y ∈ I

we have

C−1 ≤ |S′ω(x)||S′ω(y)|

≤ C.

2.8.2.1 The dimension of the attractor

The following example, due to Mauldin and Urbanski, shows that Bowen’s equation cannot be straightfor-

wardly generalised to the setting of infinite iterated function systems [MU1, Example 5.3].

Example 2.81. Take D :=(n, k) ∈N2 : k ≤ 2n2−1

and for each (n, k) ∈ D let φ(n,k) : [0, 1] → [0, 1] be an affine

map with derivative 2−(n2+n). Since ∑∞

n=1 2n2−1 · 2−(n2+n) = 12 we may choose (φn,k)(n,k)∈D so that the open interval

condition is satisfied. Then, for all s ∈ R we have,

P(−sφ) = log

(∞

∑n=1

2n2−1 · 2−(n2+n)s

),

so P(−sφ) is infinite for all s < 1 and strictly negative for all s ≥ 1.

This example shows that for infinite conformal iterated function systems the equation P(−sφ) = 0 need

not have a solution. Having said that, Mauldin and Urbanski have shown that the following generalisation of

Bowen’s equation does hold.

Theorem 2.82 (Mauldin and Urbanski). Let Sd : d ∈ D be an interval iterated function system, satisfying (OIC)

and (BDP). Then

dimHΛ = inf s : P(−sφ) ≤ 0 = sup

hµ(σ)

λµ(φ): µ ∈ Mσ(Σ), λµ(φ) < ∞

.

Thus, the Hausdorff dimension of the attractor of the conformal IFS in the previous example is one.


2.8.2.2 Measures of maximal dimension

When dealing with infinite conformal iterated function systems we cannot apply our previous strategy in de-

ducing the existence of ergodic invariant measures of full Hausdorff dimension; the symbolic space will no

longer be compact and the entropy is no longer upper semi-continuous. In fact it follows straightforwardly

from the variational principle that there can be no invariant measure of full Hausdorff dimension if the pres-

sure function doesn’t have a root. Since, as we have seen, there are examples of infinite conformal iterated

function systems for which the pressure equation has no root we know that such maximising measures need

not exist.

Nonetheless, Mauldin and Urbanski used the thermodynamic formalism for countable symbolic spaces to

prove the existence of maximising measures in the presence of a root for the pressure equation.

Proposition 2.83. Let Sd : d ∈ D be an interval iterated function system, satisfying the (OIC) and the (BDP).

Suppose further that P (−dimHΛ · φ) = 0. Then there exists a unique ergodic measure µ ∈ Eσ(Σ) with dimHµ =

dimHΛ.

2.8.2.3 Multifractal analysis of Birkhoff averages

The multifractal analysis of Birkhoff averages for infinite iterated function systems has attracted a great deal

of attention in recent years (see [JK, KMS, KS, IJ, FLM, FLWWJ, FLMW]).

The most general result in this research area is the following theorem due to Fan, Jordan, Liao and Rams

[FJLR]. We first define a shrinking family of neighbourhoods for each α ∈ R∪ −∞ ∪ +∞ by

Bn(α) :=

(−∞,−n) if α = −∞(

α− 1n , α + 1

n

)if α ∈ R

(n,+∞) if α = +∞

We also define s∞ := inf s ∈ R : P(−sφ) < ∞. We consider a countable collection of potentials ϕ = (ϕi)i∈N

and given α = (αi)i∈N ∈ RN we let Jϕ(α) denote the intersection⋂

i∈N Jϕi (αi).

Theorem 2.84. Let Sd : d ∈ D be an interval iterated function system, satisfying the (OIC). Let (ϕi)i∈N be a

countable family of uniformly continuous potentials ϕi : Σ→ R. Given α = (αi)i∈N ∈ RN we have

dimH(

Jϕ(α))= lim

n→∞sup

hµ(σ)


∫ϕidµ ∈ Bn(αi) for all i ≤ n

.

Moreover, if each map ϕi is bounded then whenever Jϕ(α) 6= ∅,

dimH(

Jϕ(α))= max

s∞, sup

hµ(σ)


∫ϕidµ = αi for all i ∈N

.

2.8.3 Self-affine Sierpinski sponges

The first class of self-affine limit sets to be studied were the self-affine limit sets of Bedford [Be] and McMullen

[Mc]. Higher dimensional analogues were subsequently considered by Kenyon and Peres [KP]. These limit

sets are known as self-affine Sierpinski sponges.


Definition 2.85 (Self-affine Sierpinski sponges). We consider the d-dimensional torus Td := Rd/Zd. Choose natural

numbers a1 > a2 > · · · > ad ≥ 2. Let f : Td → Td denote the integer valued diagonal map given by

(xq)dq=1 7→ (aqxq)

dq=1 for (xq)

dq=1 ∈ Td. (2.1)

Given a digit set D ⊆ ∏dq=1

0, · · · , aq − 1

there is a corresponding self-affine repeller Λ given by

Λ :=

(

∞

∑n=1

iq(n)an

q

)d

q=1

: (iq(n))dq=1 ∈ D for all n ∈N

. (2.2)

A limit set Λ defined in this way is referred to as a self-affine Sierpinski sponge. A two dimensional Sierpinski sponge is

known as a Bedford-McMullen carpet.

As well as being the repeller for the diagonal expanding map f : Td → Td the limit set Λ is also an attractor

for the non-conformal iterated function system (Si)i∈D where, for each i = (iq)dq=1 ∈ Dwe define Si : Td → Td

by

Si

((xq)

dq=1

):=(

xq + iq

aq

)d

q=1.

2.8.3.1 Dimension of the attractor

Given k ≤ d we let πk : Td → Td−(k−1) be the projection πk : (xq)dq=1 7→ (xq)d

q=k. We let fk : Td−(q−1) →

Td−(q−1) denote the map (xq)dq=k 7→ (aqxq)d

q=k.

For each k ≤ d we define Hk( f ) by

Hk( f ) := sup

hµπ−1

k( fk) : µ ∈ M(Λ, f )

.

The following result is due to Kenyon and Peres [KP].

Theorem 2.86 (Kenyon, Peres). Let Λ be a self-affine Sierpinski sponge. Then,

dimH(Λ) = supµ∈M(Λ, f )

hµ( f )log a1

+d

∑k=2

(1

log ak− 1

log ak−1

)h

µπ−1k( fk)

,

dimP(Λ) =H1( f )log a1

+d

∑k=2

(1

log ak− 1

log ak−1

)Hk( f ).

Bedford [Be] and McMullen [Mc] independently determined both the Hausdorff dimension and the upper

box dimension in the two dimensional setting. In [KP] Kenyon and Peres extend these results to higher dimen-

sions. It follows from [F2, Proposition 3.6] that the formula for upper box dimension also gives an expression

for the packing dimension.

Whilst the Hausdorff and packing dimension of finitely geneterated self-conformal sets, for self-affine

Sierpinski sponges we often have dimH(Λ) 6= dimP(Λ).

2.8.3.2 Measures of maximal dimension

Kenyon and Peres also gave a formula for the dimension of an ergodic measure [KP] which closely resembles

a previous result due to Ledrappier and Young [LY].


Theorem 2.87 (Kenyon and Peres). Let µ ∈ E f (Λ) be an ergodic measure on a self-affine Sierpinski sponge. Then

dimH(µ) =hµ( f )log a1

+d

∑k=2

(1

log ak− 1

log ak−1

)h

µπ−1k( fk).

Moreover there exists a measure µ ∈ E f (Λ) with dimH(µ) = dimH(Λ).


The first result concerning the multifractal analysis of Birkhoff averages for self-affine sets is due to Nielsen

[N]. Suppose Λ is a self-affine Sierpinski sponge. For x ∈ Λ we let

Γ(x) :=

((iq(ν))dq=1)ν∈N : x =

(∞

∑ν=1

iq(ν)

aνq

)d

q=1

.

Given a probability vector p = (pl)l∈D defined over a digit set D we let Nl (ω|n) := #ν ≤ n : ων = l, where

# denotes cardinality, and define

Λp :=

x ∈ Λ : ∃ω ∈ Γ(x) with limn→∞

Nl (ω|n)n

= pl for each l ∈ D

.

Let µp denote the Bernoulli measure on Λ corresponding to the probability vector p. In [N] Nielsen proved the

following formula for the Hausdorff and packing dimension of Λp in the two dimensional case. With minor

modifications the proof also applies in higher dimensions.

Theorem 2.88 (Nielsen).

dimH(Λp)= dimP

(Λp)=

hµp( f )log a1

+d

∑k=2

(1

log ak− 1

log ak−1

)hµpπk ( fk).

The more challenging problem of giving a multifractal analysis of Birkhoff averages for individual poten-

tials was first taken up by Barral and Mensi [BM1] who determined the Hausdorff dimension of the level

sets for Bedford McMullen carpets. Later this result was generalized by Barral and Feng [BF] who gave a

conditional variational principle for the dimension of the level sets on a self-affine Sierpinski sponge.

Theorem 2.89 (Barral, Feng). Let Λ be a self-affine Sierpinski sponge and a continuous potential ϕ : Λ → RN for

some N ∈N. Then, for all α ∈ RN we have

dimH(

Jϕ(α))= sup

hµ( f )log a1

+d

∑k=2

(1

log ak− 1

log ak−1

)h

µπ−1k( fk)

,

where the supremum is taken over all µ ∈ M(Λ, f ) with∫

ϕdµ = α.

2.8.4 Lalley-Gatzouras repellers

Lalley and Gatzouras generalised the self-affine limit sets of Bedford and McMullen to consider class of self-

affine iterated function systems in the plane with variable Lyapunov exponents [LG].

Definition 2.90 (Lalley-Gatzouras IFS). Take m ∈ N and choose bi ∈ (0, 1) and di ∈ [0, 1] for each i ∈ 1, · · · , m

so that ∑mi=1 bi ≤ 1, bi + di ≤ di+1 for i = 1, · · · , m− 1 and bm + dm ≤ 1. For each i ∈ 1, · · · , m choose ni ∈ N


and for each j ∈ 1, · · · , ni choose aij ∈ (0, bi] and cij ∈ [0, 1] so that ∑nij=1 aij ≤ 1 and aij + cij ≤ ci(j+1) for

j = 1, · · · , ni − 1 and aini + cini ≤ 1. Let D := (i, j) : 1 ≤ i ≤ m, 1 ≤ j ≤ ni. For each (i, j) ∈ D we define

Sij : [0, 1]2 → [0, 1]2 by

Sij(x) =

aij 0

0 bi

x +

cij

di

for x ∈ [0, 1]2.

We refer to an iterated function system formed in this way as a Lalley-Gatzouras IFS.


Lalley and Gatzouras gave a variational expression for the dimension of the attractor [LG].

Let (Sij)(i,j)∈D be a Lalley-Gatzouras IFS. Given a probability vector p = (pij)(i,j)∈D ∈ [0, 1]D with ∑(i,j)∈D pij =

1 we let pi = ∑nij=1 pij for each i ≤ m and define

D(p) :=∑(i,j)∈D pij log pij

∑(i,j)∈D pij log aij+

(1

∑mi=1 pi log bi

− 1∑(i,j)∈D pij log aij

)·

m

∑i=1

pi log bi.

Theorem 2.91 (Lalley and Gatzouras). Let Λ be the attractor of Lalley-Gatzouras IFS (Sij)(i,j)∈D . Then

dimH(Λ) = sup

D(p) : p ∈ [0, 1]D and ∑(i,j)∈D

pij = 1

.

2.8.4.2 A Bernoulli measure of maximal dimension

Lalley and Gatzouras also showed that if we take a probability vector p = (pij)(i,j)∈D ∈ [0, 1]D with ∑(i,j)∈D pij =

1 and let µp ∈ Mσ(Σ) denote the corresponding Bernoulli measure on the symbolic space Σ = DN then

dimH(µp) = D(p). Since(pij)(i,j)∈D ∈ [0, 1]D : ∑(i,j)∈D pij = 1

is compact and p 7→ D(p) is continuous it

follows that there exists a Bernoulli measure of maximal dimension. Recall that Bernoulli measures are ergodic.

2.8.4.3 The multifractal analysis of Birkhoff averages

The multifractal analysis of Birkhoff averages on a Lalley-Gatzouras repeller were studied by the present au-

thor [R2].

Given µ ∈ Mσ(Σ) we let p(µ) =(

pij(µ))(i,j)∈D ∈ [0, 1]D denote the probability vector formed by letting

pij(µ) = µ ([i, j]) for each (i, j) ∈ D and let D(µ) := D (p(µ)).

Theorem 2.92 (R.). Let Λ be the limit set for a Lalley-Gatzouras IFS. Take a continuous real valued potential ϕ : Σ→ R.

For all α ∈ R we have

dimH(

Jϕ(α))= sup

D (µ) : µ ∈ Mσ(Σ),

∫ϕdµ = α

.

2.8.5 Typical self-affine sets

Take a family of invertible matrices (Ti)i∈D ⊆ GLd(R). To each sequence a = (ai)i∈D ∈ (Rd)D we associate a

family of affine contractions by(Sa

i)

i∈D by

Sai (x) = Ti(x) + ai for x ∈ Rd.


To each sequence a ∈ (Rd)D we associate a projection Πa : Σ→ Rd defined by

Πa(ω) = limn→∞

Saω1 · · · Sa

ωn (0) .

We let Λa := Πa (Σ).


The dimension theory of self-affine sets of this form was first investigated by Falconer [F1]. A central tool in

Falconer’s analysis was the singular function ϕs : Md(R) → R. Given a matrix M ∈ Md(R) we let γ1(M) ≥

· · · ≥ γd(M) denote the singular values of M, in non-increasing order of magnitude. For each s ∈ [0, d] the

associated singular value function is defined by

ϕs(M) =

(m−1

∏j=1

γj(M)

)γm(M)s−m+1,

where m ∈N is chosen so that m− 1 ≤ s < m.

We may think of singular function ϕs, initially defined on Md(R), as a function on Σ∗ by letting

ϕs(ω) := ϕs (Tω1 · · · Tωd

),

for ω = (ωi)ni=1 ∈ Σ∗. Falconer observed that the singular value function ϕs is sub-multiplicative [F1, Lemma

2.1]. It follows that the corresponding sub-multiplicative pressure P(ϕs) is well-defined.

Theorem 2.93 (Falconer [F1]). Suppose that #D < ∞ and ||Ti||∞ < 12 for each i ∈ D. Then there exists a unique

value s for which P(ϕs) = 0 and this value gives the typical Hausdorff and packing dimension of the attractor. That is

for L almost every a ∈ (Rd)D we have dimHΛa = dimPΛa = s.

This theorem was first proved by Falconer [F1] for classes of affine maps with linear parts of norm at most

one third. Later Solomyak [S] showed that, with a minor modification, the proof may be extended to setings

where each affine map has linear part at most one half.

Note that one cannot remove the assumption that ||Ti||∞ < 12 [Ed3].

2.8.5.2 An ergodic measure of maximal dimension

For each µ ∈ Mσ(Σ). We define D(µ) to be the unique value of s for which

hµ(σ) + Λ(ϕs, µ) = 0.

Proposition 2.94 (Jordan, Pollicott, Simon [JPS]). Suppose #D < ∞ and ||Ti||∞ < 12 for each i ∈ D. Take µ ∈

Eσ(Σ). For L almost every a ∈ (Rd)D , dimH(µ) = dimP(µ) = D(µ).

By developing a thermodynamic formalism for subadditive potentials on a compact symbolic space Kaenmaki

has proven the following result [K].


Proposition 2.95 (Kaenmaki). Suppose that #D < ∞ and ||Ti||∞ < 12 for each i ∈ D. Then for L almost every

a ∈ (Rd)D we have

dimHΛa = dimPΛa = sup D(µ) : µ ∈ Mσ(Σ) .

Moreover, there exists an ergodic measure µ ∈ Eσ(Σ) such that for L almost every a ∈ (Rd)D we have

dimHµ = dimPµ = dimHΛa = dimPΛa.


Jordan and Simon have proved the following theorem for diagonal matrices in two dimensions [JS]. Given

a continuous potential ϕ : Σ → R and some α ∈ R we let Eϕ(α) denote the symbolic level set and given

a ∈ (Rd)D we let Jaϕ(α) = Πa (Eϕ(α)

).

Theorem 2.96 (Jordan, Simon). Suppose that #D < ∞ and that for each i ∈ D Ti is a 2× 2 diagonal matrix with

||Ti||∞ < 12 . Then for L almost every a ∈ (Rd)D we have

dimH

(Jaϕ(α)

)= sup

D(µ) : µ ∈ Mσ(Σ),

∫ϕdµ = α

.

2.9 Multifractal analysis for dynamical systems without a Markov partition

In this section we discuss the dimension theory of dynamical systems without a Markov partitions. Whilst the

results of the previous section were presented in the setting of iterated function systems, where the correspond-

ing dynamical system is coded by the full shift, each of the above theorems may be extended unproblematically

to the setting of dyanamical systems which are coded, via a Markov partition, by some subshift of finite type.

That is, a shift space where a certain finite collection of words is prohibited (see for example [B] and [MU2]).

However, the dimension theory of dynamical systems which cannot be coded by a subshift of finite type is

much more challenging.

The multifractal analysis of Birkhoff averages for non-Markovian dynamical systems has focused largely

upon determining the topological entropy htop(T, Jϕ(α)

)of the level sets Jϕ(α). For an introduction to the

notion of topological entropy for non-compact subsets of compact sets we refer the reader to [TV, Section 3.1].

The role of a symbolic coding is the ability to cook up points with a prescribed orbit structure. In many

situations this role can be covered by some form of specification property. Here we describe the almost speci-

fication which generalises several related notions of specification (see [Th2]).

Definition 2.97 (Almost specification). Suppose we have a topological dynamical system T : X → X. Suppose we

have a function g : (0, ∞)×N → R such that for each ε > 0 limn→∞ n−1g(ε, n) = 0. For each n ∈ N, ε > 0 and

x ∈ X we let

Bn(g, ε, x) :=

y ∈ X : #

i ∈N : d(

Ti(y), Ti(x))> ε

< g(n, ε)

.

2.9. Multifractal analysis for dynamical systems without a Markov partition 59

Suppose that for each finite sequence (εi)ki=1 ∈ (0, ∞)k there exists (N(g, εi))

ki=1 ∈ Nk so that for all finite sequences

(ni)ki=0 ∈Nk with each ni ≥ N(g, εi) and (xi)

ki=1 ∈ Xk we have

k⋂i=1

T−∑i−1j=0 nj (Bni (g, εi, xi)) 6= ∅,

where n0 = 0. Then T : X → X is said to satisfy the almost specification property. A dynamical system is said to satisfy

the specification property if g(n, ε) depends only upon ε and not on n.

Takens and Verbitsky have proven the following variational formula for the topological entropy of the level

sets for Birkhoff averages [TV].

Theorem 2.98 (Takens and Verbitsky). Suppose we have a topolocial dynamical system T : X → X on a compact

metric space X which satisfies the specification property. Let ϕ : X → R be a continuous observable. Then for all α ∈ R

we have

htop(T, Jϕ(α)

)= sup

hµ(T) : µ ∈ MT(X),

∫ϕdµ = α

.

We note that the proof may be adapted to deal with systems with the almost specification property. A

generalisation has been proven with pressure taking the place of topological entropy (see [Th1]).

Thompson has shown that for systems satisfying the almost specification property the set of points for

which Birkhoff averages do not converge has full topological entropy [Th2].

Theorem 2.99 (Thompson). Suppose we have a topolocial dynamical system T : X → X on a compact metrix space X

which satisfies the almost specification property. Let ϕ : X → R be a continuous observable. Then we have

htop

(T, X\

⋃α∈R

Jϕ(α)

)= htop (T, X) .

Recently Hofbauer determined the topological entropy htop(T, Jϕ(α)

)of the level sets Jϕ(α) for piecewise

montone interval maps [Ho]. Moreover, Hofbauer calculated the Hausdorff dimension of the level sets for

Birkhoff averages in the special case where the potential is the logarithm of the derivative. It has also been

shown by Climenhaga that for any compact topological dynamical system for which the pressure function is

suitably well-behaved the topological entropy of the level sets is equal to the Legendre transform of a suitable

pressure function [Cl2].


3

The packing spectrum for Birkhoff averages

on a self-affine repeller

3.1 Introduction

In this chapter we shall study the packing spectrum for Birkhoff averages on a self-affine Sierpinski sponge. In

the conformal setting the Hausdorff and packing spectra coincide (see Section 2.8.1.3). In addition the spectrum

satisfies various regularity properties [B]. In particular, the spectrum is continuous and when the potential is

Holder continuous the spectrum is real analytic. When the Lyapunov exponent is constant the spectrum is also

concave.

The dimension theory of non-conformal systems is much less well understood. Nonetheless there has been

some recent progress in this area. We refer to Section 2.8 for an introduction. We shall briefly recall the relevant

results on the multifractal analysis for Birkhoff averages on a self-affine Sierpinski sponge. Nielsen considered

sets of points with a prescribed digit frequency, showing that both the Hausdorff and packing dimension are

equal to the dimension of the corresponding Bernoulli measure [N]. However, Kenyon and Peres have shown

that Hausdorff and packing dimensions of self-affine Sierpinski sponges usually differ [KP]. It follows that for

potentials which are cohomologous to a constant the Hausdorff and packing spectra differ.

Barral, Mensi and Feng have determined the Hausdorff spectrum for Birkhoff averages of continuous po-

tentials on a self-affine Sierpinski sponge, expressing their results as conditional variational principle [BM1,

BF]. In this chapter we shall determine the packing spectrum for Birkhoff averages of continuous potentials

on a self-affine Sierpinski sponge. We also show that the packing spectrum is both concave and continuous.

However, we give an example showing that the spectrum need not be analytic, even for Holder continuous

potentials. We also consider the packing spectrum for pointwise dimension in the self-affine setting. We give

a formula for the packing dimension of the level sets for a very limited class of Bernoulli measures on self-

affine Sierpinski sponge and refute a conjecture of Olsen [Ol5, Conjecture 4.1.7] by showing that the packing

spectrum need not attain the full packing dimension of the repeller.

The results of this chapter have been published in Ergodic Theory and Dynamical Systems [R1].

61

62 Chapter 3. The packing spectrum for Birkhoff averages on a self-affine repeller

3.2 Notation and statement of results

Let Td denote the d-dimensional torus Rd/Zd. Choose natural numbers a1 > a2 > · · · > ad ≥ 2 and consider

the map f : Td → Td by f : (xq)dq=1 7→ (aqxq)d

q=1. Choose a digit set D ⊆ ∏dq=1

0, · · · , aq − 1

we consider

the repeller Λ given by

Λ :=

(

∞

∑n=1

iq(n)an

q

)d

q=1

: (iq(n))dq=1 ∈ D for all n ∈N

.

The repeller Λ is a self-affine set known as self-affine Sierpinski sponge (see Section 2.8.3). We shall be inter-

ested in the level sets for Birkhoff averages. Let ϕ : Λ→ R be a continuous potential. Given α ∈ R we consider

the set of points in the repeller for which the Birkhoff average converges to α,

Jϕ(α) :=

x ∈ Λ : limn→∞

1n

n−1

∑q=0

ϕ( f q(x)) = α

. (3.1)

We shall determine the packing spectrum α 7→ dimP(

Jϕ(α)).

Given k ≤ d we let πk : Td → Td−(k−1) be the projection πk : (xq)dq=1 7→ (xq)d

q=k. We let fk : Td−(q−1) →

Td−(q−1) denote the map (xq)dq=k 7→ (aqxq)d

q=k.

For each α ∈ R we define Hk( f , ϕ, α) for k = 1, · · · , d by

Hk( f , ϕ, α) := sup

hµπ−1

k( fk) : µ ∈ M f (Λ),

∫ϕdµ = α

.

Theorem 3.1. Let Λ be a self-affine Sierpinski sponge. Let ϕ : Λ → RN be some continuous potential. Then for all

α ∈ RN we have

dimP(

Jϕ(α))=

H1( f , ϕ, α)

log a1+

d

∑k=2

(1

log ak− 1

log ak−1

)Hk( f , ϕ, α).

In fact Theorem 3.1 follows from the more general Theorem 3.2. Given a Borel probability measure µ ∈

M(Λ) we define

An(µ) :=1n

n−1

∑k=0

µ f−k. (3.2)

Given x ∈ Λ we let V(x) denote the set of all weak ∗ accumulation points of the sequence of measures

(An(δx))n∈N where δx denotes the Dirac measure concentrated at x. Note that V(x) ⊆ M f (Λ) [W, Theo-

rem 6.9] for all x ∈ Λ. Given a closed convex set we let A ⊆M f (Λ) we define

X(A) : = x ∈ Λ : V(x) = A , (3.3)

Y(A) : = x ∈ Λ : V(x) ⊆ A .

We also define

A :=d⋂

k=1

(µj)

dj=1 ∈ M f (Λ)d : log ak

(1

log a1µ1 +

k

∑j=2

(1

log aj− 1

log aj−1

)µj

)∈ A

.

In [BF] Barral and Feng considered the special case in which A = µ for some µ ∈ M f (Λ). It follows that

X(µ) = Y(µ) and the Hausdorff and packing dimensions coincide. However, in general this is not the

case.

3.3. Symbolic dynamics 63

Theorem 3.2. Let Λ be a self-affine Sierpinski sponge. Suppose that A is a non-empty closed convex subset ofM f (Λ).

Then,

dimP (X(A)) = dimP (Y(A)) = sup

hµ1( f )log a1

+d

∑j=2

(1

log aj− 1

log aj−1

)h

µjπ−1j( f j) : (µj)

dj=1 ∈ A

.

Note that Theorem 3.2 is a generalisation of Theorem 3.1.

Proof of Theorem 3.1 from Theorem 3.2. Take a continuous potential ϕ : X → RN and α ∈ R. We letM(ϕ, α) ⊂

M f (Λ) denote the closed convex set defined by

M(ϕ, α) :=

µ ∈ M f (Λ) :∫

ϕdµ = α

.

It follows that Jϕ(α) = X (M(ϕ, α)) = Y (M(ϕ, α)). Moreover, one can easily show that M(ϕ, α) =M(ϕ, α)d.

Consequently, Theorem 3.1 is a special case of Theorem 3.2.

The central difficulty in determining the packing spectrum is proving the lower bound in Theorem 3.2.

Unlike the Hausdorff dimension, the packing dimension of a level set typically exceeds the supremum of the

dimensions of the invariant measures supported on that set. We construct a non-invariant measure specifically

suited to obtaining an optimal lower bound for packing dimension.

The rest of the chapter is structured as follows. We begin by restating Theorems 3.1 and 3.2 in Section 2 in

terms of the symbolic space. The proof of Theorem 3.3 is given in sections 3 and 4. In Section 3 we prove the

lower bound, and in Section 4 we prove the upper bound. In Section 5 we deduce some regularity properties

of the packing spectrum. In Section 6 we present two simple examples exhibiting some interesting features of

the packing spectrum in the two dimensional case. In Section 7 we conclude with some extensions of Theorem

3.1 which follow from Theorem 3.2 along with some open questions.

3.3 Symbolic dynamics

We begin by restating our theorem in terms of the associated symbolic space. Let Σ denote the symbolic space

DN under the usual product topology. We let Π : Σ→ Λ denote the natural projection given by

Π : (ων)ν∈N 7→(

∑ν∈N

ij(ν)

aνj

)d

j=1

where ων = (ij(ν))dj=1 for ν ∈N. (3.4)

For each k = 1, · · · , d we let ηk denote the projection of D by ηk : (ij)dj=1 7→ (ij)

dj=k and Σk := ηk(D)N. We then

define a projection χk : Σ 7→ Σk, corresponding to πk : Td → Td−(k−1) by χk : (ων)∞ν=1 7→ (ηk(ων))∞

ν=1. We let

Πk : Σk → πk(Λ) denote the natural projection given by

Πk : (τν)ν∈N 7→(

∑ν∈N

ij(ν)

aνj

)d

j=k

where τν = (ij(ν))dj=k for ν ∈N. (3.5)

Note that Πk χk = πk Π.

We let η[k denote the projection of D by η[

k : (ij)dj=1 7→ ik and define a projection Π[

k : Σ→ T by

Π[k : (ων)ν∈N 7→ ∑

ν∈N

ik(ν)

aνk

where ων = (ij(ν))dj=1 for ν ∈N. (3.6)


Note that whilst ηk and Πk take values in spaces with d− (k− 1) coordinates, their sharpened counterparts,

η[k and Π[

k map to just the first coordinate of the corresponding value.

We let σ denote the left shift on Σ and for each k, σk denotes the left shift on Σk. Note that f Π = Π σ and

for each k fk Πk = Πk σk. Given a finite sequence (ων)nν=1 ∈ ηk(D) we let [ω1 · · ·ωn] denote the cylinder set

[ω1 · · ·ωn] :=

ω′ ∈ Σk : ω′ν = ων for ν = 1, · · · , n

. (3.7)

Given ϕ : Σ→ R and n ∈N we define varn(ϕ) by

varn(ϕ) := sup |ϕ(ω)− ϕ(τ)| : ων = τν for ν = 1, · · · , n . (3.8)

We also define An(ϕ) : Σ→ R to be the map ω 7→ 1n ∑n−1

l=0 ϕ(σlω).

We are interested in the space of all Borel probability measuresM(Σ) under the weak ∗ topology. Since Σ

is compact and hence the space C(Σ) of continuous real valued functions on Σ is separable, we may choose a

countable family of potentials (ϕl)l∈N with norm one, ||ϕl ||∞ = 1, for all l ∈N, for which sets of the formν ∈ M(Σ) :

∣∣∣∣ ∫ ϕldν−∫

ϕldµ

∣∣∣∣ < ε for all l ≤ L

, (3.9)

with µ ∈ M(Σ) and L ∈N, form a neighbourhood basis ofM(Σ).

For each n ∈N we letMσn(Σ) denote the set of σn-invariant Borel probability measures, let Eσn(Σ) denote

the set of µ ∈ Mσn(Σ) which are ergodic, with respect to σn, and let Bσn(Σ) denote the set of µ ∈ Eσn(Σ) which

are also Bernoulli.

Given a Borel probability measure µ ∈ M(Σ) we define

An(µ) :=1n

n−1

∑l=0

µ σ−l . (3.10)

Given ω ∈ Σ we let V(ω) denote the set of all weak ∗ accumulation points of the sequence of measures

(An(δω))n∈N where δω denotes the Dirac measure concentrated at ω. Given a closed convex subset A ⊆

Mσ(Σ) we define

Γ(A) : = ω ∈ Ω : V(ω) = A (3.11)

Ω(A) : = ω ∈ Ω : V(ω) ⊆ A .

We also define

A :=d⋂

k=1

(µj)

dj=1 ∈ Mσ(Σ)d : log ak

(1

log a1µ1 +

k

∑j=2

(1

log aj− 1

log aj−1

)µj

)∈ A

.

We shall prove the following Theorem which implies Theorems 3.1 and 3.2.

Theorem 3.3. Suppose that A is a non-empty closed convex subset ofMσ(Σ). Then

dimPΠ(Γ(A)) = dimPΠ(Ω(A)) = sup

hµ1(σ)

log a1+

d

∑j=2

(1

log aj− 1

log aj−1

)h

µjχ−1j(σj) : (µj)

dj=1 ∈ A

.

3.4. Proof of the lower estimate 65

3.4 Proof of the lower estimate

Fix a non-empty closed convex subset A ⊆M(Σ, σ). Take ζ > 0 and choose some (µj)dj=1 ∈ A with

hµ1(σ)

log a1+

d

∑j=2

(1

log aj− 1

log aj−1

)h

µjχ−1j(σj) (3.12)

> sup

hν1(σ)

log a1+

d

∑j=2

(1

log aj− 1

log aj−1

)h

νjχ−1j(σj) : (νj)

dj=1 ∈ A

− ζ. (3.13)

Note that Γ(A) ⊆ Ω(A). Through a series of lemmas we shall prove that

dimPΠ(Γ(A)) ≥hµ1(σ)

log a1+

d

∑j=2

(1

log aj− 1

log aj−1

)h

µjχ−1j(σj). (3.14)

To this end we construct a measure allowing us to apply Proposition 2.4 to obtain a lower bound for the

packing dimension.

Let λ0 := 0 and for j = 1, · · · , d we let λj := log ad/ log aj. In order to obtain an optimal lower bound

we shall construct a measure W which, for infinitly many values of n, behaves like µj for the digits from

dλj−1ne+ 1 up to dλjne, for each j = 1, · · · , d, and use this property to show thatW Π−1 has the required

packing dimension.

We must also chooseW so that V(ω) = A on a set of largeW measure. To do this we take a sequence of

measures (mq)q∈2N in A for which the set of weak ∗ limit points is of (mq)q∈2N is precisely the set A. We shall

also constructW so that, along a subsequence of times,W behaves like (mq)q∈2N.

To obtain such a measure, W , we effectively piece together the various invariant measures that W is re-

quired to imitate. In order to carry out this procedure we must first approximate each of our invariant measures

by members of⋃

n∈N Bσn(Σ). This allows us to deal with three issues. Firstly, the invariant measures whichW

is required to mimic need not be ergodic. Nonetheless, the approximations will be ergodic for some n-shift σn,

and this allows us to apply both Birkhoff’s ergodic Theorem and the Shannon-McMillan-Breiman Theorem.

Secondly, we do not assume King’s disjointness condition (see [Ki]) and allow our approximate squares to

touch at their boundaries. As such we must insure that our measure is not too concentrated so that it behaves

well under projection by Π. For members of⋃

n∈N Bσn(Σ) we may do this simply by tweaking our measure

so that it gives each finite word some positive probability. Thirdly, the process of pieceing together measures

is greatly simplified by only working with members of⋃

n∈N Bσn(Σ). This approximation introduces an error,

both in the expected local entropy and expected Birkhoff averages. However, these error terms go to zero as

the approximation improves, so by concatenating increasingly good approximations we will obtain a measure

which not only behaves well at every given stage, but gives positive measure to the level set Γ(A) and gives

an optimal lower bound for the packing dimension of Π(Γ(A)).

Similar techniques appear in the work of Gelfert and Rams [GR], Barral and Feng [BF], Baek, Olsen and

Snigreva [BOS] and Barreira and Schmeling [BSc].

Lemma 3.4. For each j = 1, · · · , d and q ∈ 2N + 1 we may find k(q) ∈ N and νqj ∈ Bσk(q)(Σ) such that for

l = 1 · · · , q


(i)∣∣h

µqj χ−1j(σj)−

1k(q)

hν

qj χ

−1j(σ

k(q)j )

∣∣ < 1q

,

(ii)∣∣ ∫ Ak(q)(ϕl)dν

qj −

∫ϕldνj

∣∣ < 1q

,

(iii) νqj ([ω1 · · ·ωk(q)]) > 0 for all (ω1, · · · , ωk(q)) ∈ Dk(q).

Proof. Given j ∈ N and k ∈ N we let µkj denote the unique member of Bσk (Σ) which agrees with µj on

cylinders of length k. So for all (τ1, · · · , τk) ∈ ηj(D)k we have

µkj χ−1

j ([τ1 · · · τk]) = µa χ−1j ([τ1 · · · τk]).

Now by the Kolmogorov-Sinai Theorem (2.39) we have

hµjχ−1

j(σj) = − lim

k→∞

1k ∑(τ1,··· ,τk)∈η(D)k

µj χ−1j ([ω1, · · · , ωk]) log µj χ−1

j ([ω1, · · · , ωk]).

Equivalently,

hµjχ−1

j(σj) = lim

k→∞

1k

hµk

j χ−1j(σk).

Since each µj is σ invariant we have∫

Ak(ϕl)dµj =∫

ϕldµj for l = 1, · · · , k and as µj and µkj agree on cylinders

of length k we have ∣∣∣∣ ∫ Ak(ϕl)dµkj −

∫Ak(ϕl)dµj

∣∣∣∣ ≤ 1k

k−1

∑n=0

varn(ϕl).

Moreover, since each ϕl is continuous vark(ϕl)→ 0 (and hence 1k ∑k−1

n=0 varn(ϕl)→ 0) as k→ ∞. Thus, for each

l = 1, · · · , q,

limk→∞

∫Ak(ϕl)dµk

j =∫

ϕldµj.

Thus, taking νqj = µ

k(q)j , for each j, for sufficiently large k(q) ∈ N gives (i) and (ii). By slightly adjusting ν

qj we

may insure (i),(ii) and (iii) hold.

Lemma 3.5. For all q ∈ 2N we may find k(q) ∈N and mq ∈ Bσk(q)(Σ) such that for l = 1 · · · , q

(i)∣∣ ∫ Ak(q)(ϕl)dmq −

∫ϕldmq

∣∣ < 1q

;

(ii) mq([ω1 · · ·ωk(q)]) > 0 for all (ω1, · · · , ωk(q)) ∈ Dk(q).

Proof. Essentially the same as Lemma 3.4.

Choose δq > 0 for each q ∈N so that ∏q∈N(1− δq) > 0.

Lemma 3.6. For each j = 1, · · · , d and q ∈ 2N+ 1 we may find N(q) ∈N and a subset Sqj ⊆ Σ with ν

qj (S

qj ) > 1− δq

and such that for all ω ∈ Sqj and n ≥ N(q) and all l = 1, · · · , q we have


(i)∣∣∣∣ 1nk(q)

nk(q)−1

∑r=0

ϕl(σrω)−

∫ϕldµj

∣∣∣∣ < 1q

,

(ii)∣∣∣∣ 1nk(q)

log νqj χ−1

j ([ηj(ω1) · · · ηj(ωnk(q))]) + hµjχ−1

j(σj)

∣∣∣∣ < 1q

,

(iii) d ∈ D : ωr = d for some r ≤ N(q) = D.

Proof. Given q ∈ N we may apply the Birkhoff ergodic theorem and the the Shannon-Breiman-MacMillan

theorem to νqj χ−1

j ∈ Bσk(q)j

(Σj) ⊆ Eσk(q)j

(Σj) to obtain

limn→∞

1nk(q)

nk(q)−1

∑r=0

ϕl(σrω) = lim

n→∞

1n

n−1

∑r=0

Ak(q)(ϕl)(σrk(q)ω) =

∫ϕldν

qj (3.15)

limn→∞

1n

log µqj χ−1

j ([ηj(ω1) · · · η(ωnk(q))]) = −hµ

qj χ−1j(σ

k(q)j ) (3.16)

for νqj almost every ω ∈ Σj.

By Egorov’s theorem we may choose subsets Sqj ⊆ Σ with ν

qj (S

qj ) > 1− δq so that the convergences in (3.15)

and (3.16) are uniform on Sqj . Thus, by Lemma 3.4 we choose N(q) ∈N so that for all n ≥ N(q) and all ω ∈ Sq

j

we have ∣∣∣∣ 1nk(q)

nk(q)−1

∑r=0

ϕl(σrω)− α

∣∣∣∣ < 1q

(3.17)∣∣∣∣ 1nk(q) log ν

qa χ−1

j ([η(ω1) · · · η(ωnk(q))]) + hµjχ−1

j(σj)

∣∣∣∣ < 1q

. (3.18)

In light of condition (iii), for νqj almost every ω ∈ Σ we have d ∈ D : ωr = d for some r ∈ N = D. Equiva-

lently, for νqj almost every ω ∈ Σ there exists some M(ω) ∈N for which d ∈ D : ωr = d for some r ≤ M(ω) =

D. Thus, by moving to subset of Sqj of large ν

qj measure, and increasing N(q), if necessary, we may assume that

for all ω ∈ Sqj and all n ≥ N(q) we have

d ∈ D : ωl = d for some l ≤ N(q) = D. (3.19)

Lemma 3.7. For each q ∈ 2N we may find N(q) ∈N and a subset Sq ⊆ Σ with mq(Sq) > 1− δq and such that for all

ω ∈ Sq and n ≥ N(q) and all l = 1, · · · , q we have

(i)∣∣∣∣ 1nk(q)

nk(q)−1

∑r=0

ϕl(σrω)−

∫ϕldmq

∣∣∣∣ < 1q

,

(ii) d ∈ D : ωr = d for some r ≤ N(q) = D.

Proof. Essentially the same as Lemma 3.6.


We shall now construct a probability measure W on Σ. To do this we first define a rapidly increasing

sequence of natural numbers (γq)q∈N as follows. Let γ0 := 0 and for each q ≥ 1 taking some γq > (q +

1)∏dj=1(λj − λj−1)

−1(∏q+2r=1 N(r)k(r) + γq−1) so that γq − γq−1 is divisible by k(q). For each k = 1, · · · , d we

sequences of natural numbers (ϑkq)q∈2N+1 by letting ϑk

q denote the greatest integer which is divisible by k(q)

and does not exceed λkγq. For simplicity we also let ϑ0q := γq−1.

We define a measure W on Σ by first defining W on cylinders of length γ2Q for some Q ∈ N and then

extendingW to a Borel probability measure via the Daniell-Kolmogorov consistency theorem (see [W, Section

0.5] ). Given a cylinder [ω1 · · ·ωγ2Q ] of length γ2Q we let

W([ω1 · · ·ωγ2Q ]) := (3.20)

Q

∏q=1

( d

∏j=1

ν2q−1j ([ω

ϑj−12q−1+1

· · ·ωϑ

j2q−1

])× m2q([ωγ2q−1+1 · · · , ωγ2q ])

).

Define S ⊆ Σ by,

S :=∞⋂

q=1

( d⋂j=1

ω ∈ Σ : [ω

ϑj−12q−1+1

· · ·ωϑ

j2q−1

] ∩ S2q−1j 6= ∅

(3.21)

∩

ω ∈ Σ : [ωγ2q−1+1 · · · , ωγ2q ] ∩ S2q 6= ∅)

.

Lemma 3.8. W(S) > 0.

Proof.

W(S) ≥∞

∏q=1

((d

∏j=1

ν2q−1j (S2q−1

j )

)m2q(S2q)

)>

∞

∏q=1

(1− δq)d > 0. (3.22)

Lemma 3.9. For all ω ∈ S, V(ω) ⊆ A.

Proof. Choose ω ∈ S and fix Q ∈ N and ε > 0. For each q ∈ 2N with q ≥ Q take τq ∈ [ωγq−1+1 · · · , ωγq ] ∩ Sq,

which is non-empty since ω ∈ S. By Lemma 3.7, for all n ≥ k(q)N(q),∣∣∣∣ n−1

∑r=0

ϕl(σrτq)− n

∫ϕldmq

∣∣∣∣ < nq+ k(q). (3.23)

Hence, for all N(q)k(q) ≤ n ≤ γq − γq−1,

∣∣∣∣ γq−1+n−1

∑r=γq−1

ϕl(σrω)− n

∫ϕldmq

∣∣∣∣ < n

∑r=0

varr(ϕl) +nq+ k(q). (3.24)

In a similar way we can show that for all q ∈ 2N − 1, with q ≥ Q, j = 1, · · · , d and all N(q)k(q) ≤ n ≤

ϑj−1q − ϑ

jq−1 we have

∣∣∣∣ ϑj−1q +n−1

∑r=ϑ

j−1q

ϕl(σrω)− n

∫ϕldµj

∣∣∣∣ < n

∑r=0

varr(ϕl) +nq+ k(q). (3.25)


Moreover, from the definition of A, for each k = 1, · · · , d we have

λ−1k ·

(k

∑j=1

(λj − λj−1

)µj

)∈ A.

Thus, by the convexity of A, for each q ∈ 2N− 1, with q ≥ Q, k = 1, · · · , d and all N(q)k(q) ≤ n ≤ ϑk−1q − ϑk

q−1,

there exists some ρq,k,n ∈ A such that for each l ≤ Q we have

∣∣∣∣ ϑk−1q +n−1

∑r=ϑk−1

q

ϕl(σrω) +

k−1

∑j=1

ϑjq−1

∑r=ϑ

j−1q

ϕl(σrω)−

(n + ϑk−1

q − γq−1

) ∫ϕldρq,k,n

∣∣∣∣ < n

∑r=0

varr(ϕl) +nq+ k(q). (3.26)

Given any n, k, q ∈N we automatically have∣∣∣∣ k+n

∑r=k

ϕl(σrω)− n

∫ϕldmq

∣∣∣∣ < n, (3.27)

since ||ϕl ||∞ = 1.

Suppose γ2q < N < γ2q+2 where 2q− 2 ≥ Q. Now consider the sum ∑N−1r=0 ϕl(σ

rω), for l ≤ Q. First break

the sum down as follows,N−1

∑r=0

ϕl(σrω) =

γ2q−1

∑r=0

ϕl(σrω) +

N−1

∑r=γ2q

ϕl(σrω). (3.28)

To deal with the first summand, ∑γ2q−1r=0 ϕl(σ

rω), we write,

γ2q−1

∑r=0

ϕl(σrω) =

γ2q−2−1

∑r=0

ϕl(σrω)︸︷︷︸

∗

+d

∑j=1

ϑj2q−1−1

∑r=ϑ

j−12q−1

ϕl(σrω)

︸︷︷︸∗∗

+

γ2q−1

∑r=γ2q−1


∗∗∗

. (3.29)

To part ∗ we apply (3.27) whilst to each of the parts labeled ∗∗ we apply (3.26) and to the part labeled ∗ ∗ ∗ we

apply (3.24).

For the second summand, ∑N−1r=γ2q

ϕl(σrω), there are two cases. Either we have N ≤ γ2q+1 or N > γ2q+1. In

the former case we haveN−1

∑r=γ2q

ϕl(σrω) =

J

∑j=1

ϑj2q−1

∑r=ϑ

j−12q

ϕl(σrω)

︸︷︷︸∗∗

+N−1

∑r=ϑ

J2q

ϕl(σrω)

︸︷︷︸†

, (3.30)

where J is the greatest j ∈ 1, · · · , d such that ϑJ2q < N − 1. If N − ϑJ

2q − 1 < maxj∈2q,2q+1k(j)N(j) we ap-

ply (3.26) to the part labelled ∗∗ and apply (3.27) to the part labelled (†). If N−ϑJ2q− 1 ≥ maxj∈2q,2q+1k(j)N(j)

we may treat (3.30) as a single sum and apply (3.26).

In the latter case we have,

N−1

∑r=γ2q

ϕl(σrω) =

d

∑j=1

ϑj2q−1

∑r=ϑ

j−12q

ϕl(σrω)

︸︷︷︸∗∗

+N−1

∑r=γ2q+1


††

. (3.31)

Again, to the parts labeled ∗∗ we apply (3.26), and to the part labeled (††) we either apply (3.27) or (3.24), de-

pending on whether N − γ2q+1 − 1 < maxj∈2q,2q+1k(j)N(j) or N − γ2q+1 − 1 ≥ maxj∈2q,2q+1k(j)N(j).


Thus, by combining (3.29), (3.30), (3.31), in each case we see that there exists βN2q, βN

2q+2, λN1 , · · · , λN

d ∈ [0, 1]

which sum to one, depending solely on N and not on l = 1, · · · , Q, for which we have∣∣∣∣ N−1

∑r=0

ϕl(σrω)− ∑

j∈2q,2q+2NβN

j

∫ϕldmj −

d

∑j=1

NλNj

∫ϕldµj

∣∣∣∣< (2d + 1)

(N

∑r=0

varr(ϕl) +N

q− 1+ max

j∈0,1,2,3k(2q + j)

)

+

(γ2q−2 + max

j∈2q,2q+1k(j)N(j)

)< o(N) + o(γ2q) ≤ o(N),

where we use the continuity of each ϕl together with the definition of (γq)q∈N to obtain the last line. Moreover,

since A is convex, for each such N the measure ρN := ∑j∈2q,2q+2 NβNj mj + ∑d

j=1 λNj µj is a member of A.

Hence, for all sufficiently large N, we have∣∣∣∣ ∫ ϕldAN(δω)−∫

ϕldρN

∣∣∣∣ < ε, (3.32)

for l = 1, · · · , Q. Since A is also closed it follows that every weak ∗ accumulation point of the sequence

(AN(δω))N∈N is a member of A.

Lemma 3.10. For all ω ∈ S, A ⊆ V(ω).

Proof. Take ω ∈ S, ρ ∈ A. Since the set of accumulation points of (mq)q∈2N is equal to A we may extract a

subsequence (mqj)j∈N converging to ρ. Now choose ε > 0 and choose Q so large that for j with qj ≥ Q∣∣∣∣ ∫ ϕldmqj −∫

ϕldρ

∣∣∣∣ < ε. (3.33)

As in 3.24 we see that for all j with qj ≥ Q, l = 1, · · · , Q,

∣∣∣∣ γqj−1

∑r=γqj−1

ϕl(σrω)− (γqj − γqj−1)

∫ϕldmqj

∣∣∣∣ (3.34)

<

γqj−γqj−1

∑r=0

varr(ϕl) +γqj − γqj−1

qj+ k(qj). (3.35)

Hence, for all qj ≥ Q, l = 1, · · · , Q,

∣∣∣∣ γqj−1

∑r=0

ϕl(σrω)− γqj

∫ϕldmqj

∣∣∣∣ (3.36)

< γqj−1 +

γqj−γqj−1

∑r=0

varr(ϕl) +γqj − γqj−1

qj+ k(qj). (3.37)

Thus, by the definition of (γq)q∈N and the fact that each ϕl is continuous, we have∣∣∣∣ ∫ ϕldAγqj(δω)−

∫ϕldρ

∣∣∣∣ < 2ε. (3.38)

It follows that Aγqj(δω)→ ρ as j→ ∞.

Lemma 3.11. S ⊆ Γ(A).


Proof. Combine Lemmas 3.9 and 3.10.

For each q ∈ 2N + 1 we define the qth approximate square BLq (ω) to be the set

BLq (ω) :=

ω′ ∈ Σ : ηj(ω

′ν) = ηj(ων) for ν = 1, · · · , ϑ

jq, j = 1, · · · , d

.

Lemma 3.12. For all ω ∈ S

lim supq→∞

1γq

logW(BLq (ω)) ≤ −λ1hµ1(σ)−

d

∑j=1

(λj − λj−1)hµjχ−1j(σj).

Proof. Take ω ∈ S and q ∈ 2N + 1 and j ∈ 1, · · · , d. Since ϑjq − ϑ

j−1q ≥ N(q) and [ωγq−1+1 · · ·ωϑq ] ∩ Sq

j 6= ∅

it follows from the definition ofW and (γq)q∈N together with Lemma 3.6 that

logW(

σ−ϑj−1q [ηj(ωϑ

j−1q +1

) · · · ηj(ωϑjq)]

)(3.39)

= log νqj ([ηj(ωϑ

j−1q +1

) · · · ηj(ωϑjq)])

≤ −(

ϑjq − ϑ

j−1q

)h

µqj χ−1j(σj) +

ϑjq − ϑ

j−1q

q

≤ −((λj − λj−1)γq − k(q)− γq−1

)h

µqj χ−1j(σj) +

γq

q

≤ −(λj − λj−1)γqhµ

qj χ−1j(σj) + o(γq).

By the definition ofW , each of the cylinders σ−ϑj−1q [ηj(ωϑ

j−1q +1

) · · · ηj(ωϑjq)], with j = 1, · · · , d, are independent

with respect toW . Thus, letting q→ ∞ we have

lim supq→∞

1γq

logW

d⋂j=1

σ−ϑj−1q [ηj(ωϑ

j−1q +1

) · · · ηj(ωϑjq)]

(3.40)

≤ −λ1hµ1(σ)−d

∑j=1


Since BLq (ω) ⊆ ⋂d

j=1 σ−ϑj−1q [ηj(ωϑ

j−1q +1

) · · · ηj(ωϑjq)] the lemma follows.

The following lemma allows us to deal with the fact that our approximate squares may meet at their bound-

aries.

Lemma 3.13. For all ω ∈ S and q ∈N we have

B(

Π(ω), minj∈1,··· ,d

a−ϑjq−maxN(q),N(q+1)

j )∩Λ ⊆ Π(BL

q (ω)).

Proof. Fix ω = ((ijν)

dj=1)ν∈N ∈ S and let M(q) := maxN(q), N(q + 1). Clearly it suffices to show that for

each j = 1, · · · , d,

B(

Π[j (ω), min

j∈1,··· ,da−ϑ

jq−M(q)

j )∩Π[

j (Σ) ⊆ Π[j (BL

q (ω)). (3.41)

Take j ∈ 1, · · · , d and let x = Π[j (ω). We may divide the convex hull of Π[

j (BLq (ω)) into #η[

j (D)M(q) intervals

of width a−ϑ

jq−M(q)

j with disjoint ineriors, each corresponding to a possible string of digits i′ϑ

jq+1· · · i′

ϑjq+M(q)

for


τ = ((ijν)

dj=1)ν∈N ∈ BL

q (ω). Since ω ∈ S we have [ωϑ

jq+1· · ·ω

ϑjq+N(q)

] ∩ Sqj+1 6= ∅ for j = 1, · · · , d − 1 and

[ωϑdq+1 · · ·ωϑd

q+N(q+1)] ∩ Sq+1d 6= ∅. Thus, by Lemma 3.6 in the first case and Lemma 3.7

D =

d ∈ D : ωl = d for some ϑjq < l ≤ ϑ

jq + M(q)

.

It follows from η[j (D) =

d ∈ η[

j (D) : ijν = d for some ϑ

jq < l ≤ ϑ

jq + N(q)

that x is in neither the far left nor

the far right interval of the convex hull of Π[j (BL

q (ω)), for in either case (ijν)

ϑjq+N(q)

ν=ϑjq+1

would be a constant se-

quence. Since #η[j (D) > 1 it follows that x is a distance at least a

−ϑjq−M(q)

j from any point y such that y as an

aj-ary digit expansion (ijν)ν∈N with ij

ν 6= ijν for some ν ≤ ϑ

jq. Thus, (3.41) holds.

LetM :=W χ−1 denote the pushdown ofW onto Λ.

Lemma 3.14. For all ω ∈ S

lim supr→0

logM(B(χ(ω), r))log r

≥ −hµ1(σ)

log a1+

d

∑j=1

(1

log aj− 1

log aj−1

)h

µjχ−1j(σj).

Proof. Recall that for each j = 1, · · · , d we defined λj := log ad/ log aj, and for each q ∈ 2N + 1 we have

ϑjq < λjγq and so

minj∈1,··· ,d

a−ϑjq−maxN(q),N(q+1)

j ≥ a−γq−maxN(q),N(q+1)/λ1d . (3.42)

Choose ω ∈ S. By Lemma 3.13,

B(

Π(ω), a−γq−maxN(q),N(q+1)/λ1d

)∩Λ ⊆ Π(BL

q (ω)). (3.43)

Hence,

M(

B(


))≤ W

(BL

q (ω))

. (3.44)

It follows from Lemma 3.12 that

lim supq→∞

1γq

logM(

B(


))(3.45)

≤ −λ1hµ1(σ)−d

∑j=2


Thus, noting that maxN(q), N(q + 1) = o(q), by the definition of (γq)q∈N we have

lim supq→∞

logM(

B(

χ(ω), a−γq−maxN(q),N(q+1)/λ1d

))log a

−γq−maxN(q),N(q+1)/λ1d

(3.46)

≥ −hµ1(σ)

log a1+

d

∑j=2

(1

log aj− 1

log aj−1

)h

µjχ−1j(σj).

SinceM(Π(S)) ≥ W(S) > 0 we may combine Proposition 2.4 with Lemma 3.14 to see that

dimP(Π(S)) ≥ −hµ1(σ)

log a1+

d

∑j=1

(1

log aj− 1

log aj−1

)h

µjχ−1j(σj). (3.47)

This completes the proof of the lower bound.

3.5. Proof of the upper bound 73

3.5 Proof of the upper bound

Take A ⊆Mσ(Σ). Recall that for each q ∈N we defined

U(A, q) :=

µ ∈ M(Σ) : ∃ν ∈ A ∀l ≤ q∣∣∣∣ ∫ ϕldµ−

∫ϕldν

∣∣∣∣ < 1q

. (3.48)

Given N ∈N we let

Ω(A, N, q) := ω ∈ Σ : ∀n ≥ N An(δω) ∈ U(A, q) . (3.49)

Note that for each q ∈ N Π(Ω(A, N, q))N∈N is a countable cover of Π(Ω(A)). As such we shall give an

estimate for the upper box dimension of the sets Π(Ω(A, N, q)) before applying the following reformulation

of the notion of packing dimension.

Proposition 3.15. Given E ⊆ Rn we have

dimPE = inf

supn∈N

dimBEn : E ⊆⋃

n∈N

En

,

where the infimum is taken over all countable covers Enn∈N of E.

The above formula is equivalent to the usual definition of packing dimension in terms of s-dimensional

packing measures (see [Mat, Section 5.9 and Theorem 5.11]).

Recall that for each j = 1, · · · , d, λj := log ad/ log aj. We also let λ0 := 0. Given q ∈ N we define

Vq := ∏dj=1 ∏

ql=1[−1, 1] and let

Vq(A) :=

v = (vjl) ∈ Vq : ∃(νj)

dj=1 ⊂ A ∀k ≤ d ∀l ≤ q

∣∣∣∣λ−1k

k

∑j=1

(λj − λj−1)vjl −∫

ϕldνk

∣∣∣∣ < 4dλ1q

.

Given n ∈N and v ∈ Vq we define Aj,vn to be the set of all (τdλj−1ne+1, · · · , τdλjne) ∈ ηj(D)dλjne−dλj−1ne such

that for some ω ∈ χ−1j

([τdλj−1ne+1, · · · , τdλjne]

)we have

(dλjne − dλj−1ne)−1Sdλjne−dλj−1ne(ϕl)(ω) ∈(

vjl − q−1, vjl + q−1)

for each l = 1, · · · , q.

Lemma 3.16.

dimBΠ(Ω(A, N, q)) ≤ sup

d

∑j=1

lim supn→∞

log #Aj,vn

n log ad: v ∈ Vq(A)

.

Proof. Given r > 0 we let N(r) denote the minimal number of balls of radius r required to cover Π(Ω(A, N, q))

so that

dimBΠ(Ω(A, N, q)) = lim supr→0

log N(r)log(1/r)

.

For each r > 0 we take nr ∈ N so that√

2 · a−nrd < r ≤

√2 · a−nr+1

d . Given κ := (κj)dj=1 where κj =

(τjdλj−1ne+1, · · · , τ

jdλjne

) ∈ Aj,vnr for some v ∈ ∏d

j=1 ∏ql=1[−1, 1] we let B(κ) denote the approximate square

B(κ) :=d⋂

j=1

σ−dλj−1neχ−1j

([τdλj−1ne+1, · · · , τdλjne]

). (3.50)


For all sufficiently large n we have

Ω(A, N, q) ⊆⋃

v∈Vq(A)∩Dq

⋃κ∈∏d

j=1Aj,vn

B(κ). (3.51)

Indeed if we take ω ∈ Ω(A, N, q) then for all n ≥ λ−11 N there exists

(µn

j

)d

j=1⊂ A so that for k ≤ d and l ≤ q

we have ∣∣∣∣Sdλkne(ϕl)(ω)− dλkne ·∫

ϕldµnk

∣∣∣∣ < dλkneq

.

Since Dq ⊂ V is (2q)−1-dense we may choose vn ∈ Dq so that for each 1 ≤ j ≤ d and l ≤ q we have∣∣∣∣Sdλjne−dλj−1ne(ϕl)(σdλj−1neω)− (dλjne − dλj−1ne)vn

jl

∣∣∣∣ < nq

.

In particular ω ∈ B(κn) for an appropriately chosen κn ∈ ∏dj=1A

j,vn .

Also, for each 1 ≤ k ≤ d and l ≤ q we have∣∣∣∣Sdλkne(ϕl)(ω)−k

∑j=1

(dλjne − dλj−1ne)vnjl

∣∣∣∣ < dnq

.

Hence, for 1 ≤ k ≤ d and l ≤ q we have∣∣∣∣dλkne−1 ·k

∑j=1

(dλjne − dλj−1ne)vnjl −

∫ϕldµn

k

∣∣∣∣ < 2dnqdλ1ne .

Thus, choosing N0 > λ−11 N so that for n ≥ N0 and for each j ≤ k ≤ d we have

∣∣dλkne−1 · (dλjne − dλj−1ne)− λ−1k (λj − λj−1)

∣∣ < q−1

and (2dn)(qdλ1ne)−1 < (3d)(qλ1)−1. It follows that for each n ≥ N0 and for any ω ∈ Ω(A, N, q) there exists

vn ∈ Dq with κn ∈ ∏dj=1A

j,vn so that ω ∈ B(κn) and for each l ≤ q and k ≤ d we have∣∣∣∣λ−1

k ·k

∑j=1

(λj − λj−1)vnjl −

∫ϕldµn

k

∣∣∣∣ < 4dλ1q

.

That is, vn ∈ Vq(A) ∩ Dq.

Hence, for all n ≥ N0 we have

Π (Ω(A, N, q)) ⊆⋃

v∈Vq(A)∩Dq

⋃κ∈∏d

j=1Aj,vn

Π (B(κ)) . (3.52)

It follows from the definition of Π that each Π (B(κ)) has diameter no greater than

√2 · a−dλjnre

j ≤√

2 · a−nrd < r.

Thus,

N(r) ≤ ∑v∈Vq(A)∩Dq

(d

∏j=1

#Aj,vnr

)

≤ #Dq · sup

d

∏j=1

#Aj,vnr : v ∈ Vq(A)

.


Hence, since r ≤√

2 · a−nr+1d ,

lim supr→0

log N(r)log(1/r)

≤ supv∈Vq(A)

lim supr→0

log #Dq + ∑dj=1 log #Aj

nr

nr log ad + log√

2nr

nr − 1

(3.53)

≤ supv∈Vq(A)

lim supn→∞

∑dj=1 log #Aj,v

n

n log ad

.

Recall that given ν ∈ Mσk (Σ) for each k ∈N we let Ak(ν) := 1/k ∑k−1l=0 ν σ−l .

Lemma 3.17. If µ = Ak(ν) for some ν ∈ Eσk (Σ) then for each j = 1, · · · , d and all ϕ ∈ C(Σ)

(i) µ χ−1j ∈ Eσj(Σj)

(ii) hµχ−1

j(σj) = 1/kh

νχ−1j(σk

j )

(iii)∫

ϕdµ =∫

Ak(ϕ)dν.

Proof. By noting that Ak(ν) χ−1j = ∑k−1

r=1 ν χ−1j σ−r

j we see that Lemma 3.17 follows lemma in [JJOP, Lemma

2].

Lemma 3.18. Take v ∈ V, q ∈N and j ∈ 1, · · · , d. There exists a measure µ ∈ Mσ(Σ) with

lim supn→∞

1n

log #Aj,vn ≤ (λj − λj−1)h

(µ χ−1

j , σj

)+ q−1,

and for l ≤ q we have∫

ϕldµ ∈(

vjl − 2q−1, vjl + 2q−1)

.

Proof. Take v ∈ V and j ∈ 1, · · · , d. For each τ = (τdλj−1ne+1, · · · , τdλjne) ∈ Aj,vnr we choose

κτ = (ωτdλj−1ne+1, · · · , ωτ

dλjne) ∈ Ddλjne−dλj−1ne

so that (ηj(ω

τdλj−1ne+1), · · · , ηj(ω

τdλjne)

)= τ,

and for l ≤ q we have Adλjne−dλj−1ne(ϕl)(ω) ∈(

vjl − q−1, vjl + q−1)

.

We now let νn ∈ Bσdλjne−dλj−1ne(Σ) be the unique dλjne − dλj−1ne-th level Bernoulli measure satisfying

νn([ωdλj−1ne+1 · · ·ωdλjne]) :=

1

#Ajn

if (ωdλj−1ne+1, · · · , ωdλjne) = κτ for some τ ∈ Aj,vn

0 otherwise.

It follows that

νn([τdλj−1ne+1 · · · τdλjne]) :=

1

#Aj,vn

for (τdλj−1ne+1, · · · , τdλjne) = τ ∈ Ajn

0 otherwise.


Thus, h(

νn χ−1j , σ

dλjne−dλj−1nej

)= log #Aj

n (see [W, Lemma 4.26]). Also, for l ≤ q we have

∣∣∣∣ ∫ Adλjne−dλj−1ne(ϕl)dνn − vjl

∣∣∣∣ < q−1 +1

dλjne − dλj−1ne

dλjne−dλj−1ne

∑i=0

vari(ϕl).

Let µn := Adλjne−dλj−1ne(νn). By Lemma 3.17 (i) each µn is an ergodic invariant measure. By Lemma 3.17

(ii) we have

1n

log #Aj,vn =

dλjne − dλj−1nen

hµ

jnχ−1

j(σj). (3.54)

Moreover, by 3.17 (ii) for each l ≤ q we have

∣∣∣∣ ∫ ϕldµn − vjl

∣∣∣∣ < q−1 +1

dλjne − dλj−1ne

dλjne−dλj−1ne

∑i=0

vari(ϕl).

Since each ϕl is continuous we may choose n0 ∈ N so that 1dλjn0e−dλj−1n0e ∑

dλjn0e−dλj−1n0ei=0 vari(ϕl) < q−1 for

l ≤ q and also1n0

log #Aj,vn0 > lim sup

n→∞

1n

log #Aj,vn − q−1.

Thus the proof of the lemma is completed by taking µ = µn0 .

Lemma 3.19. For each q ∈N, there exists a sequence of invariant measures (µqj )

dj=1 ⊂Mσ(Σ) satisfying

dimP (Π(Ω(A))) ≤hµ

q1(σ)

log a1+

d

∑j=2

(1

log aj− 1

log aj−1

)h

µqj χ−1j(σj) +

3dq log ad

,

along with some vq ∈ Vq such that for each j ≤ d and l ≤ q we have∣∣∣∣ ∫ ϕldµqj − vq

jl

∣∣∣∣ < 2q−1,

and there exists measures (νqj )

dj=1 ⊂ A such that for each k ≤ d and l ≤ q we have

∣∣∣∣λ−1k

k

∑j=1

(λj − λj−1)vqjl −

∫ϕldν

qk

∣∣∣∣ < 4dλ1q

.

Proof. Combining Lemma 3.16 with Lemma 3.18 and the definition of Vq(A) we see that for any given N we

have a sequence of invariant measures (µq,Nj )d

j=1 ⊂Mσ(Σ) satisfying

dimB (Π(Ω(A, N, q))) ≤h

µq,N1

(σ)

log a1+

d

∑j=2

(1

log aj− 1

log aj−1

)h

µq,Nj χ

−1j(σj) +

2dq log ad

,

along with some vq,N ∈ Vq such that for each j ≤ d and l ≤ q we have∣∣∣∣ ∫ ϕldµq,Nj − vq,N

jl

∣∣∣∣ < 2q−1,

and there exists measures (νq,Nj )d

j=1 ⊂ A such that for each k ≤ d and l ≤ q we have

∣∣∣∣λ−1k

k

∑j=1

(λj − λj−1)vq,Njl −

∫ϕldν

q,Nj

∣∣∣∣ < 4dλ1q

.


Moreover since

Π(Ω(A)) ⊆⋃

N∈N

Π(Ω(A, N, q))

we may apply Proposition 3.15 to obtain

dimP (Π(Ω(A))) ≤ supN∈N

dimB (Π(Ω(A, N, q))) ,

so we may take N0 so that

dimB (Π(Ω(A, N0, q))) > dimP (Π(Ω(A)))− d(q log ad)−1.

Thus, taking each µqj := µ

q,N0j , vq := vq,N0 and ν

qj := ν

q,N0j completes the proof of the lemma.

We now complete the proof of Theorem 3.3 and hence Theorems 3.1 and 3.2.

Proposition 3.20. There exists a sequence of measures (µj)dj=1 ∈ A with

dimP (Π(Ω(A))) ≤=hµ1(σ)

log a1+

d

∑j=2

(1

log aj− 1

log aj−1

)h

µjχ−1j(σj).

Proof. This follows straightforwardly from Lemma 3.19.

We begin by taking V := ∏dj=1 ∏l∈N[−1, 1]. Note that V is compact and each Vq embeds naturally into V

by mapping the point v = (vjl)1≤l≤q1≤j≤d ∈ Vq to the point v = (vjl)

l∈N1≤j≤d ∈ V with vjl = vjl for l ≤ q and vjl = 0

for l > q. Now consider the sequences of invariant measures (µqj )q∈N ⊂Mσ(Σ) for j = 1, · · · , d, (νq

k )q∈N ⊂ A

for k = 1, · · · , d and the sequence (vq)q∈N ∈ V obtained from Lemma 3.19. That is, for each q ∈N we have

dimP (Π(Ω(A))) ≤hµ

q1(σ)

log a1+

d

∑j=2

(1

log aj− 1

log aj−1

)h

µqj χ−1j(σj) +

3dq log ad

,

for each j ≤ d and l ≤ q we have ∣∣∣∣ ∫ ϕldµqj − vq

jl

∣∣∣∣ < 2q−1,

and for each k ≤ d and l ≤ q we have∣∣∣∣λ−1k

k

∑j=1

(λj − λj−1)vqjl −

∫ϕldν

qk

∣∣∣∣ < 4dλ1q

.

Since the sets Mσ(Σ), A and V are compact we may take a subsequence such that each of these sequences

converge to limit points µj, for j = 1, · · · , d, νk for k = 1, · · · , d and v ∈ V.

Firstly, it follows from the fact that entropy is upper semicontinuous (see Theorem 2.42) that

dimP (Π(Ω(A))) ≤=hµ1(σ)

log a1+

d

∑j=2

(1

log aj− 1

log aj−1

)h

µjχ−1j(σj).

Secondly, for each j ≤ d and l ∈N we have∫

ϕldµj = vjl and for each k ≤ d and l ∈N we have λ−1k ∑k

j=1(λj−

λj−1)vjl =∫

ϕldνk. Putting these two equalities together and noting that the sequence (ϕl)l∈N is dense in

ϕ ∈ C(Σ) : ||ϕ||∞ ≤ 1 we have

λ−1k

k

∑j=1

(λj − λj−1)µj = νk,

for all k ≤ d. Since each νk ∈ A this implies that (µj)dj=1 ∈ A, which completes the proof of the upper

bound.


3.6 The shape of the spectrum

We now deduce several features of the shape of the packing spectrum.

Corollary 3.21. Let ϕ : Σ → R be a continuous real valued potential which is not cohomologous to a constant. Then,

the packing spectrum α 7→ dimPΠ(Jϕ(α)) is concave and continuous on the interval A(ϕ) = [αmin, αmax].

Proof. By Theorem 3.3 it suffices to show that for each j = 1, · · · , d, H j(σ, ϕ, α) is concave and continu-

ous. So fix j ∈ 1, · · · , d. It follows from the upper semicontinuity of entropy that H j(σ, ϕ, α) is upper

semi-continuous. Moreover, H j(σ, ϕ, α) is concave. Indeed given α−, α+ ∈ A(ϕ) and δ > 0 we may choose

µ−, µ+ ∈ Mσ(Σ) such that∫

ϕdµ− = α− and∫

ϕdµ+ = α+ hµ−χ−1

j(σj) > H j(σ, ϕ, α−) − δ, h

µ+χ−1j(σj) >

H j(σ, ϕ, α+)− δ. For each t ∈ (0, 1) we let µt := (1− t)µ+tµ+ so that∫

ϕdµt = (1− t)α− + tα+. Moreover,

since the entropy map is affine (see Theorem 2.40)

H j(σ, ϕ, (1− t)α− + tα+) ≥ hµtχ−1

j(σj)

= (1− t)hµ−χ−1

j(σj) + th

µ+χ−1j(σj)

≥ (1− t)H j(σj, ϕ, α−) + tH j(σj, ϕ, α+)− δ.

Letting δ→ 0 we see that α 7→ H j(σ, ϕ, α) is concave and hence lower semi-continuous.

The following corollary gives a sufficient condition on ϕ for the packing spectrum to be analytic.

Corollary 3.22. Suppose there is some Holder continuous potential ϕ : Σd → R such that ϕ = ϕ χd. Then α 7→

dimP Jϕ(α) is strictly concave and real analytic on the interval A(ϕ).

Proof. Note that for each j = 1, · · · , d, the projection χd χ−1j : Σj → Σd is a well defined Lipschitz function.

Hence the real valued potential ϕj : Σj → Σd, given by ϕj := ϕ χd χ−1j is Holder continuous. It follows

straightforwardly from ϕ = ϕ χd that for each j = 1, · · · , d,

H j(σ, ϕ, α) = sup

hµ(σj) : µ ∈ Mσj(Σj),∫

ϕjdµ = α

. (3.55)

One can deduce from standard results that the right hand side of (3.55) is strictly concave and analytic. Since

ϕ is not cohomologous to a constant and χj σ = σj χj it is clear that no ϕj is cohomologous to a constant.

Now fix j ∈ 1, · · · , d. By [BoEq, Theorem 1.28] it follows that, for each j, the Gibbs measure corresponding

to ϕj is not the measure of maximal entropy on Σj. Now for each α ∈ A(ϕ) consider the set

Jj(α) :=

ω ∈ Σj : lim

n→∞

1n

n

∑r=0

ϕj(σrω) = α

, (3.56)

where Σj is given the usual symbolic metric (see Section 2.4.1). By [BS, Theorem 6] dimH(

Jj(α))

is equal to a

constant multiple of the quantity on the right hand side of (3.55). Since the Gibbs measure corresponding to ϕj

is not the measure of maximal entropy it follows from [PW1, Theorem 1] that α 7→ dimH Jj(α) is strictly concave

and real analytic on (αmin, αmax). Thus the spectrum is strictly convex and real analytic on (αmin, αmax). By

Lemma 3.21 the spectrum is continuous on [αmin, αmax] and hence these properties extend to the full interval

[αmin, αmax].

3.7. Examples 79

For each j = 1, · · · , d we let bj denote the measure of maximal entropy on Σj. We conclude this section

with a necessary and sufficient condition for the packing spectrum to attain the full packing dimension of the

repeller. The proof is immediate from Theorem 3.3.

Corollary 3.23. There exists some α ∈ A(ϕ) satisfying dimP Jϕ(α) = Λ if and only if∫

ϕdµ1 =∫

ϕdµ2 = · · · =∫ϕdµd for some µ1, · · · , µd ∈ Mσ(Σ) such that µj χ−1

j = bj for each j = 1, · · · , d.

3.7 Examples

In this section we consider two simple examples exhibiting interesting features of the packing spectrum.

As noted in the introduction the packing and Hausdorff spectra need not coincide. This raises the question

of whether there are any real-valued potentials ϕ : Σ→ R supported on Bedford-McMullen repellers for which

dimH(Λ) < dimP(Λ) and yet the Hausdorff and packing spectra for ϕ coincide. Our first example shows that

this can indeed be the case. One consequence of this is that dimP Jϕ(α) = dimH Jϕ(α) ≤ dimHΛ < dimPΛ for

all α ∈ [αmin, αmax]. So the packing spectrum need not attain the full packing dimension of the repeller at any

point. This is in contrast to the situation for Hausdorff dimension where there is always some α ∈ [αmin, αmax]

for which dimH(Jϕ(α)) = dimH(Λ), namely α =∫

ϕdµ∗ where µ∗ is an invariant measure of full dimension

(see [BM1] [Be], [Mc]).

Example 3.24. Take a1 = 3, a2 = 2 and D = (0, 0), (1, 1), (2, 0) and ϕ : Λ→ R defined by

ϕ(ω) =

1 if ω1 = (1, 1)

0 if ω1 6= (1, 1).

Then, dimH(Λ) < dimP(Λ). However, for all α ∈ [0, 1],

dimPEϕΠ(α) = dimHEϕΠ(α) =−α log α− (1− α) log(1− α)

log 2+ (1− α)

log 2log 3

.

Proof. The fact that dimH(Λ) < dimP(Λ) follows from Theorem 2.86. By considering the ((1− α)/2, α, (1−

α)/2)-Bernoulli measure, it follows from Proposition 2.89 that

dimH Jϕ(α) ≥−α log α− (1− α) log(1− α)

log 2+ (1− α)

log 2log 3

.

It is easy to see that

sup

3

∑i=1−pi log pi : pi ∈ [0, 1],

3

∑i=1

pi = 1, p1 = α

= −α log α− (1− α) log

(1− α

2

)

sup

2

∑i=1−pi log pi : pi ∈ [0, 1],

2

∑i=1

pi = 1, p1 = α

= −α log α− (1− α) log(1− α).

Moreover, it follows from the fact that ϕ is locally constant (ie. ϕ(ω′) = ϕ(ω) for all ω, ω′ ∈ Σ with ω′1 = ω1)

together with the Kolmogorov-Sinai Theorem that the suprema

sup

hµ(σ) : µ ∈ Bσ(Σ),∫

ϕdµ = α

sup

h

µχ−12(σv) : µ ∈ Bσ(Σ),

∫ϕdµ = α


are both attained by Bernoulli measures. Thus, applying Theorem 3.3 we have

dimPEϕΠ(α) ≤ −α log α− (1− α) log(1− α)

log 2+ (1− α)

log 2log 3

.

For our next example we have identical a1, a2 andD, along with a potential ϕ which is prima facie very close

to our previous one. Indeed for α ≥ 12 the spectra for the two examples coincide (see Figure 2). However our

next example has a point of non-analyticity at α = 12 and for α < 1

2 the two packing spectra are very different.

In particular, the packing spectrum attains the packing dimension of Λ and so rises above the Hausdorff

spectrum.

Example 3.25. Take a = 3, b = 2 and D = (0, 0), (1, 1), (2, 0) and ϕ : Λ→ R defined by

ϕ(ω) =

1 if ω1 = (2, 0)

0 if ω1 6= (2, 0)

dimPEϕΠ(α) =

−α log α−(1−α) log(1−α)−α log 2

log 3 + 1 for α ≤ 12

−α log α−(1−α) log(1−α)log 2 + (1− α)

log 2log 3 for α > 1

2 .

Moreover, α 7→ dimP Jϕ(α) is non-analytic and attains the full packing dimension dimP(Λ) at its maximum.

Proof. Note that

sup

3

∑i=1−pi log pi : pi ∈ [0, 1],

3

∑i=1

pi = 1, p1 = α

= −α log α− (1− α) log

(1− α

2

)

sup

2

∑i=1−pi log pi : pi ∈ [0, 1],

2

∑i=1

pi = 1, p1 ≥ α

=

log 2 for α ≤ 1

2

−α log α− (1− α) log(1− α) for α > 12

.

Moreover, since ϕ is locally constant the following suprema are both attained by Bernoulli measures,

sup

hµ(σ) : µ ∈ Bσ(Σ),∫

ϕdµ = α

sup

h

µχ−12(σv) : µ ∈ Bσ(Σ),

∫ϕdµ = α

.

Thus, applying Theorem 3.3 we have

dimPEϕΠ(α) =

−α log α−(1−α) log(1−α)−α log 2

log 3 + 1 for α ≤ 12

−α log α−(1−α) log(1−α)log 2 + (1− α)

log 2log 3 for α > 1

2

.

Consequently, α 7→ dimP Jϕ(α) is non-analytic. This follows from the fact that the functions

α 7→ −α log α− (1− α) log(1− α)− α log 2log 3

+ 1

α 7→ −α log α− (1− α) log(1− α)

log 2+ (1− α)

log 2log 3

have distinct second derivatives at α = 12 . It also follows from our expression for dimP Jϕ(α), together with

2.86, that the full packing dimension is attained at α = 13 .

3.8. Generalisations and open questions 81

3.8 Generalisations and open questions

In this section we note some Corollaries to Theorem 3.2 and 3.3. The first concerns sets of divergent points.

Usually one considers sets of points for which the Birkhoff average converges to a given value. However, given

any non-empty closed convex subset A ⊆ A(ϕ) one may consider the set Jϕ(A) of points x ∈ Λ for which

the set of accumulation points for the sequence (An(ϕ)(x))n∈N is equal to A. In the conformal setting both

dimH Jϕ(A) and dimP Jϕ(A) have been well studied in a series of papers due to Olsen and Winter [Ol2, Ol3, Ol4].

This follows work by Barreira and Schmeling [BSc] showing that, given finitely many continuous potentials

ϕ1, · · · , ϕN on a conformal repeller Λ for which each A(ϕi) consists of at least two points, the Hausdorff

dimension (and hence packing dimension) of the set of all points for which none of the Birkhoff averages for

ϕ1, · · · , ϕN converge is of full Hausdorff dimension. We note that [Ol4, Theorem 4.3] implies that the set of

points x ∈ Λ for which the Birkhoff average (An(ϕ)(x))n∈N does not converge for any continuous potential

ϕ for which A(ϕ) consists of at least two points again, has full Hausdorff dimension. By a similar argument,

along with some ideas from [KP], one can extend this result to self-affine Sierpinski sponges.

One application of Theorem 3.3 is to determine the packing dimension of the sets Jϕ(A) for self-affine

Sierpinski sponges.

Theorem 3.26. Let Λ be a self-affine Sierpinski sponge. Let ϕ : Λ → RN be some continuous potential. Then given

any non-empty closed convex subset A ⊆ A(ϕ) we have

dimP Jϕ(A) = sup

hµ1( f )log a1

+d

∑j=2

(1

log aj− 1

log aj−1

)h

µjπ−1j( f j)

,

where the suprememum is taken over all sequences (µj)dj=1 ∈ M f (Λ) such that for each k = 1, · · · , d,

log ak

(1

log a1µ1 +

k

∑j=2

(1

log aj− 1

log aj−1

) ∫ϕdµj

)∈ A.

In contrast very little is known concerning the Hausdorff dimension of Jϕ(A) for self-affine Λ, aside from

the special case where A is a singleton, and it would be very interesting to see if one could obtain a formula

for dimH Jϕ(A) for arbitrary non-empty closed convex subsets ofM f (Λ).

Theorem 3.3 also implies the some results concerning the packing spectrum for the local dimension of

a Bernoulli measure on a self-affine Sierpinski sponge. To each Bernoulli measure µ on Σ we associate the

corresponding probability vector (pi1···id)(i1,··· ,id)∈D in the usual way. Given t ∈ 1 · · · , d we let pi1···it denote

the sum of all pj1···jd for which (j1, · · · , jt) = (i1, · · · , it).

For j = 1, · · · , d we define a potential Pj : Σ→ R by

Pj(ω) :=

log pχj(ω1)

/pχj+1(ω1)

log ajif j 6= d

log pχd(ω1)log ad

if j = d.(3.57)

Clearly var1(Pj) = 0 for each j and as such Pj is continuous. Let P : Σ → Rd denote the potential ω 7→

(Pj(ω))dj=1. We shall assume the Very Strong Separation Condition (see [Ol5, Condition (II)]).


Lemma 3.27. Suppose that µ is a Bernoulli measure on a self-affine Sierpinski sponge which satisfies the Very Strong

Separation Condition. Then for all x = Π(ω) ∈ Λ we have

limr→∞


=d

∑j=1

1dλjne

dλjne−1

∑l=0

Pj(σlω). (3.58)

Proof. See [Ol5, Theorem 6.2.2].

Olsen [Ol5, Conjecture 4.1.7] conjectured that the packing spectrum of a Bernoulli measure on a self-affine

Sierpinski sponge is given by the Legendre transform of an certain auxiliary function (see [Ol5, Section 3.1] for

details). In particular, this conjecture would imply that the packing spectrum for local dimension always peaks

at the full packing dimension of the attractor Λ (see [Ol5, Theorem 3.3.2 (ix)] and note that γ(0) = dimPΛ by

Theorem 2.86). Theorem 3.3 provides us with the following counterexample.

Example 3.28. Take a1 = 4, a2 = 3 and D = (0, 0), (2, 2), (3, 0) and let ν be the Bernoulli measure obtained by

taking p00 = p30 = 1/4 and p22 = 1/2. Let αmin := log 2/ log 3 and αmax := log 2/ log 3 + 1/2 and for all α ∈

[αmin, αmax] we define ρ(α) := 2 (α− log 2/ log 3) . Then, dimH(Λ) < dimP(Λ). However, for all α ∈ [αmin, αmax],

dimPDν(α) = dimH Dν(α) =−ρ(α) log ρ(α)− (1− ρ(α)) log(1− ρ(α))

log 4+

12(1− ρ(α)).

Proof. Theorem 2.86 implies dimH(Λ) < dimP(Λ). Applying Lemma 3.27 we see that for all α ∈ [αmin, αmax]

we see that Π(ω) ∈ Dν(α) if and only if

V(ω) ⊆ µ ∈ Mσ(Σ) : µ([(0, 0)]) + µ([3, 0]) = ρ(α) . (3.59)

Now proceed as in Example 1.

Theorem 3.3 also implies the following lower bound for the packing spectrum for local dimension.

Proposition 3.29. Suppose that µ is a Bernoulli measure on a self-affine Sierpinski sponge which satisfies the Very

Strong Separation Condition. Then,

dimPDµ(α) ≥ sup

H1(σ, P, α)

log a1+

d

∑k=2

(1

log ak− 1

log ak−1

)Hk( f , P, α)

,

where the supremum is taken over all α = (αj)dj=1 ∈ Rd for which ∑d

j=1 αj = α.

Proof. It follows from Lemma 3.27 that EP(α) ⊆ Dµ(α) for each α = (αj)dj=1 ∈ Rd with ∑d

j=1 αj = α. Conse-

quently, the result follows from Theorem 3.3.

For a rather limited class of Bernoulli measures we obtain an equality.

Definition 3.30. We say that a Bernoulli measure µ on a self-affine Sierpinski sponge is one dimensional if there exists

some k ∈ 1, · · · , d for which the probability vector (pi1···id)(i1,··· ,id)∈D associated to µ satisfies pi1···id+1−q /pi1···id−q =

#ηq+1(D)/#ηq(D) for all (i1, · · · , id+1−q) ∈ ηq+1(D) and all q ∈ 1, · · · , d− 1\k and each pi = 1/#ηd(D) for

i ∈ ηd(D), provided k 6= d.

3.8. Generalisations and open questions 83

Now if µ is a one dimensional Bernoulli measure on a self-affine Sierpinski sponge then for each j 6= k Pj

will be equal to an explicit constant cj. Let P denote the potential Pk + ∑j 6=k cj.

Theorem 3.31. Suppose that µ is a one dimensional Bernoulli measure on a self-affine Sierpinski sponge which satisfies

the Very Strong Separation Condition. Then

dimPDµ(α) =H1(σ, P, α)

log a1+

d

∑k=2

(1

log ak− 1

log ak−1

)Hk( f , P, α).

Proof. It follows from Lemma 3.27 that Dµ(α) = EP(α). Hence, the result follows from Theorem 3.3.

We emphasise that the class of one dimensional Bernoulli measures is really very limited and the techniques

of this chapter are insufficient for determining dimPDµ(α) for more general classes of Bernoulli measures. The

reason for this extra level of difficulty is that one is essentially dealing with a sum of Birkhoff averages taken

at multiple time scales (see Lemma 3.27). It seems unlikely that the lower bound given in Proposition 3.29

is optimal. As such it remains an open question to determine the packing spectrum for local dimension on a

self-affine Sierpinski sponge.


4

Infinite non-conformal iterated function

systems

4.1 Introduction

The first non-trivial self-affine sets for which the Hausdorff dimension was determined were the planar limit

sets of Bedford [Be] and McMullen [Mc] discussed previously. These limit sets are repellers for expanding

toral endomorphisms with a constant Lyapunov exponent. Gatzouras and Lalley subsequently generalised

Bedford and McMullen’s construction to form self-affine repellers for a collection of maps with a variable

Lyapunov exponent [LG]. We refer the reader to Section 2.8 for a more detailed introduction to the theory of

self-affine sets. A parallel tradition is the study of limit sets for a countable infinity of conformal contractions

[MU1]. We shall combine these two traditions and consider examples of iterated function systems consisting

of a countable infinity of non-conformal contractions.

Example 4.1 (Gauss-Renyi Products). Given x, y ∈ [0, 1] and n ∈ N we let an(x) ∈ N denote the nth digit in the

continued fraction expansion of x and bn(y) ∈ 0, 1 denote the nth digit in the binary expanion of y. Choose some digit

set D ⊆N× 0, 1 and define,

Λ :=(x, y) ∈ [0, 1]2 : (an(x), bn(y)) ∈ D for all n ∈N

.

Then Λ is the attractor of the iterated function system consisting of all maps of the form,

(x, y) 7→(

1a + x

,y + b

2

)for (x, y) ∈ [0, 1]2,

with (a, b) ∈ D.

This example is a member of a class of constructions which we shall refer to as INC systems (see Section

4.2 for the definition). We shall show that for all such systems the Hausdorff dimension of the limit system is

equal to the supremum over the diminsions of ergodic measures supported on compact invariant subsets of

the limit set.

85

86 Chapter 4. Infinite non-conformal iterated function systems

We shall also consider the multifractal analysis of Birkhoff averages. In recent years there has been a great

deal of interest in the multifractal analysis of Birkhoff averages for iterated function systems consisting of a

countable infinity of conformal contractions (see Section 2.8.2.3). There has been a particular focus upon level

sets for simultaneous Birkhoff averages for a countable infinity of potentials [FLM, FLMW, FJLR]. This is

motivated by questions concerning the frequencies of digits [Bes, Eg].

In this chapter we shall obtain a conditional variational principle for the level sets of a countable infinity

of Birkhoff averages on the limit set for an INC system. This generalises a previous result due to the author

in which we determined the Hausdorff dimension for the level sets for continuous potentials on a Lalley-

Gatzouras repeller which was published in Fundamenta Mathematicae [R2].

The results of this chapter have been published in The Israel Journal of Mathematics [R3].


Let I := [0, 1] denote the closed unit interval. Given a digit set B we let B :=⋃

n∈N Bn denote the space of all

finite strings. Given a sequence of maps f jj∈B indexed by B and a finite string ω = (ω1, · · · , ωn) ∈ B we

let fω denote the composition fω := fω1 · · · fωn .

Definition 4.2 (Interval Iterated Function Systems). By an interval iterated function system we shall mean a family

f j : j ∈ B of C1 maps f j : I → I, indexed over some finite or countable digit set B, which satisfies the following

assumtions.

(UCC) Uniform Contraction Condition. There exists a contraction ratio ξ ∈ (0, 1) and N ∈ N such that for all n ≥ N

and all ω ∈ Bn we have

supx∈I| f ′ω(x)| ≤ ξn.

(OIC) Open Interval Condition. For all j1, j2 ∈ B with j1 6= j2, we have

f j1((0, 1)) ∩ f j2((0, 1)) = ∅.

(TDP) Tempered Distortion Property. There exists some sequence ρn with limn→∞ ρn = 0 such that for all n ∈ N and

for all ω ∈ Bn and all x, y ∈ I we have

e−nρn ≤ | f′ω(x)|| f ′ω(y)|

≤ enρn .

If B is finite then f j : j ∈ B is said to be a finite interval iterated function system.

Definition 4.3 (INC Systems). Suppose we have a finite interval iterated function system gi : i ∈ A and for each i ∈

A we have a (finite or countable) interval iterated function system fij : j ∈ Bi with supx∈I | f ′ij(x)| ≤ infx∈I |g′i(x)|

for each j ∈ Bi. Let D := (i, j) : i ∈ A, j ∈ Bi and for each pair (i, j) ∈ D we let Sij denote the map given by

Sij(x, y) = ( fij(x), gi(y)) for (x, y) ∈ I2.

An iterated function system

Sij : (i, j) ∈ D

defined in this way shall be referred to as an INC System.

4.2. Notation and statement of results 87

We shall use the symbolic spaces Σ := DN, and Σv := AN, each of which is endowed with the product

topology. Let σ : Σ → Σ and σv : Σv → Σv denote the corresponding left shift operators. We let π : Σ → Σv

denote the projection given by π(ω) = (iν)ν∈N for ω = ((iν, jν))ν∈N ∈ Σ. We also let π(((iν, jν))nν=1) = (iν)n

ν=1

for a finite string (iν, jν)nν=1 ∈ Dn. We define a projection Π : Σ→ I2 by

Π(ω) := limn→∞

Sω1 · · · Sωn

(I2)

for ω = (ωn)n∈N ∈ Σ. (4.1)

We also define a vertical projection Πv : Σv → I by

Πv(i) := limn→∞

gi1 · · · gin (I) for i = (iν)ν∈N ∈ Σv. (4.2)

Let Λ := Π(Σ). It follows that,

Λ =⋃

(i,j)∈DSij(Λ). (4.3)

Given any finite subset F ⊂ D we let ΛF denote the unique non-empty compact set satisfying,

ΛF =⋃

(i,j)∈FSij(Λ). (4.4)

In addition we define χ ∈ Σ→ R and ψ ∈ Σv → R by

χ(ω) := − log | f ′ω1(Π(σω)) | for ω = (ων)ν∈N ∈ Σ, (4.5)

ψ(i) := − log |g′i1 (Πv(σi)) | for i = (iν)ν∈N ∈ Σv. (4.6)

Let A denote the Borel sigma algebra on Σv. We letMσ(Σ) denote the set of all σ-invariant Borel probabil-

ity measures on Σ and letMσ(Σ) denote the set of µ ∈ Mσ(Σ) which are supported on a compact subset of Σ.

Similarly we let Eσ(Σ) denote the set of µ ∈ Mσ(Σ) which are ergodic and Eσ (Σ) denote the set of µ ∈ Eσ(Σ)

which are compactly supported. Given µ ∈ Mσ(Σ) we define

D(µ) :=hµ(σ|π−1A )∫

χdµ+

hµπ−1(σv)∫ψdµ π−1 . (4.7)

The formula for D(µ) resembles Ledrappier and Young’s formula for the dimension of an ergodic invariant

measure (see [LY, Corollary D]). Indeed D(µ) gives the dimension of µ Π−1 for all µ ∈ Eσ (Σ).

Theorem 4.4. Let Λ be the attractor of an INC system. Then,

dimHΛ = sup D(µ) : µ ∈ Eσ (Σ)

= sup D(µ) : µ ∈ Mσ(Σ)

= sup dimHΛF : F is a finite subset of D .

Given a potential ϕ : Σ→ R and n ∈N we shall let Sn(ϕ) := ∑n−1l=0 ϕ σl , An(ϕ) := n−1Sn(ϕ) and define

varn(ϕ) := sup |ϕ(ω)− ϕ(τ)| : ω, τ ∈ Σ with ωl = τl for l = 1, · · · , n .


Definition 4.5 (Tempered Distortion Property). A potential ϕ : Σ → R is said to satisfy the tempered distortion

property if limn→∞ varn(An(ϕ)) = 0.

It follows from the tempered distortion property of fij : j ∈ Bi and gi : i ∈ A (see Definition 4.2 (TDP))

that both χ and ψ π satisfy the tempered distortion property in Definition 4.5. We shall focus on potentials

satisfying the tempered distortion property which are bounded on one side. That is, there exists some a ∈ R

such that either ϕ(ω) ≤ a for all ω ∈ Σ or ϕ(ω) ≥ a for all ω ∈ Σ. Note that this family includes every positive

valued uniformly continuous potential.

Suppose we have a countable family ϕ = (ϕk)k∈N of real-valued potentials ϕk : Σ → R, together with

some α = (αk)k∈N ⊂ R∪ −∞,+∞. We define,

Eϕ(α) :=

ω ∈ Σ : limn→∞

An(ϕk)(ω) = αk for all k ∈N

, (4.8)

and let Jϕ(α) := Π(Eϕ(α)). Here limits are taken with respect to the usual two point compactification of R.

Given α ∈ R∪ −∞,+∞ we define a shrinking family Bm(α)m∈N of neighbourhoods of α by,

Bm(α) :=

x : |x− α| < 1

m

if α ∈ R

(m,+∞) if α = +∞

(−∞,−m) if α = −∞.

(4.9)

Theorem 4.6. Suppose we have countably many potentials (ϕk)k∈N each of which satisfies the tempered distortion

property and is bounded on one side. Then, for all α = (αk)k∈N ∈ (R∪ −∞,+∞)N we have,

dimH Jϕ(α) = limm→∞

sup

D(µ) : µ ∈ Eσ (Σ),∫

ϕkdµ ∈ Bm(αk) for k ≤ m

= limm→∞

sup

D(µ) : µ ∈ Mσ(Σ),

∫ϕkdµ ∈ Bm(αk) for k ≤ m

.

Note that in general it is impossible to remove the dependence on m and obtain a variational principle of

the form [B, Theorem 9.1.4]. This is a consequence of lack of compactness in the symbolic space, along with

the lack of upper semi-continuity for entropy. Example 4.10 illustrates this phenomenon.

Nonetheless for the interior of the spectrum for a single potential we can use an argument from Iommi and

Jordan [IJ] to recover the usual conditional variational principle. Let αmin := inf∫

ϕdµ : µ ∈ Mσ(Σ)

and

αmax := sup∫

ϕdµ : µ ∈ Mσ(Σ)

.

Theorem 4.7. Given a non-negative potential ϕ : Σ → R satisfying the tempered distortion property and some α ∈

(αmin, αmax) we have

dimH(

Jϕ(α))= sup

D(µ) : µ ∈ M

σ(Σ),∫

ϕdµ = α

.

Proof of Theorem 4.7 given Theorem 4.6. By Theorem 4.6 it suffices to fix α ∈ (αmin, αmax) and show that

sup

D(µ) : µ ∈ Mσ(Σ),

∫ϕdµ = α

= lim

m→∞sup

D(µ) : µ ∈ M

σ(Σ),∫

ϕdµ ∈(

α− 1m

, α +1m

).


Clearly we have

sup

D(µ) : µ ∈ Mσ(Σ),

∫ϕdµ = α

≤ lim

m→∞sup

D(µ) : µ ∈ M

σ(Σ),∫

ϕdµ ∈(

α− 1m

, α +1m

).

To prove the converse inequality we follow the approach of Iommi and Jordan [IJ, Lemma 3.2]. Since α ∈

(α−, α+) we may choose µ−, µ+ ∈ Mσ(Σ) so that

∫ϕdµ− < α <

∫ϕdµ+.

Choose a sequence (µn)n∈N ⊂Mσ(Σ) with limn→∞

∫ϕdµn = α and for each n ∈N we have

limn→∞

D(µn) = limm→∞

sup

D(µ) : µ ∈ Mσ(Σ),

∫ϕdµ ∈

(α− 1

m, α +

1m

).

Now at least one of∫

ϕdµn ≤ α or∫

ϕdµn ≥ α must hold for infinitely many values of n. We shall suppose

that the former holds infinitely often.

Suppose that∫

ϕdµn ≤ α for infinitely many n. Then we may choose a subsequence (nq)q∈N so that∫ϕdµnq ≤ α for all q ∈N. It follows that for each q ∈N we may choose ρq ∈ (0, 1) with the property that

(1− ρq)∫

ϕdµnq + ρq

∫ϕdµ+ = α.

Moreover, as limq→∞∫

ϕdµnq = α we have limq→∞ ρq = 0. For each q ∈ N we let νq := (1− ρq)µnq + ρqµ+ so

that νq ∈ Mσ(Σ) and

∫ϕdνq = α. Moreover, using the fact that entropy map is affine we have

D(νq) =hνq(σ|π−1A )∫

χdνq+

hνqπ−1(σv)∫ψdνq π−1

=(1− ρq)hµnq (σ|π

−1A ) + ρqhµ+(σ|π−1A )

(1− ρq)∫

χdµnq + ρq∫

χdµ++

(1− ρq)hµnqπ−1(σv) + ρqhµ+π−1(σv)

(1− ρq)∫

ψdµq π−1 + ρq∫

ψdµ+ π−1 .

Since limq→∞ ρq = 0 we have

limq→∞

D(νq) = limq→∞

D(µq)

≥ limm→∞

sup

D(µ) : µ ∈ Mσ(Σ),

∫ϕdµ ∈

(α− 1

m, α +

1m

).

If∫

ϕdµn ≥ α for infinitely many n we proceed similarly with µ− in place of µ+.

The following examples are applications of Theorem 4.6.

Example 4.8 (Geometric Arithmetic Mean Sets). Let Λ be as in Example 4.1. For each α, β ∈ R we define,

Λ×(α, β) :=(x, y) ∈ Λ : lim

n→∞n√

a1(x) · · · an(x) = α, limn→∞

b1(y) + · · ·+ bn(y)n

= β

.

Then dimHΛ×(α, β) varies continuously as a function of (α, β) ∈ R2.

Example 4.9 (Arithmetic Mean Sets). Let Λ be as in Example 4.1. For each α, β ∈ R we define,

Λ+(α, β) :=(x, y) ∈ Λ : lim

n→∞

a1(x) + · · ·+ an(x)n

= α, limn→∞

b1(y) + · · ·+ bn(y)n

= β

.

Then dimHΛ+(α, β) varies continuously as a function of (α, β) ∈ R2.


Example 4.10 (Total Escape of Mass). Within the setting of Example 4.1 we consider the set,

Λ∞(D) :=(x, y) ∈ Λ : lim

n→∞

# l ≤ n : al(x) = mn

= 0 for all m ∈N

.

If D is finite then Λ∞(D) is clearly empty. However, if D := (n, n mod 2) : n ∈N, then dimHΛ∞(D) = 32 .

The rest of the chapter will be direceted towards proving Theorem 4.6, from which Theorems 4.7 and 4.4

follow. The proof will consist of an upper bound, contained in sections 4.3 and 4.4 and a lower bound, con-

tained in sections 4.5 and 4.7. We begin the proof of the upper bound by proving an upper estimate, in Section

4.3, for the dimension of the level sets in the special case in which we have have finitely many locally constant

potentials. It is in proving this initial upper estimate that many of the difficulties lie. We use the compactness

of the vertical symbolic space Σv = AN to partition the symbolic level sets into a countable number of sets for

which certain sequences depending only upon π(ω) converge to some prescribed value along a sequence of

good times. We then use the sequence of good times to obtain an efficient covering by approximate squares.

A Misiurewicz-type argument (see [Mi]) based on [JJOP, Ol2] is then used to extract a conditional n-th level

Bernoulli measure for each of the horizontal fibers from the covering. Note that Misiurewicz’s argument must

be adjusted to deal with the lack of compactness. By weighting the horizontal fibres according to a Bernoulli

measure derived from the frequencies of certain digits, along a subsequence of good times, we obtain an n-th

level Bernoulli measure which not only has dimension close to the exponent given by the covering, but also

integrates each of the potentials to approximately the correct value. In section 4.4 we apply a series of approx-

imation arguments to deduce the upper bound given in Theorem 4.6 from the upper estimate from Section

4.3.

To prove the lower bound we use the technique of concatenating measures applied by Gelfert and Rams

in [GR]. For each m ∈ N we obtain an compactly supported ergodic measure, with near optimal dimension,

which integrates each of the first m potentials to approximately the required value. By carefully concatenating

a sequence of such measures it is possible to obtain a measure for which typical points, with respect to that

measure, have local dimension equal to the expression in Theorem 4.6 and for which each of the countably

many Birkhoff averages converge to the required value.

4.3 The upper bound for locally constant potentials

In this section we shall make the following simplifying assumptions. Firstly, we will suppose that there exists

a contraction ratio ζ ∈ (0, 1/3????) such that for each i ∈ A, supx∈I |g′i(x)| ≤ ζ. Secondly, we will suppose

that we have finitely many potentials, ϕ1, · · · , ϕK, each of which is both locally constant and bounded below

by 1. That is, for each k = 1, · · · , K, there exists a D-sequence (ϕkij)(i,j)∈D such that ϕk(ω) = ϕk

ω1≥ 1 for all

ω = (ων)ν∈N ∈ Σ.

We shall often view the K-tuple of potentials, ϕ1, · · · , ϕK as a single vector valued potential ϕ : ω 7→

(ϕkω1)K

k=1, taking values in RK. We endow RK with the supremum metric, which we shall denote by || · ||∞, as

4.3. The upper bound for locally constant potentials 91

well as the usual partial order given by (ck)Kk=1 ≤ (dk)

Kk=1 if and only if ck ≤ dk for all k = 1, · · · , K. We also let

[c, d] :=

x ∈ RK : c ≤ x ≤ d

.

Let R ∪ −∞,+∞ denote the usual two-point compactification of R. Given a sequence of real numbers

(an)n∈N we let Ω((an)n∈N) denote its set of accumulation points in R ∪ −∞,+∞. For each k = 1, · · · , K,

we fix some (possibly infinite) interval Γk = [γkmin, γk

max] ⊆ R ∪ +∞, let Γ := ∏Kk=1 Γk = [γmin, γmax], where

γmin := (γkmin)

Kk=1 and γmax := (γk

max)Kk=1. Define,

Eϕ(Γ) := ω ∈ Σ : Ω((An(ϕ)(ω))n∈N) ⊆ Γ , (4.10)

and let Jϕ(Γ) := Π(Eϕ(Γ)).

For each (i, j) ∈ D we let bi := supx∈I |g′i(x)| and aij := supx∈I | f ′ij(x)|. We define χ : Σ → R and

χv : Σv → R by

χ(ω) := − log aω1 for ω = (ων)ν∈N ∈ Σ, (4.11)

ψ(i) := − log bi1 for i = (iν)ν∈N ∈ Σv. (4.12)

Given q ∈N and µ ∈ Mσq(Σ) we define

Dq(µ) :=hµ(σq|π−1A )∫

Sq(χ)dµ+

hµπ−1(σqv )∫

Sq(ψ)dµ π−1v

. (4.13)

In this section we shall prove the following proposition.

Proposition 4.11.

dimH Jϕ(α) ≤ limξ→0

Dq(µ) : q ∈N, µ ∈ Eσq(Σ),

∫Aq(ϕ)dµ ∈ [γmin − ξ, γmax + ξ]

.

4.3.1 Building a cover

Define

Ln(ω) := min

l ≥ 1 :

l

∏ν=1

biν ≤n

∏ν=1

aiν jν

. (4.14)

Note that this implies

1 ≤∏n

ν=1 aiν jν

∏Ln(ω)ν=1 biν

< b−1min. (4.15)

Moreover, since aij ≤ bi for all (i, j) ∈ D, Ln(ω) ≥ n.

Given (ων)nν=1 = ((iν, jν))n

ν=1 ∈ Dn we let

[ω1 · · ·ωn] := ω′ ∈ Σ : ω′ν = ων for ν = 1, · · · , n (4.16)

and

[i1 · · · in] := i′ ∈ Σv : i′ν = iν for ν = 1, · · · , n. (4.17)

Given ω = ((iν, jν))∞ν=1 ∈ Σ we let Bn(ω) denote the nth approximate square,

Bn(ω) := Π([ω1 · · ·ωn] ∩ σ−nπ−1[in+1 · · · iLn(ω)]). (4.18)


Thus,

diam(Bn(ω)) ≤√

2 ·max

(n

∏ν=1

aiν jν

),

(Ln(ω)

∏ν=1

biν

). (4.19)

We also define a map φi :⋃

n∈N∪0Dn → i ×⋃n∈N∪0 Bni for each i ∈ A by

φi : ((i′1, j′1), (i′1, j′2), · · · , (i′1, j′n)) 7→ ((i, j′ν1

), (i, j′ν2), · · · , (i, j′νni

)),

where ν1 < ν2 < · · · < νni and νlnil=1 = r ≤ n : i′r = i. Here we adopt the convention that D0 = B0

i = ∅

where ∅ denotes the empty string.

Given q ∈N we define,

Pq(A) : =

(pi)i∈Aq ∈ [0, 1]A

q: ∑

i∈Aqpi = 1

,

Qq(A) : =(pi)i∈Aq ∈ Pq(A) : pi ∈ Q\0 for each i ∈ Aq .

Each Pq(A) is given the maximum norm || · ||∞. Note that for each q ∈ N, Pq(A) is compact and Qq(A) is a

dense countable subset. We let P(A) := P1(A) and Q(A) := Q1(A).

Given p = (pi)i∈Aq ∈ Pq(A) we define,

dq(p) := ∑i∈Aq pi log pi

∑i∈Aq pi log bi=

hµp(σqv )∫

Sq(ψ)dµp, (4.20)

where µp denotes the q-th level Bernoulli measure on Σv defined by µp([i]) = pi for all i ∈ Aq. We let

d(p) := d1(p) for p ∈ P(A).

Given ρ ∈ Q(A), n ∈N, and λ = (λk)Kk=1 ∈ Q(A)K with λk = (λk

i )i∈A for each k = 1, · · · , K we define,

Bn,εi (Γ, ρ, λ) :=

(ijν)l

ν=1 : l = ρin± εn,l

∑ν=1

ϕkijν ± εn ∈ [λk

i γkmin, λk

i γkmax]

, (4.21)

for each i ∈ A and let

Bn,ε(Γ, ρ, λ) :=

(ϑi)i∈A ∈ ∏

i∈ABn,ε

i (Γ, ρ, λ) : ∑i∈A|ϑi| = n

. (4.22)

Now define,

sn,ε(Γ, ρ, λ) := inf

s : ∑(ϑi)i∈A∈Bn,ε(Γ,ρ,λ)

∏i∈A

asϑi ≤ 1

,

sε(Γ, ρ, λ) := lim supn→∞

sn,ε(Γ, ρ, λ),

δε(Γ) := sup

sε(Γ, ρ, λ) + d(ρ) : ρ ∈ Q(A), λ ∈ Q(A)K

,

δ(Γ) := lim infε→0

δε(Γ).

Lemma 4.12 (Building a Cover). dimH Jϕ(Γ) ≤ δ(Γ).

Proof. Take some ξ > 0. Note that the map p 7→ d(p) defines a continuous function on the compact space

P(A). Consequently there exists some ε > 0 such that δε(Γ) < δ(Γ) + ξ and for all p, q ∈ P(A) with ||p−

q||∞ < ε we have |d(p)− d(q)| < ξ.

We shall define a function Fξ : Σ→ Q(A)2+K in the following way. Given ω ∈ Σ we extract a subsequence

(nq)q∈N satisfying,


(i) limq→∞

∑i∈A Pi(ω|nq) log Pi(ω|nq)

∑i∈A Pi(ω|nq) log bi= lim sup

n→∞

∑i∈A Pi(ω|n) log Pi(ω|n)∑i∈A Pi(ω|n) log bi

,

(ii) limq→∞

(Pi(ω|nq))i∈A = P(ω) = (Pi(ω))i∈A,

(iii) limq→∞

(Pi(ω|Lnq(ω)))i∈A = Q(ω) = (Qi(ω))i∈A,

(iv) limq→∞

(∑j∈Bi

Pij(ω|nq)ϕkij

∑(i′ ,j′)∈D Pi′ j′(ω|nq)ϕki′ j′

)i∈A

= Rk(ω) = (Rki (ω))i∈A,

for each k = 1, · · · , K. We let R(ω) := (Rk(ω))Kk=1. Note that by (i) we always have d(Q(ω)) ≤ d(P(ω)). Since

Q(A) is dense in P(A) we may choose κ(ω) = (κi(ω))i∈A ∈ Q(A) so that κi(ω) > Qi(ω)ζξ for each i ∈ A.

We choose ρ(ω) ∈ Q(A) and λ(ω) ∈ Q(A)K so that ||P(ω) − ρ(ω)|| < ε and ||R(ω) − λ(ω)||∞ < ε. Let

Fξ(ω) := (ρ(ω), κ(ω), λ(ω)).

For each (ρ, κ, λ) ∈ Q(A)2+K we define

E(ρ,κ,λ)ϕ (Γ) := Eϕ(Γ) ∩ F−1

ξ (ρ, κ, λ),

J(ρ,κ,λ)ϕ (Γ) := Π(E(ρ,κ,λ)

ϕ (Γ)).

Since Q(A)2+K is countable, to show that

dimH(

Jϕ(Γ))≤ δ(Γ) + 6ξ,

it suffices to fix (ρ, κ, λ) ∈ Q(A)2+K and show that

dimH

(J(ρ,κ,λ)ϕ (Γ)

)≤ δ(Γ) + 6ξ.

By the definition of sε(Γ, ρ, λ) we may take N(ε) ∈N so that for all n ≥ N(ε) we have

∑(ϑi)i∈A∈Bn,ε(Γ,ρ,λ)

∏i∈A

asε(Γ,ρ,λ)+ξ

ϑi < 1, (4.23)

and hence,


∏i∈A

asε(Γ,ρ,λ)+2ξ

ϑi < ζnξ . (4.24)

Given ω ∈ E(ρ,κ,λ)ϕ (Γ) we have,

limq→∞

∑i∈A Pi(ω|Lnq(ω)) log κi

∑i∈A Pi(ω|Lnq(ω)) log bi≤ lim

q→∞

∑i∈A Pi(ω|Lnq(ω)) log Pi(ω|Lnq(ω))

∑i∈A Pi(ω|Lnq(ω)) log bi+ ξ

≤ d(Q(ω)) + ξ ≤ d(P(ω)) + ξ

< d(ρ) + 2ξ.

Thus, for all sufficiently large q we have,

diam(Bnq(ω))d(ρ)+3ξ ≤ 2(

bi1 · · · biLn(ω)

)d(ρ)+2ξ

≤ 2(

κi1 · · · κiLnq (ω)

)ζLnq (ω)ξ .


We also have,

diam(Bnq(ω)) ≤nq

∏ν=1

aiν jν ≤ ∏i∈A

aφi(ω|nq).

Moreover, by (2) and (4) we also have φi(ω|nq) ∈ Bnq ,εi (Γ, ρ, λ) for each i ∈ A and hence (φi(ω|nq))i∈A ∈

Bnq ,ε(Γ, ρ, λ) for all sufficiently large q.

Thus, if we fix some r > 0, then for each ω ∈ E(ρ,κ,λ)ϕ (Γ) we may take some n(ω) ≥ N(ε) so that,

(i) Π(ω) ∈ Bn(ω)(ω),

(ii) diam(Bn(ω)(ω)) ≤ γ,

(iii) diam(Bn(ω)(ω))d(ρ)+3ξ ≤ 2(

κi1 · · · κiLn(ω)(ω)

)ζLn(ω)(ω)ξ ,

(iv) diam(Bn(ω)(ω))sε(Γ,ρ,λ)+2ξ ≤ 2 ·∏i∈A asε(Γ,ρ,λ)+2ξ

φi(ω|n(ω)),

(v) (φi(ω|n(ω)))i∈A ∈ Bn(ω),ε(Γ, ρ, λ).

Let Br :=

Bn(ω)(ω) : ω ∈ E(ρ,κ,λ)ϕ (Γ)

. By (i) and (ii) above, Br forms a countable r-cover of J(ρ,κ,λ)

ϕ (Γ).

Note also that given ω1 = ((i1ν, j1ν))ν∈N, ω2 = ((i2ν, j2ν))ν∈N ∈ E(ρ,κ,λ)ϕ (Γ) with (i11, · · · , i1Ln(ω1)(ω)) = (i12, · · · , i2Ln(ω2)(ω))

and (φi(ω1|n(ω1)))i∈A = (φi(ω2|n(ω2)))i∈A we must have Bn(ω1)(ω1) = Bn(ω2)(ω

2). Hence,

∑B∈Bγ

diam(B)sε(Γ,ρ,λ)+d(ρ)+5ξ ,

≤ 4 · ∑l∈N

ζ lξ

∑(i1,··· ,il)∈Al

κi1 · · · κil

× ∑

n≥N(ε)∑

(ϑi)i∈A∈Bn,ε(Γ,ρ,λ)∏i∈A

asε(Γ,ρ,λ)+2ξ

ϑi

,

≤ 4 · ∑l∈N

ζ lξ × ∑n≥N(ε)

ζnξ < ∞.

Letting γ→ 0 we have that

dimH J(ρ,κ,λ)ϕ (Γ) ≤ sε(Γ, ρ, λ) + d(ρ) + 5ξ,

≤ δε(Γ) + 5ξ,

≤ δ(Γ) + 6ξ,

by our choice of ε. Since Q(A)2+K is countable, it follows that

dimH Jϕ(Γ) ≤ δ(Γ) + 6ξ.

Letting ξ → 0 proves the lemma.


4.3.2 Constructing a measure

Define An,ε(Γ, ρ) ⊆ Ad(1+2ε)ne by,

An,ε(Γ, ρ) :=

τ ∈ Ad(1+2ε)ne : Ni(τ) ≥ (1 + ε)ρin for each i ∈ A

.

Lemma 4.13. Given ρ ∈ P(A) there exists M(ρ, ε) ∈ N such that for all n ≥ M(ρ, ε) we have, Pn,ε(Γ, ρ) :=

∑τ∈An,ε(Γ,ρ) ρτ > 1− ε.

Proof. Apply Kolmogorov’s strong law of large numbers and then Egorov’s theorem.

Lemma 4.14 (Constructing a Measure).

δ(Γ) ≤ limξ→0


∫Aq(ϕ)dµ ∈ [γmin − ξ, γmax + ξ]

.

Proof. We begin by fixing some ji∗ ∈ Bi for each i ∈ A. We then let a∗ := min

aiji∗: i ∈ A

and ϕ∗ :=

max

ϕkiji∗

: i ∈ A, k ≤ K

. In what follows we shall let o(ε) denote any quantity which depends only upon the

observable ϕ, the iterated function system, our choice of (ji∗)i∈A and ε, which tends to zero as ε tends to zero.

Of course, the precise value of o(ε) will vary from line to line.

Take ξ > 0. Then there exists ε0(ξ) such that for all ε ≤ ε0(ξ) < 1/2 we have δε(Γ) > δ(Γ) − ξ. Take

ε ≤ ε0(ξ). Then there exists ρ ∈ Q(A), λ ∈ Q(A)K with s(Γ, ρ, λ) + d(ρ) > δ(Γ)− ξ.

Consequently, there exists infinitely many n ∈N for which


∏i∈A

aδ(Γ)−d(ρ)−ξ

ϑi > 1. (4.25)

In particular we may apply Lemma 4.13 and take some such n ≥ M(ρ, ε), so that Pn,ε(Γ, ρ) > 1− ε. By (4.25)

there exists a finite subset Fn,ε(Γ, ρ, λ) ⊆ Bn,ε(Γ, ρ, λ) and s > δ(Γ)− d(ρ)− ξ for which

∑(ϑi)i∈A∈Fn,ε(Γ,ρ,λ)

∏i∈A

asϑi = 1. (4.26)

Recall that we defined An,ε(Γ, ρ) ⊆ Ad(1+2ε)ne by,

An,ε(Γ, ρ) :=

τ ∈ Ad(1+2ε)ne : Ni(τ) ≥ (1 + ε)ρin for each i ∈ A

.

We now define an injective map η : An,ε(Γ, ρ) × Fn,ε(Γ, ρ, λ) → Dd(1+2ε)ne so that for all (τ, (ϑi)i∈A) ∈

An,ε(Γ, ρ)×Fn,ε(Γ, ρ, λ) and i ∈ Awe have π(η(τ, (ϑi)i∈A)) = τ and ϑi is an intial segment of φi(η(τ, (ϑi)i∈A)).

To define η we proceed as follows. Take (τ, (ϑi)i∈A) ∈ An,ε(Γ, ρ) × Fn,ε(Γ, ρ, λ). For each i ∈ A we write

ϑi = ((i, ji1), · · · , (i, jimi)) and let τ = (i1, · · · , id(1+2ε)ne). Now, for each ν ∈ 1, · · · , d(1 + 2ε)ne we choose

i ∈ A so that i = iν, and choose r so that ν is the r-th occurance of the digit i in τ. If r ≤ mi then let ην := (i, jir)

and if ν > r let ην = (i, ji∗). Write η(τ, (ϑi)i∈A) = (ην)nν=1. Note that

∏i∈A

aϑi ≥ aη(τ,(ϑi)i∈A)

≥ ∏i∈A

(aϑi × a3nε

iji∗

)≥(

∏i∈A

aϑi

)× a3#Anε∗ .


That is,

log aη(τ,(ϑi)i∈A) = log

(∏i∈A

aϑi

)+ no(ε). (4.27)

Similarly, for each i ∈ A and k ∈ 1, · · · , K we have

λki (1− ε)γk

minn ≤ ∑j∈Bi

Nij(ϑi)ϕkij,

≤ ∑j∈Bi

Nij(η(τ, (ϑi)i∈A))ϕkij,

≤ ∑j∈Bi

Nij(ϑi)ϕij + 3εnϕkiji∗

,

≤ λki (1 + ε)γmaxn + 3εnϕ∗.

Hence,

∑(i,j)∈D

Pij(η(τ, (ϑi)i∈A))ϕkij ∈ [γk

min − o(ε), γkmax + o(ε)], (4.28)

for each k = 1, · · · , K, and so,

∑(i,j)∈D

Pij(η(τ, (ϑi)i∈A))ϕij ∈ [γmin − o(ε), γmax + o(ε)]. (4.29)

We define a compactly supported d(1+ 2ε)ne-level Bernoulli measure ν on Σ in the following way. First let

ν(π−1[τ]) := ρτ for each τ ∈ Ad(1+2ε)ne. Then, given τ ∈ Ad(1+2ε)ne and κ ∈ Ad(1+2ε)ne with π(κ) = τ either

τ ∈ An,ε(Γ, ρ) in which case we let

ν([κ])

ν(π−1[τ]):=

∏i∈A as

ϑiif κ = η(τ, (ϑi)i∈A) for some (ϑi)i∈A ∈ ∏i∈A Fn,ε

i (Γ, ρ, λ),

0 if otherwise,

or τ = (iν)d(1+2ε)neν=1 ∈ Ad(1+2ε)ne\An,ε(Γ, ρ), in which case we let

ν([κ])

ν(π−1[τ]):=

1 if κ = ((iν, jiν∗ ))

d(1+2ε)neν=1 ,

0 if otherwise.

Since ν π is the (ρi)i∈A Bernoulli measure on AN we have

hνπ−1

v(σd(1+2ε)nev )∫

Sd(1+2ε)ne(ψ)dν π−1 =∑τ∈Ad(1+2ε)ne ρτ log ρτ

∑τ∈Ad(1+2ε)ne ρτ log bτ

=∑i∈A ρi log ρi

∑i∈A ρi log bi= d(ρ).

Now define

Zn,ε(Γ, ρ, λ) := ∑(ϑi)i∈A∈Fn,ε(Γ,ρ,λ)

∏i∈A

asϑi

log

(∏i∈A

aϑi

),

By (4.26) together with the fact that log aij ≤ log ζ < 0 for all (i, j) ∈ D we have

Zn,ε(Γ, ρ, λ) ≤ n log ζ.

4.4. Approximation arguments 97

By the definition of ν we have

hν(σd(1+2ε)ne|π−1A ) =

∑τ∈An,ε(Γ,ρ)

ρτ


∏i∈A

asϑi

log

(∏i∈A

asϑi

)= Pn,ε(Γ, ρ)× sZ

n,εi (Γ, ρ, λ).

Also, by (4.27) we have

∫Sd(1+2ε)ne(χ)dν =

∑τ∈An,ε(Γ,ρ)

ρτ


∏i∈A

asϑi

log

(∏i∈A

aϑi

)− no(ε)

+

1− ∑τ∈An,ε(Γ,ρ)

ρτ

d(1 + 2ε)ne log a∗

= Pn,ε(Γ, ρ)(Z

n,εi (Γ, ρ, λ)− no(ε)

)+ (1− Pn,ε(Γ, ρ)) d(1 + 2ε)ne log a∗.

Since n ≥ M(ρ, ε) we have Pn,ε(Γ, ρ) > 1− ε and consequently,

hν(σd(1+2ε)ne|π−1A )∫Sd(1+2ε)ne(χ)dν

≥ s1− o(ε)

.

Combining this with the fact that s + d(ρ) > δ(Γ)− ξ we have

Dd(1+2ε)ne(ν) ≥s

1− o(ε)+ d(ρ) (4.30)

≥ s + d(ρ)1− o(ε)

>δ(Γ)− ξ

1− o(ε).

Moreover, by the construction of ν combined with (4.29) we have,

∫Ad(1+2ε)ne(ϕ)dν ≥ Pn,ε(Γ, ρ)(γmin − o(ε)) (4.31)

≥ (1− ε)(γmin − o(ε)).

Similarly,

∫Ad(1+2ε)ne(ϕ)dν ≤ Pn,ε(Γ, ρ)(γmax + o(ε)) (4.32)

+ (1− Pn,ε(Γ, ρ)) ϕ∗

≤ γmax + o(ε).

Since we can obtain such a measure µ for all ε ≤ ε0(ξ), the lemma follows by taking ε sufficiently small.

4.4 Approximation arguments

In this section we apply Proposition 4.11 to obtain upper estimates of increasing generality until we obtain the

upper bound in Theorem 4.6.

We begin by dropping the assumption that our potentials ϕ are locally constant. Instead we assume that we

have finitely many potentials ϕ1, · · · , ϕK, with finite first level variation var1(ϕk) < ∞, for each k = 1, · · · , K.


We retain the assumption that for some ζ ∈ (0, 1) we have supx∈I |g′i(x)| ≤ ζ for each i ∈ A and also assume

that var1(χ), var1(ψ) < − log ζ. We define

Cσ(χ, ψ) : = max

− log ζ

− log ζ − var1(χ),

− log ζ

− log ζ − var1(ψ)

.

Proposition 4.11 gives the following estimate.

Lemma 4.15. Suppose we have finitely many potentials ϕ1, · · · , ϕM, with var1(ϕk) < ∞. Suppose also that for some

ζ ∈ (0, 1) we have supx∈I |g′i(x)| ≤ ζ for each i ∈ A and that var1(χ), var1(ψ) < − log ζ. Suppose that α = (αk)Mk=1

is such that for all k ≤ K ≤ M we have αk ∈ R and for K < k ≤ M, αk = ∞. Then given any m ∈N,

dimH Jϕ(α) ≤ Cσ(χ, ψ) sup

Dq(µ)

,

where the supremum is taken over all µ ∈ Eσq(Σ) for some q ∈N with |∫

Aq(ϕk)dµ− αk| < 3var1(ϕk) for k ≤ K and∫Aq(ϕk)dµ > m for K < k ≤ M.

Proof. For each k = 1, · · · , K we define a locally constant potential ϕk by

ϕk(ω) := sup

ϕk(ω′) : ω1 = ω′1

, (4.33)

for all ω = (ων)ν∈N ∈ Σ. It follows that ||ϕk − ϕk||∞ < var1(ϕk). Thus, for all ω ∈ Eϕ(α) we have

Ω(An(ϕk)) ⊆ [αk − var1(ϕk), αk + var1(ϕk)] for k ≤ K and Ω(An(ϕk)) = ∞ for K < k ≤ M, since

limn→∞ An(ϕk)(ω) = αk for all k ≤ M. Hence, Jϕ(α) ⊆ Jϕ(Γ) where Γ := ∏Kk=1[αk− var1(ϕk), αk + var1(ϕk)]×

∏Mk=K+1[m + 2var1(ϕk), ∞]. Thus, by Proposition 4.11 we have,

dimH Jϕ(α) ≤ dimH Jϕ(Γ)

≤ limξ→0

sup

Dq(µ) : q ∈N, µ ∈ Eσq(Σ),∣∣ ∫ Aq(ϕk)dµ− αk

∣∣ < var1(ϕk) + ξ,

for k ≤ K and∫

Aq(ϕk)dµ > m + 2var1(ϕk)− ξ for K < k ≤ M

≤ sup


∣∣ < 2var1(ϕk),

for k ≤ K and∫

Aq(ϕk)dµ > m + var1(ϕk) for K < k ≤ M

≤ sup


∣∣ < 3var1(ϕk),

for k ≤ K and∫

Aq(ϕk)dµ > m for K < k ≤ M

.

It is clear from the definitions of χ : Σ → R and ζ that χ(ω) ≥ χ(ω)− var1(χ)(ω) and χ(ω) ≥ − log ζ for all

ω ∈ Σ. Thus∫

Sq(χ)dµ ≥∫

Sq(χ)dµ− qvar1(χ) > 0 for each µ ∈ Mσq(Σ), since var1(χ) < − log ζ. It follows

that for each µ ∈ Mσq(Σ) we have∫Sq(χ)dµ∫Sq(χ)dµ

≤∫

Sq(χ)dµ∫Sq(χ)dµ− qvar1(χ)

≤ − log ζ

− log ζ − var1(χ)≤ Cσ(χ, ψ).

Similarly for each µ ∈ Mσq(Σ) we have∫

Sq(ψ)dµ π−1∫Sq(ψ)dµ π−1 ≤

− log ζ

− log ζ − var1(ψ)≤ Cσ(χ, ψ).


Recall that for each µ ∈ Mσq(Σ) we defined,


Sq(χ)dµ+

hµπ−1(σqv )∫

Sq(ψ)dµ π−1 (4.34)


Sq(χ)dµ+

hµπ−1(σqv )∫

Sq(ψ)dµ π−1 . (4.35)

Thus, for each µ ∈ Mσq(Σ) we have Dq(µ) ≤ Cσ(χ, ψ)Dq(µ). The lemma follows.

We now use the observation that an iterated N-system is itself an N-system to obtain a more refined estimate

which applies in a more general situation. First recall that by the Uniform Contraction Condition, for each N

system, there exists a contraction ratio ζ ∈ (0, 1) and N ∈ N such that for all n ≥ N and all ω ∈ Dn and all

i ∈ An we have

max

supx∈I| f ′ω(x)|, sup

x∈I|g′i(x)|

≤ ζn.

For each n ≥ N we let

Cnσ(χ, ψ) : = max

− log ζ

− log ζ − varn(An(χ)),

− log ζ

− log ζ − varn(An(ψ))

.

Lemma 4.16. Suppose we have finitely many potentials ϕ1, · · · , ϕM, with var1(ϕk) < ∞. Suppose that α = (αk)Mk=1

is such that for all k ≤ K ≤ M we have αk ∈ R and for K < k ≤ M, αk = ∞. Fix some m ∈N. Then for all sufficiently

large n ∈N we have,

dimH Jϕ(α) ≤ Cnσ(χ, ψ) sup

Dq(µ)

,

where the supremum is taken over all µ ∈ Eσq(Σ) for some q ∈ N with |∫

Aq(ϕk)dµ − αk| < 3varn(An(ϕk)) for

k ≤ K and∫

Aq(ϕk)dµ > m for K < k ≤ M.

Proof. First note that by the Uniform Contraction Condition, together with the Tempered Distortion Condition

applied to χ, ψ and ϕ1, · · · , ϕK we may choose N ∈N so that for all n ≥ N we have

(i) max

supx∈I | f ′ω(x)|, supx∈I |g′i(x)|≤ ζn,

(ii) max varn(An(χ)), varn(An(ψ)) < − log ζ,

(iii) max varn(An(ϕk)) : k ∈ 1, · · · , K < ∞.

For each n ≥ N we construct an associated iterated function system in the following way. Given ξ = ξ1 · · · ξn ∈

Dn we let

Sξ := Sξ1 · · · Sξn .

It follows from the fact that (Sij)(i,j)∈D is an INC-system that (Sη)η∈Dn is also an INC- system. Moreover, it

follows from conditions (i), (ii) and (iii) above that the potentials An(ϕ1), · · · An(ϕK) on (Dn)N = Σ, together


with the INC-system (Sη)η∈Dn satisfy the conditions of Lemma 4.15 with σn in place of σ, An(ϕk) in place of

ϕk, Sn(χ) in place of χ, Sn(ψ) in place of ψ, ζn in place of ζ and varn in place of var1. We let,

Eσn

An(ϕ)(α) :=

ω ∈ Σ : liml→∞

Aln(ϕk)(ω) = αk for all k ≤ K

Jσn

An(ϕ)(α) := Π(

EAn(ϕ)(α))

.

Note also that

Cσn(Sn(χ), Sn(ψ)) : = max

− log ζn

− log ζn − varn(Sn(χ)),

− log ζn

− log ζn − varn(Sn(ψ))

= max

− log ζ

− log ζ − varn(An(χ)),

− log ζ

− log ζ − varn(An(ψ))

= Cn

σ(χ, ψ).

Thus, by Lemma 4.15 we have

dimH

(Jσn

An(ϕ)(α))

≤ Cσn(Sn(χ), Sn(ψ)) sup

Dnq(µ) : q ∈N, µ ∈ Eσnq(Σ),∣∣ ∫ Anq(ϕk)dµ− αk

∣∣ < 3varn(An(ϕk)),

for k ≤ K and∫

Anq(ϕk)dµ > m for K < k ≤ M

≤ Cnσ(χ, ψ) sup


∣∣ ∫ Aq(ϕk)dµ− αk∣∣ < 3varn(An(ϕk)),

for k ≤ K and∫


.

Moreover, given ω ∈ Eϕ(α) we have, liml→∞ Al(ϕk)(ω) = αk and hence liml→∞ Aln(ϕk)(ω) = αk. Thus,

Jϕ(α) ⊆ Jσn

An(ϕ)(α). Hence,



∣∣ ∫ Aq(ϕk)dµ− αk∣∣ < 3varn(An(ϕk)),

for k ≤ K and∫


.

We now require a lemma relating σq-invariant measures to σ-invariant measures.

Lemma 4.17. Take ν ∈ Mσq(Σ) and let µ := Aq(ν). Then,

(i) µ ∈ Mσ(Σ),

(ii) If ν ∈ Eσq(Σ) then µ ∈ Eσ (Σ),

(iii) hµ(σ) = q−1hν(σk),

(iv) hµπ−1(σv) = q−1hνπ−1(σqv ),


(v) hµ(σ|π−1v Av) = q−1hν(σq|π−1A ).

Moreover, given any θ ∈ C(Σ), θv ∈ C(Σv) we have,

(vi)∫

θdµ =∫

Aq(θ)dν,

(vii)∫

θvdµ π−1 =∫

Aq(θv)dν π−1.

Proof. Parts (i), (ii), (iii) and (vi) follow from [JJOP, Lemma 2]. It is clear that µ is compactly supported. Since

π σ = σv π we have Ak(ν π−1) = Ak(ν) π−1 and hence (iv) and (vii) also follow from [JJOP, Lemma 2].

Part (v) follows from parts (iii) and (iv) combined with the Abramov Rokhlin formula 2.47.

The following proposition completes the proof of the upper bound.

Proposition 4.18. Suppose we have countably many potentials (ϕk)k∈N. Then, for all α = (αk)k∈N ∈ (R∪ ∞)N

we have,

dimH Jϕ(α) ≤ limm→∞

sup

D(µ) : µ ∈ Eσ (Σ),∫


.

Proof. It suffices to show that for each m ∈N we have,

dimH Jϕ(α) ≤ sup

D(µ) : µ ∈ Eσ (Σ),∫


.

Fix m ∈ N. Without loss of generality we may assume that there are only m potentials ϕ1, · · · , ϕm. If not, we

consider the set,

Emϕ (α) :=

ω ∈ Σ : lim

n→∞An(ϕk) = αk for k ≤ m

, (4.36)

Jmϕ (α) := Π

(Em

ϕ (α))

, (4.37)

and note that Eϕ(α) ⊆ Emϕ (α) and hence dimH Jϕ(α) ≤ dimH Jm

ϕ (α). Finally we may reorder our potentials so

that there is some K ≤ m such that for all k ≤ K ≤ M we have αk ∈ R and for K < k ≤ M, αk = ∞. Now we

are in precisely the position of Lemma 4.16, so


Dq(µ) : q ∈N, µ ∈ M

σq(Σ),∣∣ ∫ Aq(ϕk)dµ− αk

∣∣ < 3varn(An(ϕk)),

for k ≤ K and∫


.

Since limn→∞ varn(An(χ)) = limn→∞ varn(An(ψ)) = limn→∞ varn(An(ϕk)) = 0 for all k ≤ m, and hence

limn→∞ Cnσ(χ, ψ) = 1, we have,



∣∣ < 1m

,

for k ≤ K and∫


.

Recall that for γ ∈ R∪ +∞ and l ∈N we have

Bl(γ) :=

x : |x− γ| < 1l

if γ ∈ R

(l,+∞) if γ = ∞.(4.38)


So we may rewrite the above inequality as


Dq(µ) : q ∈N, µ ∈ Eσq(Σ),∫

Aq(ϕk)dµ ∈ Bm(αk) for k ≤ m

.

Finally, by Lemma 4.17, given ν ∈ Eσq(Σ) we may choose µ ∈ Eσ (Σ) with D(µ) = Dq(ν) and∫

ϕkdµ =∫Aq(ϕk)dµ. Thus,


D(µ) : µ ∈ Eσ (Σ),∫


.

This completes the proof of the upper bound.

4.5 Preliminary lemmas for the lower bound

4.5.1 Dimension lemmas

In this section we shall relate the symbolic local dimension of a measure to the local dimension of its projection.

This will enable us to apply Proposition 2.3.

Given subsets A, B ⊆ R by A ≤ B we mean x ≤ y for all x ∈ A and y ∈ B. We shall say that the digit set D

is wide if there exists (i1, j1), (i2, j2) ∈ D with i1 = i2 and j1 6= j2. It follows that there exists an i′ ∈ A together

with pairs j1−, j2−, j10 , j20 , j1+, j2+ ∈N so that

fi′ j1− fi′ j2−

([0, 1]) ≤ fi′ j10 fi′ j20

([0, 1]) ≤ fi′ j1+ fi′ j2+

([0, 1]). (4.39)

Now define,

Wn(ω) := min

l > n + 1 : ωl = (i′, j10), ωl+1 = (i′, j20)− n, (4.40)

and

Rn(ω) := max− log inf

x∈[0,1]| f ′ων

(x)| : ν ≤ n + Wn(ω) + 1

. (4.41)

Lemma 4.19. Let µ be a finite Borel measure on Σ supported on π−1(τ) for some τ ∈ Σv. Let ν := µ Π−1

be the corresponding projection on Λ. Suppose D is wide. Then for all x = Π(ω) ∈ Λ with π(ω) = τ and

limn→∞ Rn(ω)n−1 = limn→∞ Rn(ω)Wn(ω)n−1 = 0,

lim infr→0

log ν(B(x, r))log r

≥ lim infn→∞

− log µ([ω|n])Sn(χ)(ω)

.

Proof. Suppose that D is wide and fix x = (x1, x2) = Π(ω) ∈ Λ for some ω = ((iν, jν))ν∈N ∈ π−1(τ) with

limn→∞ Rn(ω)n−1 = limn→∞ Rn(ω)Wn(ω)n−1 = 0. First note that since µ is supported on π−1(τ), ν is

supported on R× x2. Define,

a0 := inf| f ′d(z)| : z ∈ [0, 1], d ∈ (i′, j1+), (i

′, j2+), (i′, j1−), (i

′, j2−)

.

Take n ∈ N. By the definition of Wn(ω) the finite string ωn+Wn(ω) = (i′, j10) and ωn+Wn(ω)+1 = (i′, j20). Now

let,

η0 := (ωn+1, · · · , ωn+Wn(ω)−1, (i′, j10), (i′, j20))

η+ := (ωn+1, · · · , ωn+Wn(ω)−1, (i′, j1+), (i′, j2+))

η− := (ωn+1, · · · , ωn+Wn(ω)−1, (i′, j1−), (i′, j2−)).

4.5. Preliminary lemmas for the lower bound 103

It follows from (4.39) that one of the following holds;

fω|n fη−([0, 1]) ≤ fω|n fη0([0, 1]) ≤ fω|n fη+([0, 1]),

fω|n fη+([0, 1]) ≤ fω|n fη0([0, 1]) ≤ fω|n fη−([0, 1]).

Clearly each interval is contained within the interval fω|n([0, 1]). Moreover, it follows from the definitions of

Wn(ω) and a0 that both fη+([0, 1]) and fω|n fη−([0, 1]) are of diameter at least infx∈[0,1] | f ′ω|n(x)|e−Wn(ω)Rn(ω)a20.

Thus, since x1 ∈ fω|n fη0([0, 1]) = fω|n+Wn(ω)+1([0, 1]) and x2 ∈ gi|n([0, 1]) we have,

B(

x1, infz∈[0,1]

| f ′ω|n(z)|e−Wn(ω)Rn(ω)a3

0

)⊆ fω|n((0, 1)), (4.42)

and hence,

ν

(B(

x, infz∈[0,1]

| f ′ω|n(z)|eWn(ω)Rn(ω)a3

0

))≤ µ ([ω|n]) , (4.43)

since ν is supported on R× x2. So let

rn := infz∈[0,1]

| f ′ω|n(z)|eWn(ω)Rn(ω)a3

0. (4.44)

By the tempered distortion property we have

log infz∈[0,1]

| f ′ω|n(z)| = log | f ′ω|n(Π(σnω))|+ o(n),

= logn

∏l=1

∣∣ f ′ωl

(fωl+1 · · · fωn(Π(σnω))

) ∣∣+ o(n),

= logn

∏l=1

∣∣ f ′ωl

(Π(σlω)

) ∣∣+ o(n),

=n

∑l=1

log∣∣ f ′ωl

(Π(σlω)

) ∣∣+ o(n),

= −Sn(χ)(ω) + o(n).

Note also that for all sufficiently large n and for all (τ1, · · · , τn) ∈ Dn, diam( fτ1 · · · fτn([0, 1])) ≤ ζn. It

follows that,

limn→∞

log rn

−Sn(χ)(ω)= lim

n→∞

log infz∈[0,1] | f ′ω|n(z)|eWn(ω)Rn(ω)a3

0

Sn(χ)(ω)

= limn→∞

−Sn(χ)(ω) + o(n) + Wn(ω)Rn(ω)

−Sn(χ)(ω)= 1.

Therefore,

lim infn→∞

log ν(B(x, rn))

log rn≥ lim inf

n→∞

log µ([ω|n])−Sn(χ)(ω)

.

To conclude the proof of the lemma we observe that

limn→∞

log rn+1

log rn= lim

n→∞

Sn+1(χ)(ω)

Sn(χ)(ω),

= limn→∞

Sn(χ)(ω) + log | f ′ωn+1(Π(σn+1ω))|

Sn(χ)(ω),

= limn→∞

Sn(χ)(ω) + O(Rn(ω))

Sn(χ)(ω)= 1.


We say that the digit set D is tall if there exists (i3, j3), (i4, j4) ∈ D with i3 6= i4. It follows that there exists

pairs i1−, i2−, i10, i20, i1+, i2+ ∈ A so that, for all x, y, z ∈ [0, 1] we have,

gi1− gi2−

(x) ≤ gi10 gi20

(y) ≤ gi1+ gi2+

(z). (4.45)

Define

Tn(τ) := min

l > n : il−1 = i10, il = i20− n. (4.46)

Lemma 4.20. Let µ be a finite Borel measure on Σv and let ν := µ Π−1v denote the corresponding projection. Suppose

D is tall. Then for all y = Πv(τ) ∈ Πv(Σv) with limn→∞ Tn(τ)n−1 = 0,

lim infr→0

log ν(B(y, r))log r

≥ lim infn→∞

− log µ([τ|n])Sn(ψ)(τ)

.

Proof. Proceed as in Lemma 4.19 with bmin := max|| − log g′i ||∞ : i ∈ A

in place of Wn(ω).

4.6 Convergence lemmas

Fix some finite subset D0 ⊆ D such that,

(i) If D is tall then there exists some j1, j2 ∈ N with (i10, j1), (i20, j1) ∈ D0, where i10, i20 ∈ A are as in the

definition of Tn(ω),

(ii) If D is wide then (i′, j10), (i′, j20) ∈ D0, where (i′, j10), (i

′, j20) ∈ D are as in the definition of Wn(ω).

We shall let B†σq(Σ) denote the set of µ ∈ Bσq(Σ) which satisfy, µ([ω1, · · · , ωq]) > 0 for all (ω1, · · · , ωq) ∈

Dq0.

Lemma 4.21. Given µ ∈ Mσ(Σ), ε > 0 and m ∈N we may obtain q ≥ m and ν ∈ B†

σq(Σ) satisfying,

(i)∣∣∣∣ hνπ−1(σ

qv )∫

Sq(ψ)dν π−1 −hµπ−1(σv)∫

ψdµ π−1

∣∣∣∣ < ε,

(ii)∣∣∣∣hν(σq|π−1A )∫

Sq(χ)dν−

hµ(σ|π−1A )∫χdµ

∣∣∣∣ < ε,

(iii)∣∣∣∣ ∫ Aq(ϕk)dν−

∫ϕkdµ

∣∣∣∣ < ε for all k ≤ m,

(iv) varn (An(ϕk)) <1m

, for all n ≥ q and all k ≤ m,

(v) max varn (An(χ)) , varn (An(ψ)) <1m

, for all n ≥ q(m).

4.6. Convergence lemmas 105

Proof. First not that since limq→∞ varq(

Aq(χ))= limq→∞ varq

(Aq(ψ)

)= 0 and limq→∞ varq

(Aq(ϕk)

)= 0 for

all k we may choose q0 ≥ m so that for n ≥ q0, max varn (An(χ)) , varn (An(ψ)) < 1m and varn (An(ϕk)) <

1m

for k ≤ m.

Fix µ ∈ Mσ(Σ). Given q ∈ N we let νq ∈ Bσq(Σ) denote the k-th level approximation of ν. That is, given a

cylinder [ω1 · · ·ωnq] of length nq we let

νq([ω1 · · ·ωnq]) :=n−1

∏l=0

ν([ωlq+1 · · ·ωlq+q]). (4.47)

By the Kolmogorov-Sinai theorem (see Theorem 2.39) we then have,

limq→∞

q−1hνq(σq) = hν(σ), (4.48)

limq→∞

q−1hνqπ−1(σqv ) = hνπ−1(σv). (4.49)

Combining these two limits and applying the Abramov Rohklin formula [AR] gives,

limq→∞

q−1hνq(σq|π−1A ) = hν(σ). (4.50)

Since µ and νq agree on cylinders of length q we have∣∣ ∫ Aq(ϕk)dνq−

∫Aq(ϕk)dµ

∣∣ ≤ varq Ak(ϕq), which tends

to zero with q by the tempered distortion property. Moreover,∫

Aq(ϕk)dµ =∫

ϕkdµ, since µ is σ-invariant.

Hence,

limk→∞

Aq(ϕk)dνq =∫

ϕdν, (4.51)

for all k ≤ m. The same argument also gives,

limk→∞

Aq(χ)dνq =∫

χdν, (4.52)

limk→∞

Aq(ψ)dνq π−1 =∫

ψdν π−1. (4.53)

Consequently, by taking q ≥ q0 sufficiently large, we may obtain q ∈ N and ν ∈ Bσq(Σ) satisfying (i), (ii) and

(iii) from the lemma. To obtain ν ∈ B0σq(Σ) satisfying (i), (ii) and (iii) we peturb ν slightly to obtain ν ∈ Bσq(Σ)

with νq([ω1 · · ·ωq]) > 0 for each (ω1, · · · , ωq) ∈ Dq0 whilst using continuity to insure that (i), (ii) and (iii) still

hold. Since q ≥ q0, (iv) and (v) also hold.

Recall that we defined A to be the Borel sigma algebra on Σv. Given any Borel probability measure ν ∈

M(Σ) and ω ∈ Σ we let νπ−1Aω denote the conditional measure at ω [EW, Section 5.3]. Since π−1A is countably

generated there exists Σ′ ⊆ Σ with ν(Σ′) = 1 such that for all ω1, ω2 ∈ Σ′ with τ = π(ω1) = π(ω2) we have

νπ−1Aω1 = νπ−1A

ω2 and νπ−1Aω1

(π−1τ

)= 1 [EW, Theorem 5.14]. It follows that we can take a family of measures

νττ∈Σv ⊂M(Σ) with ντ(π−1τ

)= 1 for all τ ∈ Σv and

ν =∫

ντdν π−1(τ). (4.54)

We shall make use of the Shannon-McMillan-Breimann Theorem 2.48.

Lemma 4.22. Given ν ∈ E†σq(Σ), the following convergences hold for ν almost every ω ∈ Σ,


(i) limn→∞ Anq(ψ)(τ) =∫

Aq(ψ)dν π−1,

(ii) limn→∞ Anq(χ)(ω) =∫

Aq(χ)dν,

(iii) limn→∞ Anq(ϕk)(ω) =∫

Aq(ϕk)dν for all k ≤ m,

(iv) limn→∞−n−1 log ν π−1([π(ω)1 · · ·π(ω)nq]) = hνπ−1(σqv ),

(v) limn→∞−n−1 log νπ(ω)([ω1 · · ·ωnq]) = hν(σq|π−1A ),

(vi) limn→∞ n−1Tn(τ) = 0, provided D is tall,

(vii) limn→∞ n−1Wn(ω) = 0, provided D is wide.

Proof. Limits (i)-(iii) follow from Birkhoff’s ergodic theorem. Indeed since σq is ergodic with respect to ν, σqv is

ergodic with respect to ν π−1.

If we let ξv denote the partition of Σv into cylinder sets of length q andN := Σv, ∅ denote the null sigma

algebra, then for each τ ∈ Σv we have,

Iνπ−1

(n−1∨l=0

σ−lqξv

∣∣∣∣ N)(τ) = log ν π−1([τ1 · · · τnq]).

Thus (i) follows from Lemma 2.48. Similarly if we let ξh denote the partition of Σ into cylinder sets of length q

then for each ω ∈ Σ we have,

Iν

(n−1∨l=0

σ−lqξh

∣∣∣∣π−1A

)(ω) = log νπ(ω)([ω1 · · ·ωnq]),

so (ii) also follows from Lemma 2.48.

For each (η1, · · · , ηq) ∈ Dq0 we have,

limn→∞

1n

#

l < n : (ωlq+1, · · · , ωlq+q) = (η1, · · · , ηq)

= ν([η1, · · · , ηq]) > 0.

Limits (vi) and (vii) follow.

Lemma 4.23. Given ν ∈ E†σq(Σ), supported on some compact set K, along with constants δ, ε > 0 and m ∈ N, there

exists N ∈N and U ⊆ Σv with ν π−1(U) > 1− δ such that for all τ ∈ U and all n ≥ N we have,

(i)∣∣∣∣ log ν π−1([τ1 · · · τnq])

Snq(ψ)(τ)+

hνπ−1(σqv )∫

Sq(ψ)dν π−1

∣∣∣∣ < ε,

(ii) Tn(τ) < nε, provided D is tall.

Moreover, for each τ ∈ U there exists Vτ ⊆ π−1τ ∩ K with ντ(Vτ) > 1− δ and for all ω ∈ Vτ and n ≥ N we have,

4.6. Convergence lemmas 107

(iii)∣∣∣∣ log ντ([ω1 · · ·ωnq])

Snq(χ)(ω)+

hν(σq|π−1A )∫Sq(χ)dν

∣∣∣∣ < ε,

(iv) Wn(ω) < nε, provided D is wide,

(v)∣∣∣∣Anq(ϕk)(ω)−

∫Aq(ϕk)dν

∣∣∣∣ < ε for all k ≤ m.

Proof. By Lemma 4.22 (i), (iv) and (vi) combined with Egorov’s theorem, there exists a set U′′ ⊂ Σv with

ν π−1(U′′) > 1− δ/2, such that for all τ ∈ U′′ and all n ≥ N′ we have,

(i)∣∣∣∣ log ν π−1([τ1 · · · τnq])

Snq(ψ)(τ)+

hνπ−1(σqv )∫

Sq(ψ)dν π−1

∣∣∣∣ < ε,

(ii) Tn(τ) < nε, provided D is tall.

By Lemma 4.22 (ii), (iii), (v) and (vii) we may take U′ ⊂ U′′ with ν π−1(U′) = ν π−1(U′′) such for all τ ∈ U′,

ντ is supported on π−1τ ∩ K and for ντ almost all ω ∈ π−1τ ∩ K we have,

(iii)’ limn→∞

log ντ([ω1 · · ·ωnq])

Snq(χ)(ω)=


,

(iv)’ limn→∞

n−1Wn(ω) = 0, provided D is wide,

(iv)’ limn→∞

Anq(ϕk)(ω) =∫

Aq(ϕk)dν for all k ≤ m.

Applying Egorov’s theorem once more, we obtain for each τ ∈ U′ a set Vτ ⊆ π−1τ ∩ K with ντ(Vτ) > 1− δ

and some N(τ) ≥ N′ such that for all ω ∈ Vτ and n ≥ N(τ) we have,

(iii)∣∣∣∣ log ντ([ω1 · · ·ωnq])

Snq(χ)(ω)+


∣∣∣∣ < ε,

(iv) Wn(ω) < nε, provided D is wide,

(iv)∣∣∣∣Anq(ϕk)(ω)−

∫Aq(ϕk)dν

∣∣∣∣ < ε for all k ≤ m.

Now choose U ⊆ U′ with ν π−1(U) > ν π−1(U′)− δ/2 > 1− δ for which,

N := max N(τ) : τ ∈ U < ∞.


4.7 Proof of the lower bound

Throughout the proof of the lower bound we shall fix some α = (αk)k∈N ⊂ R∪ ∞. We define,

δ(α) := limm→∞

sup

D(µ) : µ ∈ Mσ(Σ),


.

In this section we shall prove the following.

Proposition 4.24. dimH Jϕ(α) ≥ δ(α).

Clearly we may assume that δ(α) ∈ [0, 2]. Thus, by a simple compactness argument there exists δh(α), δv(α) ∈

R with δh(α) + δv(α) = δ(α), along with a sequence of measures µmm∈N ⊂Mσ(Σ) with

A(i)hµmπ−1(σv)∫ψdµm π−1 > δv(α)−

13m

,

A(ii)hµm(σ|π−1A )∫

χdµm> δh(α)−

13m

,

A(iii)∫

ϕkdµm ∈ B3m(αk) for k ≤ m.

Now choose δm > 0 for each m ∈N in such a way that ∏∞m=1(1− δm) > 0.

By Lemma 4.21, for each m ∈N, there exists q(m) ≥ m and νm ∈ B†σq(m)(Σ) satisfying,

B(i)hνmπ−1(σ

q(m)v )∫

Sq(m)(ψ)dνm π−1 > δv(α)−1

2m,

B(ii)hνm(σ

q(m)|π−1A )∫Sq(m)(χ)dνm

> δh(α)−1

2m,

B(iii)∫

Aq(m)(ϕk)dνm ∈ B2m(α) for all k ≤ m,

B(iv) varn (An(ϕk)) <1m

, for all n ≥ q(m) and k ≤ m,

B(v) varn (An(χ)) , varn (An(ψ)) <1m

, for all n ≥ q(m).

Since νm ∈ B†σq(m)(Σ) is compactly supported there is a finite digit set Dm ⊂ D such that νm is supported on

DNm . We define,

A(m) := sup(− log inf

x∈[0,1]| fd| : d ∈ Dm

∪|var1(ϕk)(ω)| : ω1, · · · , ωq(m) ∈ Dm

).

Note that A(m) is finite since Dm is finite and var1(ϕk) is finite for all k ≤ m.

For each m we define, A(m) := ∏m+1l=1 A(l) + 1.

By Lemma 4.23, for each m ∈ N we may take N(m) ∈ N and U(m) ⊆ Σv with νm π−1(U(m)) > 1− δm

such that for all τ ∈ U(m) and all n ≥ N(m) we have,

4.7. Proof of the lower bound 109

C(i)− log νm π−1([τ1 · · · τnq])

Snq(ψ)(τ)> δv(α)−

1m

,

C(ii)Tn(τ)

n<

1m

, provided D is tall.

Moreover, for each τ ∈ U(m) there exists Vτ(m) ⊆ π−1τ ∩ DNm with ντ

m(Vτ(m)) > 1 − δm and for all

ω ∈ Vτ(m) and n ≥ N(m) we have,

C(iii)− log ντ

m([ω1 · · ·ωnq])

Snq(χ)(ω)> δh(α)−

1m

,

C(iv) A(m)Wn(ω)

n<

1m

, provided D is wide,

C(v) Anq(ϕk)(ω) ∈ Bm(αk) for all k ≤ m.

We now define a rapidly increasing sequence (γm)m∈N∪0 of natural numbers by γ0 = 2N(1), γ1 = 2N(2)

and for m > 1 we let

γm := (m + 1)! · γm−1

(m+1

∏l=1

N(l)

)(m+1

∏l=1

A(l)

)(m+1

∏l=1

q(l)

)+ γm−1. (4.55)

We now define a measureW on Σv by first definingW on a semi-algebra of cylinders and then extending

W to a Borel probability measure on Σv via the Daniell-Kolmogorov consistency theorem (see 2.5). Given a

cylinder [τ1 · · · τγM ] of length γM for some M ∈N we define

W([τ1 · · · τγM ]) :=M

∏m=1

νm π−1([τγm−1+1 · · · τγm ]). (4.56)

Define U ⊆ Σv by

U :=∞⋂

m=1

τ ∈ Σv : [τγm−1+1 · · · τγm ] ∩U(m) 6= ∅

. (4.57)

For each τ ∈ U and m ∈N we choose τm ∈ [τγm−1+1 · · · τγm ] ∩U(m) and define a measure Zτ on Σ by

Zτ([ω1 · · ·ωγM ]) :=M

∏m=1

ντm

m ([ωγm−1+1 · · ·ωγm ]). (4.58)

Lemma 4.25. For all τ ∈ U we have Zτ(π−1τ

)= 1.

Proof. For each m ∈N we have,

ντm

m

(π−1[τγm−1+1 · · · τγm ]

)= ντm

m

(π−1τm

)= 1. (4.59)

Hence, for each M ∈N we have

Zτ(π−1[τ1 · · · τγM ]) =M

∏m=1

ντm

m

(π−1[τγm−1+1 · · · τγm ]

)= 1. (4.60)

The lemma follows.


For each τ ∈ U we define Vτ ⊆ π−1τ by

Vτ :=∞⋂

m=1

ω ∈ Σ : [ωγm−1+1 · · ·ωγm ] ∩Vτ(m) 6= ∅

. (4.61)

We also define,

S :=

ω ∈ Σ : π(ω) ∈ U and ω ∈ Vπ(ω)

. (4.62)

To complete the proof of the lower bound in theorem 4.6 shall show that S ⊆ Eϕ(α) and dimHΠ(S) ≥ δ(α).

Lemma 4.26. S ⊆ Eϕ(α).

Proof. Note that it suffices to take ω ∈ Σ with π(ω) ∈ U and ω ∈ Vπ(ω) and show that for each k ∈N we have

limn→∞ An(ϕk)(ω) = αk.

For each m ∈N we choose ωm ∈ [ωγm−1 · · ·ωγm ] ∩Vτ(m).

Given n ∈N we choose m(n) ∈N so that γm(n) ≤ n < γm(n)+1 and l(n) ∈N so that γm(n) + l(n)q(m(n) +

1) ≤ n < γm(n) + (l(n) + 1)q(m(n) + 1). Since limn→∞ m(n) = ∞ we may choose n(k) ∈ N so that m(n) ≥ k

for all n ≥ n(k).

First lets suppose that αk ∈ R. Given n ≥ n(k), either l(n) ≤ N(m(n) + 1), in which case

∣∣Sn(ϕk)(ω)− nαk∣∣

≤∣∣Sγm(n)−γm(n)−1(ϕk)(σ

γm(n)−1 ω)− (γm(n) − γm(n)−1)αk∣∣

+γm(n)−1 A(m(n)− 1) + (N(m(n) + 1) + 1)q(m(n) + 1)A(m(n) + 1)


γm(n)−1 ω)− (γm(n) − γm(n)−1)αk∣∣+ 2γm(n)

m(n)

≤∣∣Sγm(n)−γm(n)−1(ϕk)(ω

m(n))− (γm(n) − γm(n)−1)αk∣∣

+(

γm(n) − γm(n)−1

)varγm(n)−γm(n)−1

(Aγm(n)−γm(n)−1(ϕk)

)+

2nm(n)

≤2(γm(n) − γm(n)−1)

m(n)+

2nm(n)

≤ 4nm(n)

.

On the other hand, if l(n) > N(m(n) + 1) then we have,

∣∣Sn(ϕk)(ω)− nαk∣∣



γm(n)−1 ω)− (γm(n) − γm(n)−1)αk∣∣

+∣∣Sl(n)q(m(n)+1)−γm(n)

(ϕk)(σγm(n)ω)− (l(n)q(m(n) + 1)− γm(n))αk

∣∣+γm(n)−1 A(m(n)− 1) + q(m(n) + 1)A(m(n) + 1)


γm(n)−1 ω)− (γm(n) − γm(n)−1)αk∣∣

+∣∣Sl(n)q(m(n)+1)−γm(n)

(ϕk)(σγm(n)ω)− (l(n)q(m(n) + 1)− γm(n))αk

∣∣+ 2γm(n)

m(n)

≤∣∣Sγm(n)−γm(n)−1(ϕk)(ω

m(n))− (γm(n) − γm(n)−1)αk∣∣

+∣∣Sl(n)q(m(n)+1)−γm(n)

(ϕk)(ωm(n)+1)− (l(n)q(m(n) + 1)− γm(n))αk

∣∣+(γm(n) − γm(n)−1)varγm(n)−γm(n)−1 Aγm(n)−γm(n)−1(ϕk)

+(l(n)q(m(n) + 1)− γm(n))varl(n)q(m(n)+1)−γm(n)Al(n)q(m(n)+1)−γm(n)

(ϕk) +2γm(n)

m(n)

≤2(γm(n) − γm(n)−1)

m(n)+

2(l(n)q(m(n) + 1)− γm(n))

m(n)+

2γm(n)

m(n)≤ 6n

m(n).

Thus, for all n ≥ n(k) we have, ∣∣An(ϕk)(ω)− αk∣∣ ≤ 6

m(n).

Since limn→∞ m(n) = ∞ the lemma holds when αk is finite.

Now suppose that αk = ∞. Given n ≥ n(k), either l(n) ≤ N(m(n) + 1), in which case,

Sn(ϕk)(ω) ≥ Sγm(n)−γm(n)−1(ϕk)(σγm(n)−1 ω)

≥ Sγm(n)−γm(n)−1(ϕk)(ˆωm(n))

−(


)varγm(n)−γm(n)−1

(Aγm(n)−γm(n)−1(ϕk)

)≥ (γm(n) − γm(n)−1)m(n)−


m(n)

≥ nm(n)− (γm(n)−1 + l(n)q(m(n) + 1))m(n)− nm(n)

≥ nm(n)−2γm(n)

m(n)− n

m(n)

≥ nm(n)− 3nm(n)

.

On the other hand, if l(n) > N(m(n) + 1) then we have,

Sn(ϕk)(ω) ≥ Sγm(n)−γm(n)−1(ϕk)(σγm(n)−1 ω) + Sl(n)q(m(n)+1)−γm(n)

(ϕk)(σγm(n)ω)

≥ Sγm(n)−γm(n)−1(ϕk)(ωm(n)) + Sl(n)q(m(n)+1)−γm(n)

(ϕk)(ωm(n)+1)

−(γm(n) − γm(n)−1)varγm(n)−γm(n)−1 Aγm(n)−γm(n)−1(ϕk)

−(l(n)q(m(n) + 1)− γm(n))varl(n)q(m(n)+1)−γm(n)Al(n)q(m(n)+1)−γm(n)

(ϕk)

≥(


)m(n) +

(l(n)q(m(n) + 1)− γm(n)

)m(n)− 2n

m(n)

≥ nm(n)− 4nm(n)

.

Thus, for all n ≥ n(k) we have,

An(ϕk)(ω) ≥ m(n)− 4m(n)

.


Letting n→ ∞ proves the lemma.

Lemma 4.27. W(U) > 0 and for each τ ∈ U, Zτ (Vτ) > 0.

Proof.

W(U) ≥∞

∏m=1

νm π−1(U(m)) >∞

∏m=1

(1− δm) > 0.

Similarly for each τ ∈ U we have,

Zτ(Vτ) ≥∞

∏m=1

ντm

m (Vτ(m)) >∞

∏m=1

(1− δm) > 0.

Lemma 4.28. For all τ ∈ U and all ω ∈ Vτ we have,

(i) lim infn→∞

− logW([τ1 · · · τn])

Sn(ψ)(τ)≥ δv(α),

(ii) lim infn→∞

− log Zτ([ω1 · · ·ωn])

Sn(χ)(ω)≥ δh(α).

Proof. We prove (ii). The proof of (i) is similar. Take τ ∈ U and ω ∈ Vτ . Given n ∈ N we choose m(n) ∈ N so

that γm(n) ≤ n < γm(n)+1 and l(n) ∈N so that γm(n) + l(n)q(m(n) + 1) ≤ n < γm(n) + (l(n) + 1)q(m(n) + 1).

If l(n) ≤ N(m(n) + 1) then by C(iii) we have,

− log Zτ([ω1 · · ·ωn]) ≥ − log ντm

m

([ωγm(n)−1+1 · · ·ωγm(n) ]

)≥ Sγm(n)−γm(n)−1(χ)(ω

m(n))

(δh(α)−

1m(n)

)where

ωm(n) ∈ [ωγm(n)−1+1 · · ·ωγm(n) ] ∩Vπ(ω)(m).

Moreover, using B(v) combined with γm(n) ≤ n we have,

Sγm(n)−γm(n)−1(χ)(ωm(n))

≥ Sγm(n)−γm(n)−1(χ)(σγm(n)−1 ω)− (γm(n) − γm(n)−1)varγm(n)−γm(n)−1 Aγm(n)−γm(n)−1(χ)

≥ Sγm(n)−γm(n)−1(χ)(σγm(n)−1 ω)−


m(n)

≥ Sn(χ)(ω)− nm(n)

.

On the other hand, if l(n) > N(m(n) + 1) then by C(iii) we have

− log Zτ([ω1 · · ·ωn])

≥ − log ντm

m

([ωγm(n)−1+1 · · ·ωγm(n) ]

)− log ντm

m

([ωγm(n)+1 · · ·ωγm(n)+l(n)q(m(n)+1)]

),

≥(

Sγm(n)−γm(n)−1(χ)(ωm(n)) + Sl(n)q(m(n)+1)(χ)(ω

m(n)+1))(

δh(α)−1

m(n)

),


ωm(n) ∈ [ωγm(n)−1+1 · · ·ωγm(n) ] ∩Vπ(ω)(m(n)),

ωm(n)+1 ∈ [ωγm(n)+1 · · ·ωγm(n+1) ] ∩Vπ(ω)(m(n) + 1).

As before, using B(v) combined with γm(n) + l(n)q(m(n) + 1) ≤ n we have,

Sγm(n)−γm(n)−1(χ)(ωm(n)) ≥ Sγm(n)−γm(n)−1(χ)(σ

γm(n)−1 ω)− nm(n)

Sl(n)q(m(n)+1)(χ)(ωm(n)+1) ≥ Sl(n)q(m(n)+1)(χ)(σ

γm(n)ω)− nm(n)

.

It follows that,

Sγm(n)−γm(n)−1(χ)(ωm(n)) + Sl(n)q(m(n)+1)(χ)(ω

m(n)+1)

≥ Sγm(n)−γm(n)−1(χ)(σγm(n)−1 ω) + Sl(n)q(m(n)+1)(χ)(σ

γm(n)ω)− 2nm(n)

≥ Sn(χ)(ω)− 2nm(n)

.

Thus, for all n ∈N we have,

− log Zτ([ω1 · · ·ωn]) ≥(

Sn(χ)(ω)− 4nm(n)

)(δh(α)−

1m(n)

).

Since lim infn→∞ n−1Sn(χ)(ω) ≥ ξ > 0 and limn→∞ m(n) = ∞, the lemma holds.

Lemma 4.29. For all τ ∈ U and all ω ∈ Vτ we have,

(i) limn→∞ n−1Rn(ω) = 0, provided D is wide,

(ii) limn→∞ n−1Rn(ω)Wn(ω) = 0, provided D is wide,

(iii) limn→∞ n−1Tn(τ) = 0, provided D is tall.

Proof. We shall prove (i) and (ii) simultaneously. The proof of (iii) is similar to that of (ii).

Suppose that D is wide and take ω ∈ Vτ with τ ∈ U. Given n ∈ N we choose m(n) to be the maximal

natural number with γm(n)−1 + N(m(n)) ≤ n. Now suppose n(1 + (A(m(n))m(n))−1) < γm(n). Then we may

choose ωm(n) ∈ [ωγm(n)−1+1 · · ·ωγm(n) ] ∩Vτ(m(n)). It follows from C(iv) that,

Wn−γm(n)−1(ωm(n)) <

n− γm(n)−1

A(m(n))m(n)(4.63)

<n

A(m(n))m(n)(4.64)

≤ γm(n) − n. (4.65)

It follows that n + Wn(ω) ≤ γm(n) and hence,

Rn(ω) ≤ maxl≤m(n)

A(l) ≤ A(m(n)) <2γm(n)−1

m(n)− 1<

2nm(n)− 1

.

Also,

Rn(ω)Wn(ω) ≤ A(m(n))n− γm(n)−1

A(m(n))m(n)<

nm(n)

.


On the other hand, suppose that n(1 + (A(m(n))m(n))−1) ≥ γm(n). In particular γm(n) ≤ 2n.

By C(iv) we have,

WN(m(n)+1)(ωm(n)+1) <

N(m(n) + 1)A(m(n) + 1)(m(n) + 1)

≤ γm(n)+1 − γm(n)+1.

Thus, there is some l ∈

n + 1, · · · , n(1 + (A(m(n))m(n))−1)

with ωl = (i′, j10) and ωl+1 = (i′, j20). It follows

that,

n +Wn(ω) ≤ γm(n) + N(m(n) + 1)(

A(m(n) + 1)(m(n) + 1))−1

< γm(n)+1.

Hence,

Rn(ω) ≤ maxl≤m(n)+1

A(l) ≤ A(m(n) + 1)

<2γm(n)

m(n)<

4nm(n)

.

In addition, we have,

Wn(ω) ≤ γm(n) +N(m(n) + 1)

A(m(n) + 1)(m(n) + 1)− n ≤ n + N(m(n) + 1)

A(m(n))m(n).

Hence,

Rn(ω)Wn(ω) ≤ n + N(m(n) + 1)m(n)

≤n + γm(n)

m(n)≤ 3n

m(n).

Thus, for all n ∈N we have,

max

Rn(ω)

n,

Rn(ω)Wn(ω)

n

≤ 4

m(n).

Letting n→ ∞, and hence m(n)→ ∞, proves the lemma.

To complete the proof of the lower bound we apply Proposition 2.10.

Lemma 4.30. dimHΠ(S) ≥ δ(α).

Proof. Recall that, S :=

ω ∈ Σ : π(ω) ∈ U and ω ∈ Vπ(ω)

. It follows that, Π(S) =

⋃τ∈U Π(Vτ). Thus, for

each y = Πv(τ) ∈ Πv(U) with τ ∈ U we have Π(S) ∩ (R× y) = Π(Vτ), since Vτ ⊆ π−1τ. Hence, by

Lemma 2.10 it suffices to prove that dimHΠv(U) ≥ δv(α) and for each τ ∈ U we have dimHΠ(Vτ) ≥ δh(α).

To see that dimHΠv(U) ≥ δv(α) we consider two cases. Either δv(α) = 0, in which case the supposition

is trivial since U 6= ∅ by Lemma 4.27, or δv(α) > 0. It follows from A(i) that for some µ ∈ Mσ(Σ) we

have hµπ−1(σv) > 0. Consequently D must be tall. Thus, by Lemma 4.29 the hypotheses of Lemma 4.20 are

satisfied, and so by Lemma 4.20 combined with Lemma 4.28 (i) for all y = Πv(τ) ∈ Π(U) we have,

lim infr→0

logW Π−1v (B(y, r))

log r≥ lim inf

n→∞

− logW([τ|n])Sn(ψ)(τ)

≥ δv(α).

Since, by Lemma 4.27,W Π−1v (Πv(U)) ≥ W(U) > 0, by Proposition 2.3 we have dimHΠv(U) ≥ δv(α).


Now fix τ ∈ U. To show that dimHΠ(Vτ) ≥ δh(α) we proceed similarly. If δh(α) = 0 then by Lemma 4.27

Vτ 6= ∅ and so the supposition is trivial. If on the other hand δh(α) > 0 then by A(ii) we have hµ(σ|π−1A ) > 0

for some µ ∈ Mσ(Σ) and consequently D must be wide. Thus, by Lemma 4.29 the hypotheses of Lemma 4.19

are satisfied, and so by Lemma 4.19 combined with Lemma 4.28 (ii) for all x = Π(ω) ∈ Π(Vτ) we have,

lim infr→0

log Zτ Π−1(B(x, r))log r

≥ lim infn→∞

− log Zτ([ω|n])Sn(χ)(ω)

≥ δh(α).

Again, by Lemma 4.27, Zτ Π−1(Π(Vτ)) ≥ W(Vτ) > 0, by Proposition 2.3 we have dimHΠ(Vτ) ≥ δh(α).

Thus, by Proposition 2.10 the lemma holds.

To complete the proof of the lower bound we note that by Lemma 4.26 Π(S) ⊆ Jϕ(α). Therefore, by Lemma

4.30 we have,

dimH Jϕ(α) ≥ limm→∞

sup

D(µ) : µ ∈ Mσ(Σ),


.


5

Multifractal analysis for typical infinitely

generated self-affine sets

5.1 Introduction

In this chapter we shall investigate the dimension theory of “typical” infinitely generated self-affine limit sets

in the plane in the tradition of Falconer [F1]. We refer the reader to Section 2.8 for an introduction to the theory

of self-affine sets.

We shall show that the dimension of a typical infinitely generated self-affine planar limit set is the infimal

value for which the pressure function passes below zero which is equal to the supremum of the dimensions of

its finitely generated subsystems. In addition we show that if the pressure function has a root then there exists

an ergodic measure of full Hausdorff dimension. We also consider the multifractal analysis for a countable

infinity of potentials on a typical infinitely generated self-affine limit set.

All of our results are underpinned by an investigation into the thermodynamic formalism of quasi-multiplicative

potentials on a symbolic space with a countable alphabet. In order to apply the thermodynamic formalism we

will require the associated singular value function to be quasi-multiplicative. This in turn requires a certain

irreducibility constraint which holds generically in the planar setting.

The results of this chapter are part of a joint project with Antti Kaenmaki.

The remainder of this chapter is organized as follows. In Section 5.2, we describe and motivate our results,

and in Section 5.3–5.6, we provide the reader with all the necessary details.


5.2.1 Thermodynamic formalism for sub-multiplicative potentials

Define Σ = NN to be the set of all infinite words constructed from the integers. Let Σn = Nn for all n ∈N and

Σ∗ =⋃

n∈N Σn be the collection of all finite word. If ω ∈ Σ∗ and τ ∈ Σ∗ ∪Σ, then ωτ denotes the concatenation

of ω and τ. Furthermore, if ω ∈ Σ∗ ∪ Σ and n ∈N, then ω|n is the unique word in Σn for which there is τ ∈ Σ

117

118 Chapter 5. Multifractal analysis for typical infinitely generated self-affine sets

so that ω|nτ = ω. If ω, τ ∈ Σ∗ ∪ Σ, then by ω ∧ τ we mean the common beginning of ω and τ. Given n ∈ N

and ω ∈ Σn we set |ω| = n and define the cylinder set given by ω to be [ω] = ωτ : τ ∈ Σ. We denote the

left shift operator by σ and letMσ(Σ) be the set of all σ-invariant Borel probability measures on Σ.

We equip Σ with the discrete topology and call it a shift space. If the shift space is constructed by using a

finite alphabet, i.e. Σ = IN for some finite set I ⊂ N, then we say that the shift space is finitely generated. The

shift space is compact if and only if it is finitely generated. Moreover, the cylinder sets are open and closed

and they generate the Borel σ-algebra.

We shall consider maps ϕ : Σ∗ → (0, ∞). We refer to such maps as potentials. We say that a potential ϕ is

sub-multiplicative if

ϕ(ωτ) ≤ ϕ(ω)ϕ(τ).

for all ω, τ ∈ Σ∗. A sub-multiplicative ϕ potential is said to be quasi-multiplicative if there exist a constant c ≥ 1

and a finite subset Γ ⊂ Σ∗ such that for any given pair ω, τ ∈ Σ∗ there exists κ ∈ Γ with

ϕ(ω)ϕ(τ) ≤ cϕ(ωκτ). (5.1)

We also define K = max|ω| : ω ∈ Γ+ 1. A sub-multiplicative ϕ potential is said to be almost-multiplicative if

there exists a constant c > 0 such that

ϕ(ω)ϕ(τ) ≤ cϕ(ωτ).

for all ω, τ ∈ Σ∗. We note that quasi-multiplicativity is significantly less restrictive than the conditon of almost-

multiplicativity which also appears in the literature; see e.g. Iommi and Yayama [IY].

If ϕ is a sub-multiplicative potential, then we define the pressure P(ϕ) by setting

P(ϕ) = limn→∞

1n log Zn(ϕ) = inf

n∈N

1n log Zn(ϕ),

where Zn(ϕ) = ∑ω∈Σn ϕ(ω) for all n ∈ N. Note that by the sub-multiplicativity, the pressure is well-defined,

although it may not be finite. It is immediate that P(ϕ) = ∞ if and only if Zn(ϕ) = ∞ for all n ∈N. Thus, if the

shift space is finitely generated, then P(ϕ) < ∞. Observe that even if the shift space is finitely generated, the

pressure can be negative infinity. Let ψ : Σ∗ → (0, ∞) be a sub-multiplicative potential so that P(ψ) < ∞ and

Zn+m(ψ) ≥ cZn(ψ)Zm(ψ) for some constant c > 0. If the shift space is finitely generated, then the potential

ψ ≡ 1 satisfies these assumptions. Now defining ϕ : Σ∗ → (0, ∞) by setting ϕ(ω) = (cZn(ψ)n!)−1ψ(ω) for all

ω ∈ Σ∗, it is easy to see that ϕ is sub-multiplicative with P(ϕ) = − limn→∞1n log n! = −∞.

We letMσ(Σ) denote the set of all σ-invariant Borel probability measures on Σ. Given µ ∈ Mσ(Σ) along

with a sub-multiplicative potential ϕ, we define the measure-theoretical pressure Pµ(ϕ) by setting

Pµ(ϕ) = infn∈N

1n ∑

ω∈Σn

µ([ω]) logϕ(ω)

µ([ω]). (5.2)

We adopt the usual convention according to which 0 log(x/0) = 0 log 0 = 0 for all x > 0.

Lemma 5.1. If ϕ is a sub-multiplicative potential and µ ∈ Mσ(Σ), then

Pµ(ϕ) = limn→∞

1n ∑

ω∈Σn

µ([ω]) logϕ(ω)

µ([ω]).


Proof. The proof follows from the standard theory of sub-additive sequences by the sub-multiplicativity of ϕ,

the concavity of the function H(x) = −x log x, and the invariance of µ.

Furthermore, we define the Lyapunov exponent for ϕ and the entropy of µ by setting

Λµ(ϕ) = limn→∞

1n ∑

ω∈Σn

µ([ω]) log ϕ(ω) = infn∈N

1n ∑

ω∈Σn

µ([ω]) log ϕ(ω) ≤ log ‖ϕ‖,

hµ = limn→∞

1n ∑

ω∈Σn

−µ([ω]) log µ([ω]) = infn∈N

1n ∑

ω∈Σn

−µ([ω]) log µ([ω]) ≥ 0,(5.3)

respectively. Similarly as in the proof of Lemma 5.1, we see that the Lyapunov exponent and the entropy are

well-defined by the sub-multiplicativity of ϕ and the invariance of µ.

Lemma 5.2. If ϕ is a sub-multiplicative potential, then

P(ϕ) ≥ Pµ(ϕ)

for all µ ∈ Mσ(Σ). Furthermore, if hµ < ∞ or Λµ(ϕ) is finite, then Pµ(ϕ) = hµ + Λµ(ϕ).

Proof. To show the first claim, we may assume that Pµ(ϕ) > −∞ and P(ϕ) < ∞. Thus ∑ω∈Σn µ([ω]) log ϕ(ω)/µ([ω]) >

−∞ for all n ∈ N and there is n0 ∈ N so that Zn(ϕ) < ∞ for all n ≥ n0. For each n ≥ n0 and Cn ⊂ Σn we use

the concavity of the function H(x) = −x log x to obtain

∑ω∈Cn

µ([ω])

(log

ϕ(ω)

µ([ω])− log ∑

ω∈Cn

ϕ(ω)

)= ∑

ω∈Cn

β(ω)H(µ([ω])/β(ω)

)≤ H

(∑

ω∈Cn

β(ω)µ([ω])/β(ω)

)∈ [0, 1

e ],(5.4)

where β(ω) = ϕ(ω)/ ∑ω∈Cn ϕ(ω). Dividing by n before letting n→ ∞ proves the first claim.

To show the second claim, we first assume that Λµ(ϕ) is finite. Notice first that if hµ < ∞, then also

Pµ(ϕ) = hµ + Λµ(ϕ) is finite. On the other hand, if Pµ(ϕ) < ∞, then there is n0 ∈N so that

−∞ < Λµ(ϕ) ≤ 1n ∑

ω∈Σn

µ([ω]) log ϕ(ω) and 1n ∑

ω∈Σn

µ([ω]) logϕ(ω)

µ([ω])≤ Pµ(ϕ) + 1 < ∞

for all n ≥ n0. Thus

1n ∑

ω∈Σn

−µ([ω]) log µ([ω]) ≤ Pµ(ϕ)−Λµ(ϕ) + 1

for all n ≥ n0 and hµ < ∞. Therefore, if hµ = ∞, then Pµ(ϕ) = ∞ and the desired equality holds.

Finally, we notice that the proof of the second claim in the case hµ < ∞ is similar.

Our first main result is the following variational principle. The proof of the result can be found in the end

of Section 5.3.2.

Theorem 5.3. If ϕ is a quasi-multiplicative potential, then

P(ϕ) = supPµ(ϕ) : µ ∈ Mσ(Σ).

Moreover, if P(ϕ) < ∞, then there exists a unique invariant measure µ for which P(ϕ) = Pµ(ϕ).


If the shift space is finitely generated, then we always have Pµ(ϕ) = hµ + Λµ(ϕ). Moreover, the variational

principle holds for all sub-multiplicative potentials; see Kaenmaki [K, Theorem 2.6] and Cao, Feng, and Huang

[CFH, Theorem 1.1]. Quasi-multiplicativity has been a crucial property in the study of Lyapunov exponents

for products of matrices; see e.g. Feng and Lau [FLW], Feng [FLWWJ], and Feng and Kaenmaki [FK]. It has

also been used in connection with finitely generated self-affine sets; see Feng [Fe3] and Falconer and Sloan

[FS]. Finally, we remark that in the infinitely generated setting, Iommi and Yayama [IY, Theorem 3.1] have

recently verified the variational principle for almost-multiplicative potentials.

5.2.2 Infinitely generated self-affine sets

Let (Ti)i∈N ∈ GLd(R)N be such that supi∈N ‖Ti‖ < 1. Define A = ([0, 1]d)N and note that by the Kol-

mogorov extension theorem A supports a natural probability measure LA = (Ld|[0,1]d)N. To each sequence

a = (ai)i∈N ∈ A we associate a projection πa : Σ→ Rd defined by

πa(ω) =∞

∑j=1

Tω|j−1aj.

Here Tω = Tω1 · · · Tωn for all ω = ω1 · · ·ωn ∈ Σn and n ∈N. The set F = Fa = πa (Σ) is termed self-affine.

The dimension theory of self-affine sets of this form was first investigated in the finitely generated setting

by Falconer [F1]. A central tool in Falconer’s analysis was the singular function ϕs. Given a matrix T ∈ GLd(R)

we let 1 > γ1(T) ≥ · · · ≥ γd(T) > 0 denote the singular values of T (the square roots of the eigenvalues of

T∗T), in non-increasing order of magnitude. Thus γ1(T) = ‖T‖ and γd(T) = ‖T−1‖−1. If 0 ≤ s = m + δ ≤ d

with m ∈ Z and 0 < δ ≤ 1, then we define the singular value function to be

ϕs(T) = γ1(M) · · · γm(T)γm+1(T)δ.

When s ≥ d, we set ϕs(t) = |det(T)|s/d for completeness. Given (Ti)i∈N ∈ GLd(R)N the singular value

function introduces a potential by setting

ϕs(ω) = ϕs(Tω) (5.5)

for all ω ∈ Σ∗. Note that ‖ϕs‖ ≤ 1 for all 0 ≤ s ≤ d. Also singular values γi introduce potentials in a similar

way. For example, if s ≥ 0, then γs1 is the sub-multiplicative potential ω 7→ ‖Tω‖s.

Falconer [F1, Lemma 2.1] showed that the singular value function is ϕs is sub-multiplicative. It follows that

the corresponding sub-multiplicative pressure P(ϕs) is well-defined. Following the proof of [KV, Lemma 2.1],

we see that the function s 7→ P(ϕs) is strictly decreasing and thus finite on an interval I of [0, ∞). Furthermore,

it is convex on connected components of I \ 1, . . . , d. Note that also the functions s 7→ Pµ(ϕs) and s 7→ Λµ(ϕs)

are strictly decreasing and continuous for all µ ∈ Mσ(Σ).

Falconer [F1, Theorem 5.3] proved that given finitely many affine contractions with contraction ratios at

most 13 the Hausdorff dimension of the corresponding self-affine set Fa is given by the unique zero of s 7→ P(ϕs)

for almost every translation vector a. Later Solomyak extended Falconer’s proof to self-affine sets with the

contraction ratios up to 12 ; see [S, Proposition 3.1(i)]. See Kaenmaki [K, Theorem 4.5] and Jordan, Pollicott


and Simon [JPS, Theorem 1.7] for corresponding results for measures. In order to extend Falconer’s result to

infinitely generated self-affine sets we have to assume that the singular value function is quasi-multiplicative.

Proposition 5.4. The singular value function ϕs is quasi-multiplicative for all 0 ≤ s ≤ d if (Ti)i∈N ∈ GLd(R)N

satisfies one of the following conditions:

(1) Suppose that d = 2 and for every line ` ∈ R2 there is i ∈N with Ti(`) 6= `.

(2) Suppose that d = 2 and the matrices Ti have strictly positive entries so that the ratio of the smallest and largest

entry of Ti is uniformly bounded away from zero for all i ∈N.

(3) Suppose that d ∈N and Ti = diag(ti1, . . . , ti

d), where 1 > |ti1| > · · · > |ti

d| > 0 for all i ∈N.

Proof. Assuming (1), [Fe2, Proposition 2.8] shows that the potential γ1 is quasi-multiplicative. Observe that

the proof of [Fe2, Proposition 2.8] applies verbatim in the infinite case. Similarly, assuming (2), [IY, Lemma

7.1] shows that γ1 is quasi-multiplicative. The claim in both of these cases follows now by recalling that the

determinant is the product of singular values. Finally, assuming (3), the quasi-multiplicativity of the singular

value is immediate.

Remark

1 The assumption (1) in Proposition 5.4 is equivalent to the property that the matrices do not have a com-

mon eigenvector. Thus, if the 2× 2 matrices cannot simultaneously be presented (in some coordinate

system) as upper triangular matrices, then the singular value function ϕs is quasi-multiplicative for all

0 ≤ s ≤ d.

2 The set of (Ti)i∈N ∈ GL2(R)N satisfying the assumption (1) in Proposition 5.4 is open and dense set

under the product topology. Indeed the set of pairs (T1, T2) ∈ GL2(R)2 for which there is no common

eigenvector is easily seen to be an open and dense set of full Lebesgue measure.

3 Falconer and Sloan [FS] have introduced a certain condition under which the singular value function is

quasi-multiplicative also in higher dimensions; see [FS, Corollary 2.3].

Example Two strictly positive 2× 2 matrices having a common eigenvector show that strict positivity does

not imply irreducibility. Furthermore, if

T1 =

10 0

0 1

, T2 =

0 −1

10 11

,

then (T1, T2) is irreducible, but it is easy to see that there is no coordinate system in which the matrices are

simultaneously strictly positive.

In our second main theorem, we generalize Falconer’s dimension result to infinitely generated self-affine

sets. The proof of the result can be found in Section 5.4.


Theorem 5.5. If (Ti)i∈N ∈ GLd(R)N satisfies supi∈N ‖Ti‖ < 12 and the singular value function ϕs is quasi-

multiplicative for all 0 ≤ s ≤ d, then

dimH(Fa) = mind, infs : P(ϕs) ≤ 0 = supdimH(πa(IN)) : I ⊂N is finite

for LA-almost all a ∈ A.

5.2.3 Multifractal analysis of Birkhoff averages

We shall consider Birkhoff averages of functions Φ : Σ → RN. The vector space RN is endowed with the

product topology, so a sequence (α(n))n∈N with α(n) = (αi(n))i∈N ∈ RN converges to α = (αi)i∈N if

limn→∞ αi(n) = αi for each i ∈N. Given a function φ : Σ→ R we define the variation varnφ for each n ∈N by

varnφ = sup|φ(ω)− φ(τ)| : [ω|n] = [τ|n].

A function φ : Σ→ R is said to have summable variations if ∑∞n=1 varnφ < ∞.

We take a sequence Φ = (φi)i∈N of functions φi : Σ → R, each with summable variations, which we think

of as a function from Σ to RN. We define the Birkhoff sum for each n ∈N by

SnΦ =n−1

∑j=0

Φ σj

and the Birkhoff average by AnΦ = n−1SnΦ. We define Snφ and Anφ similarly when φ : Σ → Rk for some

k ∈N. We let the symbolic level set to be

EΦ(α) = ω ∈ Σ : limn→∞

AnΦ(ω) = α

for all α = (αi)i∈N ∈ RN, where R = R∪ −∞,+∞.

Suppose we have a self-affine set Fa, that is, (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 12 , a =

(ai)i∈N ∈ A, and Fa = πa(Σ) is the projection of the shift space. Let us denote the affine maps x 7→ Tix + ai

by fi. If the sequence a is such that there is a compact set X ⊂ Rd so that fi(X) ⊂ X for all i ∈ N and

fi(X) ∩ f j(X) = ∅ for i 6= j, then the projection πa gives a conjugacy between the left shift σ : Σ → Σ and the

well-defined map g :⋃

i∈N fi(X) → X for which g(x) = f−1i (x) = T−1

i x − ai for x ∈ fi(X). This leads us to

consider the projections of symbolic level sets,

JaΦ(α) = πa(EΦ(α)),

for as many a ∈ A as possible.

Next we state our main results concerning multifractal formalism in this paper. For each k ∈ N we let

Mσk (Σ) denote the set of all σk-invariant Borel probability measures and defineM∗σk (Σ) to be the collection

of all measures µ ∈ Mσk (Σ) which are compactly supported. If k ∈ N and µ ∈ M∗σk (Σ), then we let Dk(µ) to

be the unique s ≥ 0 satisfying

∑ω∈Σk

µ([ω]) logϕs(ω)

µ([ω])= 0.

5.3. Thermodynamic formalism for quasi-multiplicative potentials 123

Recall that the potential ϕs is the singular value function defined in (5.5). We also set

D(µ) = infs : Pµ(ϕs) ≤ 0 and Λµ(ϕs) > −∞

for all µ ∈ Mσ(Σ) and call it a Lyapunov dimension of µ. Given α ∈ R we define an indexed family of neigh-

bourhoods by

Bn(α) =

(−∞,−n) , if α = −∞,(

α− 1n , α + 1

n

), if α ∈ R,

(−∞,−n) , if α = ∞.

We have two main results concerning multifractal analysis of Birkhoff averages. In the first one, we consider

general potentials and in the second one, we restrict our analysis to bounded potentials.

Theorem 5.6. If (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 12 , the singular value function ϕs is quasi-

multiplicative for all 0 ≤ s ≤ d, Φ : Σ→ RN has summable variations, and α ∈ RN, then

dimH(JaΦ(α)) = min

d, lim

n→∞limk→∞

sup

Dk(µ) : µ ∈ M∗σk (Σ) so that∫

Ak(φi)dµ ∈ Bn(αi) for all i ∈ 1, . . . , n


The proof of Theorem 5.6 is given in Section 5.5.

Let s∞ = inf s : P(ϕs) < ∞. We also let P(Φ) :=∫

Φdµ : µ ∈ Mσ(Σ)

and let P(Φ) denote the closure

of P(Φ) with respect to the pointwise topology.

The following result generalises the theorem of Fan, Jordan, Liao and Rams [FJLR, Theorem 1.2] to the

self-affine setting.

Note that for α /∈ P(Φ) we have JaΦ(α) = ∅ (see Lemma 5.31).

Theorem 5.7. If (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 12 , the singular value function ϕs is quasi-

multiplicative for all 0 ≤ s ≤ d, Φ : Σ→ RN is bounded with summable variations, and α ∈ P(Φ), then

dimH(JaΦ(α)) = min

d, max

s∞, supD(µ) : µ ∈ Mσ(Σ) so that

∫Φdµ = α


The proof of Theorem 5.7 is presented in Section 5.6.

5.3 Thermodynamic formalism for quasi-multiplicative potentials

5.3.1 Existence of Gibbs measures

Suppose we have a sub-multiplicative potential ϕ along with a subset I ⊂N. We define the pressure P(ϕ, I) by

P(ϕ, I) = limn→∞

1n log Zn(ϕ, I) = inf

n∈N

1n log Zn(ϕ, I),


where Zn(ϕ, I) = ∑ω∈In ϕ(ω) for all n ∈ N. Thus Zn(ϕ, N) = Zn(ϕ) and P(ϕ, N) = P(ϕ). Observe that

Zn(ϕ, I) ≤ Zn(ϕ, I) and hence also P(ϕ, J) ≤ P(ϕ, I) for all J ⊂ I ⊂ N. If C ≥ 1, then an invariant probability

measure µ ∈ Mσ(Σ) is said to be a C-Gibbs measure for the potential ϕ on I if it is supported on IN, the pressure

P(ϕ, I) is finite, and

C−1 ≤ µ([ω])

ϕ(ω) exp(−nP(ϕ, I))≤ C

for all ω ∈ In and n ∈N. An invariant measure µ ∈ Mσ(Σ) is said to be a Gibbs measure for the potential ϕ on I

if there exists some C ≥ 1 such that µ is a C-Gibbs measure for the potential ϕ on I. Finally, µ ∈ Mσ(Σ) is said

to be a Gibbs measure for the potential ϕ if µ is a Gibbs measure for the potential ϕ on N.

In this section, our main goal is to show that if ϕ is a quasi-multiplicative potential with finite pressure,

then ϕ has a Gibbs measure. We remark that this is not the case for all sub-multiplicative potentials; see [KV,

Example 6.4] for a counter-example in a finitely generated self-affine set. For a given quasi-multiplicative

potential, throughout the section, we let Γ ⊂ Σ∗, K ∈ N, and c ≥ 1 be as in the definition of the quasi-

multiplicative potential; see (5.1).

Lemma 5.8. If ϕ is a quasi-multiplicative potential and I ⊂N is so that Γ ⊂ ⋃Kk=1 Ik, then

enP(ϕ,I) ≤ Zn(ϕ, I) ≤ cK max1, eKP(ϕ)enP(ϕ,I).

for all n ∈N. In particular, P(ϕ, I) > −∞.

Proof. Since the left-hand side inequality follows immediately from the definition of the pressure, it suffices

to show the right-hand side inequality. Fix n, m ∈ N and ωi ∈ In for all i ∈ 1, . . . , m. By the quasi-

multiplicativity, there are κ1, . . . , κm−1 so that

ϕ(ω1) · · · ϕ(ωm) ≤ cm−1 ϕ(ω1κ1ω2κ2 · · ·ωm−1κm−1ωm).

Denoting ξn,m(ω1 · · ·ωm) = ω1κ1ω2κ2 · · ·ωm−1κm−1ωm for all ω = ω1 · · ·ωm ∈ (In)m defines a mapping

ξn,m : (In)m → ⋃K(m−1)`=1 Inm+l which is at most Km−1 to one. Hence

Zn(ϕ, I)m =

(∑

ω∈Inϕ(ω)

)m

= ∑ω∈(In)m

m

∏i=1

ϕ(ωi) ≤ cm−1 ∑ω∈(In)m

ϕ(ξn,m(ω))

≤ (cK)m−1K(m−1)

∑`=1

∑ω∈Inm+l

ϕ(ω) = (cK)m−1K(m−1)

∑`=1

Znm+l(ϕ, I).

Consequently, for each m ∈N there is `m ∈N with nm ≤ `m ≤ (n+K)m satisfying Zn(ϕ, I)m ≤ m(cK)mZ`m(ϕ, I).

Hence

Zn(ϕ, I) ≤ m1/mcK(Z`m(ϕ, I)1/`m

)`m/m →

cKe(n+K)P(ϕ,I), if P(ϕ, I) > 0,

cKenP(ϕ,I), if P(ϕ, I) ≤ 0,

by letting m→ ∞. The proof follows since P(ϕ, I) ≤ P(ϕ).

The following proposition is a finite approximation property for the pressure. It is a crucial property in our

analysis since it makes it possible to construct a Gibbs measure on an infinitely generated shift space via its

finitely generated sub-spaces.


Proposition 5.9. If (I`)`∈N is a sequence of non-empty finite sets I` ⊂ N with I` ⊂ I`+1 for all ` ∈ N so that

N =⋃`∈N I`, then

P(ϕ) = lim`→∞

P(ϕ, I`)

for all quasi-multiplicative potentials ϕ. In particular, P(ϕ) = supP(ϕ, I) : I ⊂N is finite.

Proof. Recall that P(ϕ, I`) ≤ P(ϕ, I`+1) ≤ P(ϕ) for all ` ∈N. Fix $ < P(ϕ), n ∈N, and let P = lim`→∞ P(ϕ, I`).

Since $ < 1n log Zn(ϕ), we may choose ` ∈ N so that Γ ⊂ ⋃K

k=1 Ik` and $ < 1

n log Zn(ϕ, I`). By Lemma 5.8, we

have Zn(ϕ, I`) ≤ cK max1, eKP(ϕ)enP and thus $ < 1n(log cK + K|P(ϕ)|

)+ P. The proof is finished by letting

n→ ∞.

Lemma 5.10. If ϕ is a quasi-multiplicative potential with P(ϕ) < ∞, then there exists a constant C ≥ 1 such that for

each I ⊂N with Γ ⊂ ⋃Kk=1 Ik we have

C−1e(n+m)P(ϕ,I)ϕ(ω) ≤ ∑κ∈In

∑τ∈Im

ϕ(κωτ) ≤ Ce(n+m)P(ϕ,I)ϕ(ω)

for all m, n ∈N and ω ∈ ⋃n∈N In.

Proof. The right-hand side inequality follows immediately since

∑κ∈In

∑τ∈Im

ϕ(κωτ) ≤ ∑κ∈In

∑τ∈Im

ϕ(κ)ϕ(ω)ϕ(τ) = ϕ(ω)Zn(ϕ, I)Zm(ϕ, I)

≤(cK max1, eKP(ϕ)

)2e(n+m)P(ϕ,I)ϕ(ω)

by Lemma 5.8.

To show the left-hand side inequality, we first notice that the quasi-multiplicativity implies

ϕ(ω)ϕ(κ) ≤ cK

∑k=1

∑α∈Ik

ϕ(ωακ) (5.6)

for all ω, κ ∈ Σ∗. Applying Lemma 5.8, along with (5.6), we obtain

e(n+m)P(ϕ,I)ϕ(ω) ≤ Zn(ϕ, I) ∑κ∈Im

ϕ(ω)ϕ(κ) ≤ cZn(ϕ, I) ∑τ∈Im

K

∑k=1

∑α∈Ik

ϕ(ωατ)

= cZn(ϕ, I)K

∑k=1

∑α∈Ik

∑τ∈Im

ϕ(ωτα) ≤ cZn(ϕ, I)K

∑k=1

Zk(ϕ, I) ∑τ∈Im

ϕ(ωτ)

≤ c2K

∑k=1

Zk(ϕ, I)K

∑k=1

∑α∈Ik

∑κ∈In

∑τ∈Im

ϕ(καωτ)

≤ c2( K

∑k=1

Zk(ϕ, I))2

∑κ∈In

∑τ∈Im

ϕ(κωτ).

The proof is now finished since

K

∑k=1

Zk(ϕ, I) ≤ cK max1, eKP(ϕ)K

∑k=1

ekP(ϕ) < ∞

by Lemma 5.8.

We are now ready to show that every finite sub-space carries a Gibbs measure. Observe that, to be able to

extend the result into infinitely generated shift space, it is crucial to find a uniform constant.


Proposition 5.11. If ϕ is a quasi-multiplicative potential with P(ϕ) < ∞, then there is C ≥ 1 so that ϕ has a C-Gibbs

measure for ϕ on I for all finite subsets I ⊂N with Γ ⊂ ⋃Kk=1 Ik.

Proof. Let I ⊂N be a finite subset with Γ ⊂ ⋃Kk=1 Ik. Given a finite word ω ∈ ⋃n∈N In we choose ω ∈ [ω]∩ IN

and let δω denote the point mass concentrated at ω. For each n ∈ N we define a probability measure νn on Σ

by

νn = Z3n(ϕ, I)−1 ∑ω∈I3n

ϕ(ω)δω.

Note that νn is supported on IN. If m, ` ∈ 1, . . . , n and ω ∈ Im, then

νn σ−`([ω]) = ∑κ∈I`

∑τ∈I3n−`−m

νn([κωτ]) = Z3n(ϕ, I)−1 ∑κ∈I`

∑τ∈I3n−`−m

ϕ(κωτ).

According to Lemmas 5.8 and 5.10 there exists a constant C ≥ 1 so that

C−1e−mP(ϕ,I)ϕ([ω]) ≤ νn σ−`([ω]) ≤ Ce−mP(ϕ,I)ϕ(ω) (5.7)

for all finite subsets I ⊂ N with Γ ⊂ ⋃Kk=1 Ik. Observe that the above estimate remains true if we replace

νn σ−` by the probability measure

µn = 1n

n

∑`=1

νn σ−`. (5.8)

Since I is finite and each µn is supported on the compact set IN, there is a convergent subsequence (µnk )k∈N

converging to some limit µ in the weak∗ topology. It follows from (5.8) that µ is a σ-invariant probability

measure. Moreover, by (5.7), µ is a C-Gibbs measure for ϕ on I.

Theorem 5.12. If ϕ is a quasi-multiplicative potential with P(ϕ) < ∞, then ϕ has a Gibbs measure µ. Moreover,

there is C ≥ 1 so that for each ` ∈ N there are a finite set I` ⊂ N and a C-Gibbs measure µ` for ϕ on I` such that

P(ϕ, I`)→ P(ϕ) and µ` → µ in the weak∗ topology.

Proof. Let (I`)`∈N be a sequence of non-empty finite sets I` ⊂ N with I` ⊂ I`+1 and Γ ⊂ ⋃Kk=1 Ik

` for all ` ∈ N

such that N =⋃`∈N I`. Recalling Proposition 5.9, we have lim`→∞ P(ϕ, I`) = P(ϕ). By Proposition 5.11, there

exist a constant C ≥ 1 and for each ` ∈N a σ-invariant probability measure µ` ∈ Mσ(Σ) so that

C−1 ≤ µ`([ω])

ϕ(ω) exp(−nP(ϕ, I`))≤ C (5.9)

for all ω ∈ In` and ` ∈ N. It suffices to show that the sequence (µ`)`∈N is tight, that is, for each ε > 0 there

exists a compact set K ⊂ Σ for which µ`(K) > 1− ε for all ` ∈N. Then the sequence (µ`)`∈N has a converging

subsequence and it follows from (5.9) that the limit measure of that subsequence is a Gibbs measure for ϕ.

Fix ε > 0 and notice that ∑i∈N ϕ(i) = Z1(ϕ) ≤ CeP(ϕ) < ∞ by Lemma 5.8. Thus, for each k ∈ N there is a

finite subset Ik ⊂N so that

∑i∈N\Ik

ϕ(i) < ε2−kC−1eP(ϕ,I1) ≤ ε2−kC−1eP(ϕ,I`)


for all ` ∈N. We define K = ω ∈ Σ : ωk ∈ Ik for all k ∈N. It follows from (5.9) that

µ`(K) = µ`

(Σ \

⋃k∈N

ω ∈ Σ : ωk /∈ Ik)= 1− ∑

k∈N

µ`(ω ∈ Σ : ωk /∈ Ik)

= 1− ∑k∈N

∑i∈N\Ik

µ`

(σ−k([i])

)= 1− ∑

k∈N

∑i∈N\Ik

µ`([i])

≥ 1− ∑k∈N

∑i∈N\Ik

Ce−P(ϕ,I`)ϕ(i) > 1− ∑k∈N

2−kε = 1− ε.

for all ` ∈N.

5.3.2 Variational principle

We shall study the properties of the Gibbs measure found in Theorem 5.12. At the end of this section, we prove

Theorem 5.3.

Proposition 5.13. If ϕ is a quasi-multiplicative potential with P(ϕ) < ∞ and µ is a Gibbs measure for ϕ, then µ is

ergodic. In particular, µ is the only Gibbs measure for ϕ.

Proof. By (5.6), it is straightforward to see that a C-Gibbs measure µ satisfies

K

∑p,q=1

µ([ω] ∩ σ−(n+p+q)([τ])

)≥ c−2C−1e−2K|P(ϕ)|µ([ω])µ([τ])

for all ω, τ ∈ Σ and n ≥ |ω|. The proof follows now by standard arguments; see e.g. [FLW, Theorem 3.2].

Lemma 5.14. If ϕ is a quasi-multiplicative potential with P(ϕ) < ∞ and µ is the Gibbs measure for ϕ on a set I ⊂ N,

then P(ϕ, I) = Pµ(ϕ).

Proof. By the definition of a Gibbs measure, we get

Pµ(ϕ) = limn→∞

1n ∑

ω∈Inµ([ω]) log

ϕ(ω)

µ([ω])= lim

n→∞1n ∑

ω∈Inµ([ω]) log enP(ϕ,I) = P(ϕ, I)

as desired.

Lemma 5.15. If ϕ is a quasi-multiplicative potential with P(ϕ) < ∞ and µ is the Gibbs measure for ϕ, then any measure

ν ∈ Mσ(Σ) with P(ϕ) ≤ Pν(ϕ) is absolutely continuous with respect to µ.

Proof. To prove the claim, we follow the ideas of [BoEq] and [KV, Theorem 3.6]. Let µ be a C-Gibbs measure.

Assume to the contrary that there exist a measure ν ∈ Mσ(Σ) with P(ϕ) ≤ Pµ(ϕ) and a Borel set B ⊂ Σ so that

µ(B) = 0 and ν(B) > 0. Since the semi-algebra of cylinder sets generates the Borel σ-algebra we may choose

a sequence of sets (Bn)n∈N such that each Bn is a union of cylinders of length n with (µ + ν)(Bn4B) → 0 as

n→ ∞. Let B′n = ω ∈ Σn : [ω] ⊂ Bn. Hence, by (5.2) and (5.4), we have

0 ≤ ∑ω∈B′n

ν([ω]) logϕ(ω)

ν([ω])+ ∑

ω∈Σ\B′nν([ω]) log

ϕ(ω)

ν([ω])− nP(ϕ)

≤ ν(Bn) log ∑ω∈B′n

ϕ(ω) + ν(Σ \ Bn) log ∑ω∈Σ\B′n

ϕ(ω)− nP(ϕ) + 2e

≤ ν(Bn) log µ(Bn) + ν(Σ \ Bn) log µ(Σ \ Bn) + log C + 2e

(5.10)


for all n large enough. Since ν(Bn)→ ν(B) and µ(Bn)→ 0 the right-hand side of (5.10) tends to−∞ as n→ ∞.

This contradiction finishes the proof.

We are now ready to prove Theorem 5.3.

Proof of Theorem 5.3. Let us first assume that P(ϕ) = ∞. Let (I`)`∈N be a sequence of non-empty finite sets

with I` ⊂N and Γ ⊂ ⋃Kk=1 Ik

` for all ` ∈N such that N =⋃`∈N I`. Recalling Proposition 5.11, let µ` be a Gibbs

measure for ϕ on I` for all ` ∈N. Now

P(ϕ) = supP(ϕ, I`) : ` ∈N = supPµ`(ϕ) : ` ∈N ≤ supPµ(ϕ) : µ ∈ Mσ(Σ)

by Proposition 5.9 and Lemma 5.14.

If P(ϕ) < ∞, then it suffices to prove that a Gibbs measure µ is the only invariant measure for which

P(ϕ) = Pµ(ϕ). Theorem 5.13 shows that µ is ergodic and Lemma 5.14 shows that it satisfies P(ϕ) = Pµ(ϕ). If

ν ∈ Mσ(Σ) is an invariant measure satisfying P(ϕ) = Pµ(ϕ), then ν is absolutely continuous with respect to µ

by Lemma 5.15. It follows from the proof of [W, Theorem 6.10(iii)] that ν = µ.

5.3.3 Differentiation of pressure

Given a pair of potentials ϕ1, ϕ2 : Σ∗ → [0, ∞) we let ϕ1 · ϕ2 denote the potential defined by ω 7→ ϕ1(ω)ϕ2(ω)

for all ω ∈ Σ∗. Given a function φ : Σ→ R we define an associated potential eφ : Σ∗ → [0, ∞) by setting

eφ(ω) = exp(supSnφ(τ) : τ ∈ [ω]

)for all ω ∈ Σn and n ∈N. Recall that Snφ(τ) = ∑n−1

j=0 φ(σj(τ)) for all τ ∈ Σ.

Lemma 5.16. If ϕ is a quasi-multiplicative potential and φ : Σ → R has summable variations, then the potential ϕ · eφ

is quasi-multiplicative.

Proof. If c = exp(∑∞n=1 varn(φ)), then

c−1eφ(ω)eφ(κ) ≤ eφ(ωκ) ≤ eφ(ω)eφ(κ)

for all ω, κ ∈ Σ∗. The claim follows from the quasi-multiplicativity of ϕ.

Lemma 5.17. If ϕ is a sub-multiplicative potential with P(ϕ) < ∞ and φ : Σ → R is bounded with summable varia-

tions, then the function q 7→ P (ϕ · e(qφ)) is convex.

Proof. If q, p ∈ R and 0 ≤ λ ≤ 1, then

ϕ(ω)e(λq+(1−λ)p)φ(ω) ≤(

ϕ(ω)eqφ(ω))λ(

ϕ(ω)epφ(ω))1−λ

for all ω ∈ Σ∗. Thus, by Holder’s inequality, we have

∑ω∈Σn

ϕ(ω)e(λq+(1−λ)p)φ(ω) ≤(

∑ω∈Σn

ϕ(ω)eqφ(ω)

)λ(∑

ω∈Σn

ϕ(ω)epφ(ω)

)1−λ

.

Taking logarithms, dividing by n, and letting n→ ∞ gives the claim.

5.4. Dimension of infinitely generated self-affine sets 129

Lemma 5.18. If ϕ is a quasi-multiplicative potential with P(ϕ) < ∞, µ is the Gibbs measure for ϕ, and φ : Σ → R is

bounded with summable variations, then the function q 7→ P(

ϕ · eqφ

)is differentiable at zero with derivative

∂P(

ϕ · eqφ

)∂q

∣∣∣∣q=0

=∫

φdµ.

Proof. To prove the claim, we use some of the ideas used in the proof of [KV, Thorem 4.4]. It suffices to show

that the right derivative exists at zero and equals to∫

φdµ since applying this result with−φ in place of φ gives

limq↑0

1q(

P(ϕ · eqφ)− P(ϕ))= − lim

q↓01q(

P(ϕ · eq(−φ))− P(ϕ))=∫

φdµ.

Throughout the proof of the lemma, to simplify notation, we write P(q) in place of P(

ϕ · eqφ

). By Lemma 5.17,

the function q 7→ P(q) is convex and hence there is a well-defined right derivative at zero. We shall denote it

by P′+(0).

To prove that P′+(0) ≤∫

φdµ, take β >∫

φdµ. Define

C′n = ω ∈ Σn : Snφ(τ) > nβ for some τ ∈ [ω]

for all n ∈N and let Cn =⋃

ω∈C′n [ω]. Since µ is a Gibbs measure for ϕ there is C ≥ 1 so that

ϕ(ω) ≤ CenP(0)µ([ω]) (5.11)

for all ω ∈ Σn and n ∈ N. By Theorem 5.13, µ is ergodic and thus, we may apply Birkhoff’s ergodic theorem,

Egorov’s theorem, and the fact that φ has summable variations, to obtain limn→∞ µ(Cn) = 0.

Fix γ > 0. Since P(q) is convex we have P(γ/n) ≥ P(0) + γ/nP′+(0). Using the sub-multiplicativity of

ϕ · eγ/nφ and (5.11), we have

enP(0)+γP′+(0) ≤ enP(γ/n) ≤ ∑ω∈Σn

ϕ(ω) exp(γ/n‖Sn(φ)|[ω]‖)

≤ ∑ω∈Σn\C′n

ϕ(ω)eγβ + ∑ω∈C′n

ϕ(ω)eγ‖φ‖

≤ CeγβenP(0)(1− µ(Cn)) + Ceγ‖φ‖enP(0)µ(Cn).

Dividing by enP(0), letting n→ ∞, and then γ→ ∞ gives P′+(0) ≤ β as desired.

To show that P′+(0) ≥∫

φdµ, we use Lemma 5.2 for the sub-multiplicative potential ϕ · eqφ and Lemma 5.14

for the quasi-multiplicative potential ϕ to obtain

P(q) ≥ Pµ(ϕ · eqφ) ≥ Pµ(ϕ) + q∫

φdµ = P(0) + q∫

φdµ

for all q ≥ 0. The proof follows.

5.4 Dimension of infinitely generated self-affine sets

In this section, we prove Theorem 5.5, that is, we show that the dimension of a typical infinitely generated self-

affine set is a supremum of dimensions of its finitely generated subsets. We also examine when the projection

of the Gibbs measure is a measure of maximal dimension. The reader is prompted to recall notation from

Section 5.2.2.


Proof of Theorem 5.5. Define s0 = infs : P(ϕs) ≤ 0 and let (I`)`∈N be a sequence of non-empty finite sets

I` ⊂ N with I` ⊂ I`+1 and Γ ⊂ ⋃Kk=1 Ik

` for all ` ∈ N such that N =⋃`∈N I`. Fix ` ∈ N and let 0 < s` ≤ s0

be such that P(ϕs` , I`) = 0. To show that s0 ≤ sup`∈N s`, take s < s0. Since P(ϕs) > 0 and P(ϕs, I`) → P(ϕs)

by Proposition 5.9, we may choose `0 ∈ N so that P(ϕs, I`0) > 0. Therefore s`0 > s, and, consequently,

s0 = sup`∈N s`.

Since dimH(πa(IN` )) = mind, s` for LA-almost all a ∈ A by [F1, Theorem 5.3] and

⋃`∈N πa(IN

` ) ⊂ Fa, we

have mind, s0 ≤ dimH(Fa). To show that dimH(Fa) ≤ s0, take s < dimH(Fa). Choose m ∈ Z and 0 < δ ≤ 1

so that s = m + δ and let ∆ be a closed ball such that fi(∆) ⊂ ∆ for all i ∈ N. It follows from the deinition of

singular values that for each ω ∈ Σ∗ we may cover fω(∆) with at most a constant times

γ1(ω)

γm+1(ω)

γ2(ω)

γm+1(ω)· · · γm(ω)

γm+1(ω)

balls of radius γm+1(ω). Thus there exists c ≥ 1 so that

Hs2−k (Fa) ≤ ∑

ω∈Σk

Hs2−k ( fω(∆)) ≤ c ∑

ω∈Σk

ϕs(ω)

for all k ∈ N. It follows that ∑ω∈Σkϕs(ω) ≥ 1 for all k ∈ N large enough. Thus P(ϕs) ≥ 0 and s ≥ s0 which

finishes the proof.

Considering the projection πa, we denote the pushforward measure of µ ∈ Mσ(Σ) by πaµ.

Proposition 5.19. If (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 12 , the singular value function ϕs is quasi-

multiplicative for all 0 ≤ s ≤ d, there exists 0 ≤ s0 ≤ d so that P(ϕs0) = 0, µ is the Gibbs measure for ϕs0 so that

Λµ(ϕs0) > −∞, and F′a ⊂ Fa with πaµ(F′a) > 0, then

dimH(F′a) = dimH(Fa)


Proof. Let s < t < s0 and recall that by Lemma 5.14, Theorem 5.3, and Lemma 5.2, the measure µ is ergodic

and satisfies hµ + Λµ(ϕs0) = 0. Hence, by Shannon-McMillan Theorem and Kingman’s sub-additive ergodic

theorem, we have

limn→∞

log µ([ω|n])log ϕt(ω|n)

> 1

for µ-almost all ω ∈ Σ. Applying Egorov’s theorem, we find for each ε > 0 a compact set C ⊂ Σ and n0 ∈ N

so that µ(C) > 1− ε and µ([ω|n]) ≤ ϕt(ω|n) for all ω ∈ C and n ≥ n0. Now, according to [S, Proposition

3.1(i)], we have ∫A

∫C

∫Σ

dµ(ω)dµ(τ)da|πa(ω)− πa(τ)|s

≤ c′∫

C

∫Σ

ϕs(ω ∧ τ)−1dµ(ω)dµ(τ)

= c′∞

∑n=0

∑ω∈Σn

ϕs(ω)−1µ([ω])µ(C ∩ [i])

≤ c∞

∑n=0

∑ω∈Σn

ϕs(ω)−1 ϕt(ω)µ([ω])

≤ c∞

∑n=0

2−(t−s)n < ∞

5.4. Dimension of infinitely generated self-affine sets 131

for some constants c, c′ > 0. Observe that [S, Proposition 3.1(i)] is a refinement of [F1, Lemma 3.1] and it

generalises immediately to the infinite case. It follows that

lim infr↓0

log πaµ(B(πa(τ), r))log r

= sup

t ≥ 0 :∫

Σ

dµ(ω)

|πa(ω)− πa(τ)|t< ∞

≥ s

for µ-almost all τ ∈ C and for LA-almost all a ∈ A. The proof is finished by recalling [F2, Proposition 2.3(a)]

and Theorem 5.5.

To finish this section, we provide the reader with a sufficient condition to guarantee the finiteness of the

Lyapunov exponent in Theorem 5.19. Recall that s∞ = infs : P(ϕs) < ∞.

Lemma 5.20. If (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 1, the singular value function ϕs is quasi-

multiplicative for all 0 ≤ s ≤ d, s0 > s∞, and µ is the Gibbs measure for ϕs0 , then Λµ(ϕs0) > −∞.

Proof. Observe that since P(ϕs0) < ∞ the Gibbs measure µ for ϕs0 exists by Theorems 5.12 and 5.13. To prove

the claim, let m ∈ Z be so that m < s0 ≤ m + 1. By the Gibbs property there is a constant C ≥ 1 so that

µ([ω]) ≤ Cϕs0(ω)e−nP(ϕs0 )

for all ω ∈ Σn and n ∈N. Thus

logϕs0(ω)

µ([ω])≥ nP(ϕs0)− log C (5.12)

for all ω ∈ Σn and n ∈N.

If maxs∞, m < t < s0, then P(ϕt) < ∞ and Zn(ϕt) < ∞ for all n ∈ N by Lemma 5.8. As in (5.4), Jensen’s

inequality gives

∑ω∈Σn

µ([ω]) logϕt(ω)

µ([ω])≤ log

(∑

ω∈Σn

ϕt(ω)

)= log Zn(ϕt). (5.13)

Since ϕs0(ω) = γm+1(ω)s0−t ϕt(ω) for all ω ∈ Σ∗ we have, by (5.12) and (5.13), that

(nP(ϕs0)− log C

)≤ ∑

ω∈Σn

µ([ω])

(log γm+1(ω)s0−t + log

ϕt(ω)

µ([ω])

)≤ ∑

ω∈Σn

µ([ω]) log γm+1(ω)s0−t + log Zn(ϕt).

Hence,

1n ∑

ω∈Jnµ([ω]) log ϕs0(ω) ≥ 1

n ∑ω∈In

µ([ω]) log γm+1(ω)m+1

≥(m + 1)

(nP(ϕs0)− log C− log Zn(ϕt)

)n(s0 − t)

.

Letting n→ ∞ we have

Λµ(ϕs0) ≥(m + 1)

(P(ϕs0)− P(ϕt)

)s0 − t

> −∞.


5.5 Multifractal analysis of Birkhoff averages

The aim of this section is to prove Theorem 5.6. The upper bound is proved in Proposition 5.22 and the lower

bound in Theorem 5.24. It is worth mentioning that the upper bound in Theorem 5.6 holds for all a ∈ A.

5.5.1 Proof of the upper bound in Theorem 5.6

In this subsection we shall prove the upper bound in Theorem 5.6. The reader is prompted to recall notation

from Section 5.2.2 and Section 5.2.3. We begin with a lemma relating the dimension of JΦ(α) to the singular

value function. Define

AΦ(α, n, k) =

ω ∈ Σk : Akφi(τ) ∈ Bn(αi) for all τ ∈ [ω] and i ∈ 1, . . . , n

for all n, k ∈N.

Lemma 5.21. If (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 1, Φ : Σ→ RN has summable variations, α ∈ RN,

a ∈ A, s < dimH(JaΦ(α)), and n ∈N, then there is k0 ∈N such that

∑ω∈AΦ(α,n,k)

ϕs(ω) > 1

for all k ≥ k0.

Proof. Let δ = supi∈N ‖Ti‖ and set

DΦ(α, n, k) =

ω ∈ Σk : there is τ ∈ [ω] such that Akφi(τ) ∈ B2n(αi) for all i ∈ 1, . . . , n

for all n, k ∈ N. Fix n ∈ N. Since limk→∞ vark (Akφi) = 0 we may choose k1 so that vark (Akφi) < (2n)−1 for

all i ∈ 1, . . . , n and all k ≥ k1. Thus we have DΦ(α, n, k) ⊂ AΦ(α, n, k) for all k ≥ k1. Since

JΦ(α) ⊂⋃

l∈N

∞⋂k=l

⋃ω∈DΦ(α,n,k)

πa ([ω])

and dimH(JaΦ(α)) > s there is l ∈N with

dimH

( ∞⋂k=l

⋃ω∈DΦ(α,n,k)

πa ([ω])

)> s.

Hence, continuing as in the proof of Theorem 5.5, we find a constant c ≥ 1 so that

Hsδk

( ⋃ω∈DΦ(α,n,k)

πa ([ω])

)≤ c ∑

ω∈DΦ(α,n,k)ϕs(ω)

for all k ∈N. The proof follows.

Proposition 5.22. If (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 1, Φ : Σ→ RN has summable variations, and

α ∈ RN, then

dimH(JaΦ(α)) ≤ lim

n→∞limk→∞

sup


Aiφidµ ∈ Bn(αi) for all i ∈ 1, . . . , n

for all a ∈ A.

5.5. Multifractal analysis of Birkhoff averages 133

Proof. Fix a ∈ A, s < dimH(JaΦ(α)), and n ∈N. According to Lemma 5.21, there is k0 ∈N such that

∑ω∈AΦ(α,n,k)

ϕs(ω) > 1

for all k ≥ k0. Let k ≥ k0 and choose a finite subset FΦ(α, n, k) ⊂ AΦ(α, n, k) with F(k) = ∑ω∈FΦ(α,n,k) ϕs(ω) ≥

1. Define a compactly supported k-th level Bernoulli measure µ ∈ Mσk (Σ) by setting

µ([ω]) =

ϕs(ω)/F(k), if ω ∈ FΦ(α, n, k),

0, if ω ∈ Σk \ FΦ(α, n, k)

for all ω ∈ Σk. It follows immediately that

∑ω∈Σk

µ([ω]) logϕs(ω)

µ([ω])= log F(k) ≥ 0

yielding s ≤ Dk(µ). Since µ is supported on⋃

ω∈FΦ(α,n,k)[ω] and Akφi(τ) ∈ Bn(αi) for all ω ∈ FΦ(α, n, k),

τ ∈ [ω], and i ∈ 1, . . . , n we also have ∫Akφidµ ∈ Bn(αi)

for all i ∈ 1, . . . , n. These observations imply the proof.

5.5.2 Symbolic tree structure in level sets

The following proposition contains the essence of the proof of the lower bound in Theorem 5.6.


multiplicative for all 0 ≤ s ≤ d, Φ : Σ→ RN has summable variations, α ∈ RN, and

s < limn→∞

limk→∞

sup



,

then there exists a set S ⊂ EΦ(α) and a Borel probability measure µ, supported on S, and a constant C, such that

µ([ω]) ≤ Cϕs(ω) for all ω ∈ Σ∗.

In fact, with this proposition, the lower bound in Theorem 5.6 follows almost immediately.


multiplicative for all 0 ≤ s ≤ d, Φ : Σ→ RN has summable variations, and α ∈ RN, then

dimH(JaΦ(α)) ≥ min

d, lim

n→∞limk→∞

sup




Proof. Let s > 0 be as in Proposition 5.23. Applying the measure given by Proposition 5.23 in the proof of

Theorem 5.19, we get dimH(πa(S)) ≥ s for LA-almost all a ∈ A, where S ⊂ EΦ(α) is as in Proposition 5.23.

Thus dimH(JaΦ(α)) ≥ s for LA-almost all a ∈ A.


In the course of the proof of Proposition 5.23, we shall rely on the concept of M-trees. This approach is

inspired by a similar notion discussed by Furstenberg in [Fu2]. We shall now define all the required concepts.

If ω, τ ∈ Σ∗ ∪ Σ so that ω ∧ τ = ω, then we write ω 4 τ. This defines a partial order on Σ∗. Let X ⊂ Σ∗ be

an antichain with respect to 4. If there is a functionMX : X→ [0, 1] so that

∑ω∈X

MX(ω) = 1,

then the ordered pair (X,MX) is called anM-tree. AnM-tree (X,MX) is said to be finite if X is a finite set. If

(X,MX) and (Y,MY) areM-trees so that

MX(ω) = ∑τ∈κ∈Y:ω4κ

MY(τ)

for all ω ∈ X, then we write (X,MX) 4 (Y,MY). This defines a partial order on the collection of allM-trees.

Next we shall define a limit for certainM-tree sequences. If ((X,MX))n∈N is a chain of finiteM-trees so

that limn→∞ min|ω| : ω ∈ Xn = ∞, then the limit of that sequence is defined to be

limn→∞

(Xn,MXn) = (X∞,M∞),

where

X∞ = τ ∈ Σ : for each n ∈N there is ω ∈ Xn so that ω 4 τ

andM∞ is a Borel probability measure supported on X∞ defined as follows. Observe first that since each Xn

is a finite antichain, it is readily checked that the collection

A(X∞) = ∅, X∞ ∪ [ω] ∩X∞ : ω ∈⋃

n∈N

Xn

is a semi-algebra of subsets of X∞. Moreover, since limn→∞ min |ω| : ω ∈ Xn = ∞, it is clear that this semi-

algebra generates the Borel σ-algebra restricted to X∞. We define M∞ on A(X∞) by setting M∞(∅) = 0,

M∞(X∞) = 1, and

M∞ ([ω] ∩X∞) =MXn(ω).

for all ω ∈ Xn and n ∈ N. It follows from the fact that (Xn,MXn) 4(Xn+1,MXn+1

)for each n ∈ N, that this

set function is well-defined and countably additive. ThusM∞ extends to a measure on Σ.

Finally, given a subset Ω ⊂ Σ∗ ∪ Σ let D(Ω) ⊂ N be the collection of all digits contained within words

from Ω, that is, D(Ω) = l ∈N : there is ω ∈ Ω with ωi = l for some i.

Proof of Proposition 5.23. We begin by noting that, without loss of generality, we may assume that if φ is in the

sequence Φ, then also−φ is in Φ. Indeed, if Φ = (φi)i∈N, α = (αi)i∈N, and the right-hand side of the inequality

in the formulation of Proposition 5.23 is denoted by D(Φ, α), then we clearly have D(Φ, α) = D(Φ′, α′), where

Φ′ = (φ′i)i∈N and α′ = (α′i)i∈N are defined so that φ′2i = φi, φ′2i−1 = −φi, α′2i = αi, α′2i−1 = −αi for all i ∈N.

Choose s < t < D(Φ, α) and define

Ai = sup|φ(ω)| : ω ∈ [τ] and τ ∈ D(Γ ∪ 1) (5.14)


for all i ∈ N. For each n ∈ N we choose k = k(n) ≥ 4Kn (maxi≤n Ai + 1) so that vark(n)Ak(n)φi <1

2n for all

i ∈ 1, . . . , n and there exists νn ∈ M∗σk (Σ) with Dk(νn) > t and∫

Akφidνn ∈ B2n(αi) (5.15)

for all i ∈ 1, . . . , n. Let ρn ∈ M∗σk (Σ) be the compactly supported k(n)-th level Bernoulli measure given by

ρn([ω1 · · ·ωk(n)q]) =q−1

∏j=0

νn

([ωjk(n)+1 · · ·ω(j+1)k(n)]

).

For each potential φi we define a k-th level locally constant potential Akφi by

Akφi(ω) = inf1

k

k−1

∑l=0

φi(σlτ) : τj = ωj for j ∈ 1, . . . , k

.

Note that Akφi(ω) − vark Akφi ≤ Akφi(ω) ≤ Akφi(ω) for all ω ∈ Σ and i ∈ N. Since vark Ak(φi) < 12n it

follows from (5.15) that ∫Akφidρn ∈ Bn(αi)

for all i ∈ 1, . . . , n. Moreover, it is immediate from Dk(νn) > t that

∑ω∈Σk(n)

ρn(ω) logϕt(ω)

ρn(ω)> 0.

We let D(n) =

ω ∈ Σk(n) : ρn(ω) > 0

. Since ρn ∈ M∗σk (Σ) the number of words in D(n) is finite. Hence,

for each n there is a finite set of digits D∗(n) ⊂N defined by

D∗(n) = D

(n+2⋃l=1

D(l) ∪ 1 ∪ Γ)

.

Since we also have var1φi < ∞ the quantities

A(n) = sup|φi(τ)| : τ ∈ [ω] for some ω ∈ D∗(n) and i ∈ 1, . . . , n

,

B(n) = supρn(ω)−1 : ω ∈ D(n)(5.16)

are both finite. By Kolmogorov’s strong law of large numbers, we have

limN→∞

1N

N−1

∑j=0

logϕt(ωjk(n)+1 · · ·ω(j+1)k(n))

ρn([ωjk(n)+1 · · ·ω(j+1)k(n)])= ∑

τ∈Σk(n)

ρn(τ) logϕt(τ)

ρn([τ])> 0

for ρn-almost all ω ∈ Σ, and

limN→∞

1N

N−1

∑j=0

Ak(n)φi(σjk(n)ω) ∈ Bn(αi)

and for all i ∈ 1, . . . , n. By Egorov’s theorem we find Sn ⊂ supp(ρn) with ρn(Sn) > 1/2 so that each of the

above convergences are uniform upon Sn. Hence there is L(n) ∈N such that

N−1

∏j=0

ϕt(ωjk(n)+1 · · ·ω(j+1)k(n)) >N−1

∏j=0

ρn([ωjk(n)+1 · · ·ω(j+1)k(n)]),

1N

N−1

∑j=0

Ak(n)φi(σjk(n)ω) ∈ Bn(αi) for i ∈ 1, . . . , n.

(5.17)

for all ω ∈ Sn and all N ≥ L(n). We also let M(n) = max

ϕt(τ)−1 : [τ] ∩ supp(ρn) 6= ∅

.


For every α, β ∈ Σ∗, according to the quasi-multiplicativity of ϕt, there exists ω ∈ Γ such that

ϕt (αωβ) ≥ c(t)ϕt(α)ϕt(β),

where c(t) > 0 is a constant depending only on t. We let α ? β denote the word αωβ, so ϕt (α ? β) ≥

c(t)ϕt(α)ϕt(β). Note that for any given α, β ∈ Σ∗ there are at most K = max |ω| : ω ∈ Γ finite words

β′ ∈ Σ∗ with α ? β′ = α ? β (including β itself). We also write α ? β ? ω = (α ? β) ? ω.

Our aim is to construct a sequence ofM-trees ((Tn,Mn))n∈N∪0 with (Tn−1,Mn−1) 4 (Tn,Mn) for all

n ∈ N, along with functions (Ψn)n∈N of the form Ψn : Tn → Σ∗, together with a sequence (γn)n∈N with the

property that every τ ∈ Tn satisfies γn − K ≤ |Ψn(τ)| ≤ γn.

We begin by letting T0 = ∅, M0(∅) = 1, Ψ0 = ∅ 7→ ∅, and γ0 = 0. Suppose we have defined

(Mn−1, Tn−1), Ψn−1 : Tn−1 → Σ∗, and γn−1 with the required properties. For each ω ∈ Tn−1 we let

Zn−1(ω) = τ ∈ Tn−1 : Ψn−1(τ) 4 Ψn−1(ω) or Ψn−1(ω) 4 Ψn−1(τ) .

We shall construct (Mn, Tn), Ψn : Tn → Σ∗, and γn as follows. First take qn ∈N so that

qn >4M(n)L(n)M(n + 1)L(n+1)

minϕt(Ψn−1(ω)) : ω ∈ Tn−1+ max#Ψ−1

n−1(ω) : ω ∈ Tn−1

+ nL(n + 1)(A(n) + 1)(B(n) + 1)(γn−1 + 4K + k(n + 1) + k(n + 2) + 1)

+ #D(n)#D(n + 1)max #Zn−1(ω) : ω ∈ Tn−1+ qn−1

+ #D(n + 1)#(1 ∪ Γ ∪D(D(n)) ∪D(D(n + 1))

)5K+k(n+1)+k(n)+2,

(5.18)

where A(n) and B(n) are as in (5.16).

Let Fn = τ ∈ Σk(n)qn : [τ] ∩ Sn 6= ∅ and η(n) = ∑τ∈Fn ρn([τ]) ≥ ρn(Sn) > 1/2. Define

F ln = τ ∈ Σk(n)l : there is β ∈ Σk(n)(qn−l) with τβ ∈ Fn

for all l ∈ 1, . . . , qn. In the process of constructing (Tn,Mn), Ψn, and γn, we shall construct a sequence of

intermediaryM-trees ((Tln,Ml

n))qnl=0 so that (Tl

n,Mln) 4 (Tl+1

n ,Ml+1n ) and

(Tn−1,Mn−1) 4 (Tln,Ml

n) 4 (Tn,Mn) (5.19)

for all l ∈ 1, . . . , qn. In addition, we construct intermediary maps Ψln : Tl

n → Σ∗ and(

γln

)qn

l=0so that γl

n −

K ≤ |Ψln(τ)| ≤ γl

n for all τ ∈ Tln and Ψl

n(ωl) 4 Ψl+1

n (ωl+1) for all ωl ∈ Tln, ωl+1 ∈ Tl+1

n , and ωl 4 ωl+1.

First take (T0n,M0

n) = (Tn−1,Mn−1), Ψ0n = Ψn−1, and γ0

n = γn−1. Clearly (T0n,M0

n), Ψ0n, and γ0

n satisfy

the required properties. For each l ∈ 1, . . . , qn we let

Tln = κτ : κ ∈ Tn−1 and τ ∈ F l

n.

For each ω ∈ Tln we take the (unique) pair κ ∈ Tn−1 and τ ∈ F l

n with ω = κτ and let

Mln(ω) =

Mn−1(κ)

η(n) ∑β∈α:τα∈Fn

ρn([τβ]).


It is clear that (Tln,Ml

n) 4 (Tl+1n ,Ml+1

n ) and if we let (Tn,Mn) = (Tqnn ,Mqn

n ), we have shown (5.19). We shall

construct the functions (Ψln)

qnl=1 and numbers (γl

n)qnl=1 recursively.

Suppose l ∈ 1, . . . , qn and we have constructed Ψl−1n and γl−1

n satisfying the required properties. Define

γln = γl−1

n + 2K + k(n) and let ω = κτ ∈ Tln so that κ ∈ Tn−1 and τ ∈ F l

n. Choose τ′ ∈ F l−1n so that τ′ 4 τ

and set ω′ = κτ′ ∈ Tl−1n . Thus there exists τl ∈ Σk(n) so that ω = ω′τl . The function Ψl

n is defined by setting

Ψln(ω) = Ψl−1

n (ω′) ? τl ? 1 · · · 1,

where the length of 1 · · · 1 is γln − K − |Ψl−1

n (ω′) ? τl |. Since γl−1n − K ≤ |Ψl−1

n (ω′)| ≤ γl−1n we have γl−1

n −

K + k(n) ≤ |Ψln(ω

′) ? τl | ≤ γl−1n + k(n) + K. Thus Ψl

n(ω) is well-defined, the length of 1 · · · 1 is at most 2K,

and γln − K ≤ |Ψl

n(ω)| ≤ γln. Moreover, if we let c0 = c(t)2 ϕt(1 · · · 1), where the length of 1 · · · 1 is 2K, then a

simple induction gives

ϕt(Ψn−1(κ))l

∏j=1

ϕt(τj) ≤ c−l0 ϕt(Ψl

n(κτ)), (5.20)

where each τj has length k(n). We emphasise that c0 is independent of n and l. Recalling that Tn = Tqnn , we

set Ψn = Ψqnn .

To finish the construction ofM-trees ((Tn,Mn))n∈N∪0, functions (Ψn)n∈N, and the sequence (γn)n∈N,

we shall show that

max#Zln(ω) : ω ∈ Tl

n ≤ qn−1(2K)l+qn−1 (5.21)

for all l ∈ 1, . . . , qn, and n ∈N, where

Zln(ω) = ω′ ∈ Tl

n : Ψln(ω) 4 Ψl

n(ω′) or Ψl

n(ω′) 4 Ψl

n(ω)

for all ω ∈ Tln, l ∈ 1, . . . , qn, and n ∈N.

Fix ω ∈ Tln and choose κ ∈ Tn−1 and τ ∈ F l

n so that ω = κτ. Write τ = τ1 · · · τl , where each τj ∈ Σk(n).

Now suppose ω′ ∈ Zln(ω) and similarly take κ′ ∈ Tn−1 and τ′ = τ′1 · · · τ′l ∈ F

ln so that ω′ = κ′τ′ and

τ′j ∈ Σk(n). It is clear that either Ψn−1(κ) 4 Ψn−1(κ′) or Ψn−1(κ

′) 4 Ψn−1(κ). Thus we have κ′ ∈ Zn−1(κ).

Now since either Ψln(ω) 4 Ψl

n(ω′) or Ψl

n(ω′) 4 Ψl

n(ω) and |Ψl−1n (ω′′)| ≤ γl−1

n < γln − K ≤ |Ψl

n(ω)|, we have

Ψl−1n (ω′′) 4 Ψl

n(ω), where ω′′ = κ′τ′1 · · · τ′l−1 ∈ Tl−1n . Thus, for j ≤ l − 1 there is a subword of Ψl

n(ω) which

starts between positions γj−1n − K and γ

j−1n + K (or between positions γn−1 − K and γn−1 + K if j = 1). Also,

τ′l ∈ D(n). This shows that

#Zln(ω) ≤ #Zn−1(κ)(2K)l−1#D(n).

Hence


n ≤ max#Zn−1(κ) : κ ∈ Tn−1(2K)l−1#D(n)

= max#Zqn−1n−1 (κ) : ω ∈ T

qn−1n−1 (2K)l−1#D(n).

Iterating this inequality and applying the definition of qn−1 we obtain


n ≤ max#Zn−2(κ) : κ ∈ Tn−2(2K)qn−1+l−2#D(n− 1)#D(n)

≤ qn−1(2K)qn−1+l .


Let (T, ν) = limn→∞ (Tn,Mn). Then T consists of all ω ∈ Σ such that there is a sequence (ωn)n∈N ∈

∏n∈N Tn such that ωn 4 ωn+1 4 ω for all n ∈ N. It follows from the construction of (Ψn)n∈N that Ψn(ωn) 4

Ψn+1(ωn+1) and |Ψn(ωn)| < |Ψn+1(ωn+1)|. Thus, there is a unique infinite word Ψ(ω) ∈ Σ with Ψn(ωn) 4

Ψ(ω) for all n ∈ N. This defines a map Ψ : T → Σ. We let S = Ψ(T) and µ = ν Ψ−1 and let S∗ = ω ∈ Σ∗ :

[ω] ∩ S 6= ∅.

In Lemmas 5.25–5.30 we shall verify that S and µ defined above have the required properties. The proof of

Proposition 5.23 thus follows.

Lemma 5.25. In the setting of the proof of Proposition 5.23, let n ∈ N, l ∈ 1, . . . , qn, and ω = κτ ∈ Tln so that

κ ∈ Tn−1 and τ ∈ F ln. Then

ν([ω]) ≤ 2ν([κ])M(n)L(n)Cl0

( ϕt(

Ψln(ω)

)ϕt(Ψn−1(κ))

).

Proof. Since ω ∈ Tln and κ ∈ Tn−1 we have ν([κ]) =Mn−1(κ) and

ν([ω]) =Mln(ω) =

Mn−1(κ)


ρn([τβ])

=ν([κ])


ρn([τ])ρn([β]) ≤ 2ν([κ])ρn([τ]).

So it suffices to show that

ρn([τ]) =l

∏j=1

ρn([τj]) ≤ M(n)L(n)Cl0

( ϕt(

Ψln(κτ)

)ϕt(Ψn−1(κ))

).

Thus, by (5.20), it suffices to show that

l

∏j=1

ρn([τj]) ≤ M(n)L(n)l

∏j=1

ϕt([τj]). (5.22)

Now either l ≥ L(n), in which case it follows from τ ∈ F ln and (5.17) that

l

∏j=1

ρn([τj]) ≤l

∏j=1

ϕt([τj]),

or l < L(n), in which case we have

l

∏j=1

ϕt([τj])−1 ≤ M(n)l ≤ M(n)L(n).

This shows (5.22) and thus completes the proof of the lemma.

Lemma 5.26. In the setting of the proof of Proposition 5.23, let n ∈N, l ∈ 1, . . . , qn, and ω ∈ Tln. Then

ν([ω]) ≤ qn−1Cqn−1+l0 ϕt(Ψl

n(ω)).

Proof. Since ω ∈ Tln we may take κ ∈ Tn−1 and τ ∈ F l

n so that ω = κτ, so by Lemma 5.25, we have

ν([ω]) ≤ 2ν([κ])M(n)L(n)Cl0

( ϕt(

Ψln(ω)

)ϕt(Ψn−1(κ))

).


Moreover, since κ ∈ Tn−1 = Tqn−1n−2 there exists κ− ∈ Tn−2 and τ− ∈ F qn−1

n−1 so that κ = κ−τ−. Applying Lemma

5.25 once more, we obtain

ν([κ]) ≤ 2ν([κ−])M(n− 1)L(n−1)Cqn−10

( ϕt(

Ψqn−1n−1 (κ)

)ϕt(Ψn−2(κ−))

).

Combining these two estimates we get

ν([ω]) ≤ 4M(n− 1)L(n−1)M(n)L(n)

ϕt(Ψn−2(κ−))Cqn−1+l

0 ϕt(Ψln(ω)

).

Noting that the definition of qn−1 implies

qn−1 ≥4M(n− 1)L(n−1)M(n)L(n)

min ϕt(Ψn−2(κ′)) : κ′ ∈ Tn−2

completes the proof.

Lemma 5.27. In the setting of the proof of Proposition 5.23, let n ∈N, l ∈ 1, . . . , qn, and ω ∈ Tln. Then,

µ([Ψln(ω)]) ≤ q2

n−1#D(n)(2KC0)qn−1+l ϕt(Ψl

n(ω)).

Proof. Note that µ([Ψln(ω)]) = ν Ψ−1

([Ψl

n(ω)])

. Moreover, Ψ−1([Ψl

n(ω)])⊂ ⋃

[η], where the union is

taken over all η ∈ Tl+1n satisfying Ψl

n(ω) 4 Ψl+1n (η). This follows from the fact that every η ∈ Tl+1

n maps to a

string Ψl+1n (η) of length |Ψl+1

n (η)| ≥ γl+1n − K ≥ γl

n ≥ |Ψln(ω)|. Since

ν([η]) ≤ qn−1Cqn−1+l+10 ϕt(Ψl+1

n (η))≤ qn−1Cqn−1+l+1

0 ϕt(Ψln(ω)

)for all η ∈ Tl+1

n satisfying Ψln(ω) 4 Ψl+1

n (η), it suffices to show that

#η ∈ Tl+1n : Ψl

n(ω) 4 Ψl+1n (η) ≤ #D(n)qn−1(2K)qn−1+l .

Now if η ∈ Tl+1n satisfies Ψl

n(ω) 4 Ψl+1n (η), then there exists some η− ∈ Tl

n with η− 4 η and Ψln(ω) 4 Ψl

n(η−)

or Ψln(η−) 4 Ψl

n(ω). By Lemma 5.21, there are at most qn−1(2K)qn−1+l such η−. Moreover, each such η− is

continued by at most #D(n) strings in Tl+1n . This finishes the proof.

Lemma 5.28. In the setting of the proof of Proposition 5.23, let τ ∈ Σ∗. If n = n(τ) is minimal so that |τ| ≤ γln − K

for some l ∈ 1, . . . , qn and let l = l(τ) be the least such l. Then |τ| > γln/2 and

µ([τ]) ≤ q3n−1(2KC0)

qn−1+l ϕt (τ) .

Proof. Since |τ| ≤ γln − K every ω ∈ Tl

n satisfies |τ| ≤ |Ψln(ω)|. Hence µ([τ]) ≤ ∑ µ

([Ψl

n(ω)])

, where the

sum is taken over all ω ∈ Tln with τ 4 Ψl

n(ω). By Lemma 5.27, for each ω ∈ Tln with τ 4 Ψl

n(ω), we have

µ([Ψln(ω)]) ≤ q2

n−1#D(n)(2KC0)qn−1+l ϕt(Ψl

n(ω))≤ q2

n−1#D(n)(2KC0)qn−1+l ϕt (τ) .

As such, we must estimate the number of ω ∈ Tln with τ 4 Ψl

n(ω). Either l > 1, in which case |τ| > γl−1n − K,

or l = 1, in which case |τ| > γqn−1−1n−1 − K. In either case |τ| > γl

n − (5K + k(n) + k(n− 1) + 2) ≥ γln/2. Now


each ω ∈ Tln with τ 4 Ψl

n(ω) is of length no more than γln and each of the final |Ψl

n(ω)| − |τ| digits is chosen

from, 1 ∪ Γ ∪D(D(n− 1)) ∪D(D(n)). Thus, there are at most

# (1 ∪ Γ ∪D(D(n− 1)) ∪D(D(n)))5K+k(n)+k(n−1)+2

words ω ∈ Tln with τ 4 Ψl

n(ω).

By the choice of qn−1, we have

qn−1 > #D(n)# (1 ∪ Γ ∪D(D(n− 1)) ∪D(D(n)))5K+k(n)+k(n−1)+2

and the claim follows.

Lemma 5.29. In the setting of the proof of Proposition 5.23, there exists a constant C ≥ 1 with µ([τ]) ≤ Cϕs(τ) for all

τ ∈ Σ∗.

Proof. Clearly we may assume τ ∈ S∗ since otherwise µ([τ]) = 0. Since each Tn consists of finitely many

elements all of length at least γn, the set S∗ ∩ Σm is finite for all m ∈ N. As such, it suffices to show that there

is N ∈N so that µ([τ]) ≤ ϕs(τ) for all τ ∈ S∗ with |τ| > N. Choose N so that (2KC0)2/(N−1) < (3/2)(t−s) and

N3(3/4)N(t−s) < 1 and let M = maxN, γN.

Given τ ∈ S∗ with |τ| > N we let n = n(τ) to be minimal so that |τ| ≤ γln − K for some l ∈ 1, . . . , qn and

take l = l(τ) to be the least such l. Then, by Lemma 5.28, we have |τ| > γln/2 and

µ([τ]) ≤ q3n−1(2KC0)

qn−1+l ϕt (τ) .

Hence 2|τ| ≥ γln ≥ lk(n) + qn−1k(n + 1) ≥ (l + qn−1)(n− 1) and

µ([τ]) ≤ |τ|3(2KC0)|τ|/(n−1)ϕt (τ) .

Since supi∈N ‖Ti‖ < 12 we have ϕt (τ) ≤ 2−|τ|(t−s)ϕs (τ) and so

µ([τ]) ≤ |τ|3((2KC0)

2/(n−1)2−(t−s))|τ|ϕs (τ) .

Since γn ≥ γnl ≥ |τ| > M ≥ γN we have n > N, so (2KC0)

2/(n−1)(1/2)(t−s) < (3/4)(t−s). Moreover, since

|τ| > N we have |τ|3(3/4)|τ|(t−s) < 1 and

µ([τ]) ≤ |τ|3((2KC0)

2/(n−1)(1/2)(t−s))|τ|ϕs (τ) ≤ |τ|3(3/4)|τ|(t−s)ϕs (τ) ≤ ϕs (τ)

finishing the proof.

Lemma 5.30. In the setting of the proof of Proposition 5.23, S ⊂ EΦ(α).

Proof. Recall that we previously made the assumption that if φi is in the sequence Φ, then also −φi is in Φ. As

such it suffices to fix φi and show that for each τ ∈ S we have

lim infm→∞

1m

m−1

∑j=0

φi(σj(τ)) ≥ αi. (5.23)


Given m ≥ γi we choose n = n(m) to be maximal so that γn ≤ m and choose l = l(m) ≤ qn+1 to be maximal so

that γln+1 ≤ m. Since τ ∈ S = Ψ(T) there is ω ∈ Tl

n+1 with Ψln+1(ω) 4 τ. It follows from the construction of

Tln+1 that ω = κω1ω2, where κ ∈ Tn−1, ω1 ∈ T

qnn and ω2 ∈ Tl

n. We deal with these three segments seperately.

Since l is maximal we have m < γl+1n+1, or m < γ1

n+2 if l = qn+1. This implies that m − |Ψln+1(ω)| ≤

2K + k(n + 1) + k(n + 2) + 1. It also follows that τ|m, the initial segment of τ of length m, consists entirely of

digits from D∗(n) = D(⋃n+2

l=1 D(l) ∪ 1 ∪ Γ). Hence, for all j ∈ 1, . . . , m we have

−φi(σj(ω)) ≤ A(n), (5.24)

where A(n) is as in (5.16). Since m ≥ γn ≥ qn > nA(n)(γn−1 + 2K + k(n + 1) + k(n + 2) + 1), we thus get

|Ψn−1(κ)|−1

∑j=0

φi(σj(τ)) +

m−1

∑j=|Ψl

n+1(ω)|φi(σ

j(τ)) ≥ −γn−1A(n)− (2K + k(n + 1) + k(n + 2) + 1)A(n)

≥ −2mn

. (5.25)

Observe that qn > L(n) and ω1 ∈ Tqnn imply

1qn

qn−1

∑j=0

Ak(n)φi(σjk(n)(ω1)) ∈ Bn(αi)

for all ω1 ∈ [ω1]. Here we have used the fact that Ak(n)φi is constant on cylinders of length k(n). Write ω1 in

the form ω1 = ω11 · · ·ω1

qn where each ω1ν ∈ D(n) ⊂ Σk(n). It follows from the construction of Ψn, along with

the fact that Ψn(κω1) 4 τ, that some set A ⊂ [γn−1, γn] ∩N of cardinality qn has the property that for each

j ∈ A there is ν ∈ 1, . . . , qn such that σjτ ∈ [ω1ν]. Let A = A + 0, 1, . . . , k(n)− 1. We may choose A so

that τν ∈ 1 ∪D (Γ) for all integers ν ∈ [γn−1 + 1, γn] \ (A + 1). Since σj(τ) ∈ [ω1ν] for each j ∈ A, we have

1qnk(n) ∑

j∈A

φi(σj(τ)) ≥ 1

qn

qn−1

∑j∈A

Ak(n)φi(σj(τ)) ≥ 1

qn

qn−1

∑ν=0

Ak(n)φi(σνk(n)(ω1)) ∈ Bn(αi).

By the construction of Ψn, the cardinality of [γn−1, γn − 1] \A is at most 4Kqn, k(n) ≥ 4Kn, and m ≥ γn ≥

k(n)qn ≥ 4Knqn. Thus #A ≥ γn − γn−1 − 4Kqn ≥ γn − 2mn and

∑j∈A

φi(σj(τ)) >

γn

(αi − 1

n

), if αi ≤ 1

n ,(γn − 2m

n) (

αi − 1n

), if αi >

1n ,(

γn − 2mn)

n, if αi = ∞.

As noted, for each j ∈ [γn−1, γn − 1] \A there is η ∈ 1 ∪D (Γ) so that σjτ ∈ [η]. Thus, for all such j,

we have φi(σj(τ)) ≥ −Ai, where Ai is as in (5.14). Moreover, since k(n) ≥ 4KnAi for i ∈ 1, . . . , n and

m ≥ γn ≥ k(n)qn, we get

∑j∈[γn−1,γn−1]\A

φi(σj(τ)) ≥ −4KAiqn ≥ −

mn

.

Putting these inequalities together we have

γn−1

∑j=γn−1

φi(σj(τ)) >

γn

(αi − 1

n

)− m

n , if αi ≤ 1n ,(

γn − 2mn) (

αi − 1n

)− m

n , if αi >1n ,(

γn − 2mn)

n− mn , if αi = ∞.

(5.26)


For the sum ∑|Ψl

n+1(ω)|−1j=γn

φi(σj(τ)), there are two cases. Either l ≥ L(n + 1), in which case there exists

ω2 ∈ [ω2] with1l

l−1

∑j=0

Ak(n+1)(φi)(σjk(n+1)ω2) ∈ Bn(αi), (5.27)

so we may proceed as in the previous case to deduce

|Ψln+1(ω)|−1

∑j=γn

φi(σj(τ)) >

(|Ψl

n+1(ω)| − γn

) (αi − 1

n

)− m

n , if αi ≤ 1n ,(

|Ψln+1(ω)| − γn − 2m

n

) (αi − 1

n

)− m

n , if αi >1n ,(

|Ψln+1(ω)| − γn − 2m

n

)n− m

n , if αi = ∞.

Recall that m − |Ψln+1(ω)| ≤ 2K + k(n + 1) + k(n + 2) + 1 ≤ γn/n ≤ m/n, so we may combine the above

inequalities to obtain

|Ψln+1(ω)|−1

∑j=γn

φi(σj(τ)) >

(m− γn)

(αi − 1

n

)− m

n , if αi ≤ 1n ,(

m− γn − 3mn) (

αi − 1n

)− m

n , if αi >1n ,(

m− γn − 3mn)

n− mn , if αi = ∞.

(5.28)

Combining (5.28) with (5.25) and (5.26) we conclude that whenever l ≥ L(n + 1), we have

1m

m−1

∑j=0

φi(σj(τ)) >

(αi − 1

n

)− 3

n , if αi ≤ 1n ,(

1− 3n) (

αi − 1n

)− 3

n , if αi >1n ,(

1− 3n)

n− 3n , if αi = ∞.

(5.29)

If l ≤ L(n + 1), then we apply (5.24) once more to obtain

|Ψln+1(ω)|−1

∑j=γn

φi(σj(τ)) ≥ −k(n + 1)L(n + 1)A(n) ≥ −γn

n≥ −m

n. (5.30)

Notice that if l ≤ L(n + 1), then |Ψln+1(ω)| − γn ≤ (4K + k(n + 1))L(n + 1) ≤ m/n. We also have m −

|Ψln+1(ω)| ≤ m/n, so m− γn ≤ 2m/n which combined with (5.26) gives

γn−1

∑j=γn−1

φi(σj(τ)) >

m(

αi − 1n

)− m

n , if αi ≤ 1n ,(

m− 4mn

) (αi − 1

n

)− m

n , if αi >1n ,(

m− 4mn

)n− m

n , if αi = ∞.

(5.31)

Combining (5.31) with (5.30) and (5.25) gives

1m

m−1

∑j=0

φi(σj(τ)) >

(αi − 1

n

)− 3

n , if αi ≤ 1n ,(

1− 4n

) (αi − 1

n

)− 3

n , if αi >1n ,(

1− 4n

)n− 3

n , if αi = ∞.

(5.32)

Since either (5.29) or (5.32) holds for all m ≥ γi and n(m) → ∞ as m → ∞ we have shown (5.23) and thus

finished the proof.

5.6. Conditional variational principle for bounded potentials 143

5.6 Conditional variational principle for bounded potentials

In this section we shall prove Theorem 5.7.

The progression from Theorem 5.6 to Theorem 5.7 is relies upon the thermodynamic formalism developed

in Section 5.3. The main challenge is to prove the upper bound which is given in Sections 5.6.2-5.6.5.

We begin by proving some elementary lemmas in Section 5.6.1. The upper bound for the interior points

of the spectrum for finitely many potentials is given in Section 5.6.2. In Section 5.6.3 we prove an upper-

semicontinuity lemma. In Section 5.6.4 we prove a lemma which allows us to extend the upper bound to the

boundary of the section. The proof of the upper bound, for all points of the spectrum and for a countabe

infinity of potentials is given in Section 5.6.5.

The lower bound in Theorem 5.7 follows reasonably straightforwardly from Theorem 5.6 and it is proved

in Section 5.6.6.

5.6.1 Space of integrals with respect to invariant measures

Recall that we defined P(Φ) to be the set

P(Φ) :=∫

Φdν : ν ∈ Mσ(Σ)

.

Lemma 5.31. Suppose that α ∈ RN satisfies EΦ(α) 6= ∅. Then α ∈ P(Φ).

Proof. It suffices to take α ∈ RN with EΦ(α) 6= ∅ and show that for each q ∈ N there exists a measure

µ ∈ Mσ(Σ) such that for all i ≤ q we have∫φidµ ∈

(αi −

1q

, αi +1q

).

Now since each φi is uniformly continuous there exists some N ∈ N for which varn(An(φi)) < (2q)−1 for

i ≤ q and all n ≥ N0. Moreover, EΦ(α) 6= ∅ so we may take ω ∈ EΦ(α). In particular there exists N1 ∈ N

such that for all i ≤ q and all n ≥ N1 we have An(φi)(ω) ∈(αi − (2q)−1, αi + (2q)−1). From these two facts it

follows that if we take N := max N0, N1 then τ ∈ Σ denote the σN fixed point with σlN(τ) ∈ [ω|N] for all

l ∈N∪ 0 then for all i ≤ q we have

AN(φi)(τ) ∈(

αi −1q

, αi +1q

).

Thus, if µ := N−1 ∑N−1i=0 δσiτ then for each i ≤ q∫

φidµ ∈(

αi −1q

, αi +1q

).

Moreover, τ is a fixed point for σN so µ is σ-invariant.

For the rest of this section we restrict our attention to potentials taking values in some finite dimensional

vector space Φ : Σ→ RN . We begin by recalling an elementary lemma concerning convex sets of RN .

If κ ∈ −1, 1N , then we define the open κ-orthant O(κ) to be the set

O(κ) = (xi)Ni=1 : κi · xi > 0 for each i.


Lemma 5.32. If C is a convex set, then a ∈ RN lies in the interior of C if and only if C ∩ (a + O(κ)) 6= ∅ for all

κ ∈ −1, 1N .

Suppose we have a collection Φ = (φi)Ni=1, consisting of finitely many bounded functions φi : Σ → R with

summable variations. We write∫

Φdν = (∫

φ1dν, . . . ,∫

φNdν) for all ν ∈ M(Σ). The space of integrals with

respect to invariant measures is P(Φ) = ∫

Φdν : ν ∈ Mσ(Σ) ⊂ RN .

Lemma 5.33. The set P(Φ) is bounded and convex. Moreover, either P(Φ) is contained within some (N − 1)-

dimensional hyperplane or P(Φ) ⊂ int (P(Φ)).

Proof. The first statement follows immediately from the fact that the mapping ν 7→∫

Φdν defined onMσ(Σ)

is bounded and affine and Mσ(Σ) is convex. The second statement follows from elementary properties of

convex sets in Euclidean spaces.

If I ⊂N is finite, then we define P(Φ, I) ⊂ P(Φ) by

P(Φ, I) = Φ(ν) : ν ∈ Mσ(Σ) and ν(IN) = 1,

Pe(Φ, I) = Φ(ν) : ν ∈ Mσ(Σ) is ergodic and ν(IN) = 1.

Lemma 5.34. It holds that

P(Φ) ⊂⋃IPe(Φ, I),

int (P(Φ)) ⊂⋃I

int (P(Φ, I)) ,

where the unions are taken over all finite subsets I ⊂N.

Proof. Let ν ∈ Mσ(Σ), α =∫

Φdν ∈ P(Φ) and ε > 0. Since each φi has summable variations we may

choose n ∈ N with varn(Anφi) < ε. For each ω ∈ Nn we let ω ∈ Σ denote the unique periodic point with

σqn(ω) ∈ [ω] for all q ∈N∪ 0.

Note that since ν is σ-invariant,∫

Anφidν =∫

φidν = αi for each i ∈ 1, . . . , N. Hence, as varn(Anφi) < ε

for each i, we have

∑ω∈Σn

ν([ω]) infτAnφi(τ) > αi − ε,

∑ω∈Σn

ν([ω]) supτAnφi(τ) < αi + ε.

(5.33)

Given a finite set I ⊂ N we let c(ν, I) = ∑ω∈In ν([ω]). Note that I may be chosen so that c(ν, I) is arbitrarily

close to one. Hence, (5.33) implies that there exists a finite set I ⊂N such that

c(ν, I)−1 ∑ω∈In

ν([ω]) infτAnφi(τ) > αi − ε,

c(ν, I)−1 ∑ω∈In

ν([ω]) supτAnφi(τ) < αi + ε.

(5.34)


for all i ∈ 1, . . . , N. Let µ′ be the unique nth level Bernoulli measure which satisfies µ′([ω]) = c(ν, I)−1ν([ω])

for all ω ∈ In. By (5.34), we have ∫Anφidµ′ ∈ (αi − ε, αi + ε) .

Now let µ = 1n ∑n−1

j=0 µ′ σ−j. Since µ′ is σn-invariant and ergodic with respect to σn, the measure µ is σ-

invariant and ergodic with respect to σ. It is also clear that µ is supported on IN. Moreover, since

∫φidµ =

1n

n−1

∑j=0

∫φidµ′ σ−j =

∫Anφidµ′ ∈ (αi − ε, αi + ε)

for all i ∈ 1, . . . , N, we have shown the first claim.

To prove the second claim, we apply Lemma 5.32. Indeed, if α ∈ int (P(Φ)), then P(Φ) ∩ (α + O(κ)) 6= ∅

for all κ ∈ −1, 1N . Since each set α + O(κ) is open it follows from the first claim that for each κ ∈ −1, 1N

there is a finite set I(κ) ⊂N with P(Φ, I(κ)) ∩ (α + O(κ)) 6= ∅. Letting I =⋃

κ∈−1,1N I(κ), we obtain a finite

set with P(Φ, I) ∩ (α + O(κ)) 6= ∅ for all κ ∈ −1, 1N . Moreover, since P(Φ, I) is convex it follows from

Lemma 5.32 that α ∈ int (P(Φ, I)). This completes the proof.

5.6.2 Upper bound for interior points of the spectrum

In this section we give the proof of the upper bound in Theorem 5.7 for interior points of the spectrum in the

special case where there are finitely many potentials.

Proposition 5.35. If (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 1, a ∈ A, the singular value function ϕs is

quasi-multiplicative for all 0 ≤ s ≤ d, Φ : Σ→ RN is bounded with summable variations, and α ∈ int(P(Φ)) then

dimH(JaΦ(α)) ≤ min

d, max


∫Φdµ = α

.

The proof of Proposition 5.35 requires two lemmas. Lemma 5.36 uses Lemma 5.21 to relate the dimension

of JΦ(α) to the pressure. Then Lemma 5.38 proves the upper bound in Theorem 5.7 proves the existence of an

appropriate maximising measure.

Lemma 5.36. If (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 1, a ∈ A, Φ : Σ → RN is bounded with summable

variations, α ∈ RN , and s < dimH(JaΦ(α)), then

P(ϕs · e〈q,Φ−α〉) ≥ 0

for all q ∈ RN .

Proof. Fix s < dimH(JΦ(α)) and q ∈ RN . By Lemma 5.21, for each n ∈N there exists k(n) ∈N such that

∑ω∈AΦ(α,n,k)

ϕs(ω) > 1


for all k ≥ k(n). Thus

∑ω∈Σk

ϕs(ω) supτ∈[ω]

exp (Sn 〈q, Φ− α〉 (τ)) = ∑ω∈Σk


exp( N

∑i=1

qi (Snφi − nαi)

)

≥ ∑ω∈AΦ(α,n,k)


exp( N

∑i=1

qi (Snφi − nαi)

)≥ ∑

ω∈AΦ(α,n,k)ϕs(ω) · e−Nqk/n > e−Nqk/n,

where q = maxi∈1,...,N qi. Hence

P(ϕs · e〈q,Φ−α〉) = limk→∞

1k log

(∑

ω∈Σk


exp (Sn 〈q, Φ− α〉 (τ)))≥ −Nq

n.

Letting n→ ∞ completes the proof of the lemma.


multiplicative for all 0 ≤ s ≤ d, Φ : Σ → RN is bounded with summable variations, α ∈ RN , q ∈ RN , and s > s∞,

then the potential ϕs · e〈q,Φ−α〉 is quasi-multiplicative and P(ϕs · e〈q,Φ−α〉) < ∞. Moreover, if µ is the Gibbs measure for

ϕs · e〈q,Φ−α〉, then Λµ(ϕs) > −∞.

Proof. Observe that the quasi-multiplicativity follows immediately from Lemma 5.16. Since Φ is bounded we

have B = sup|〈q, Φ(ω)− α〉| : ω ∈ Σ < ∞. This together with P(ϕs) < ∞ gives P(ϕs · e〈q,Φ−α〉) = P < ∞.

Thus, by Theorems 5.12 and 5.13, the Gibbs measure µ for ϕs · e〈q,Φ−α〉 exists.

To prove the last claim, let m ∈ Z be so that m < s ≤ m + 1. By the Gibbs property of µ there is a constant

C ≥ 1 so that

µ([ω]) ≤ Cϕs(ω)e〈q,Φ−α〉(ω)e−nP ≤ Cϕs(ω)en(B−P)

for all ω ∈ Σn and n ∈N. Now, following the proof of Lemma 5.20, this implies Λµ(ϕs) > −∞.


multiplicative for all 0 ≤ s ≤ d, Φ : Σ → RN is bounded with summable variations, α ∈ int(P(Φ)) ⊂ RN , and

s > s∞ satisfies

infq∈RN

P(ϕs · e〈q,Φ−α〉) ≥ 0,

then there exists an ergodic invariant measure µ ∈ Mσ(Σ) with∫

Φdµ = α and D(µ) ≥ s.

Proof. We shall consider the function F : RN → R defined by F(q) = P(ϕs · e〈q,Φ−α〉). Since α ∈ int (P(Φ))

we may apply the latter claim of Lemma 5.34 to obtain a finite subset I ⊂ N with α ∈ int (P(Φ, I)). Since I

is finite and all of the matrices Ti are non-singular we have c = minγd(Ti) : i ∈ I > 0, ϕs(ω) ≥ cn for all

ω ∈ Σn, and Λµ(ϕs) ≥ log c > −∞ for all µ ∈ Mσ(Σ) with µ(IN) = 1. Since α ∈ int (P(Φ, I)) there exists

ε > 0 with Bε(α) ⊂ P(Φ, I). Hence for each q ∈ RN \ 0 we have α + εq/‖q‖ ∈ P(Φ, I) and there exists a

measure νq ∈ Mσ(Σ) with Λνq(ϕs) ≥ log c satisfying

∫Φdνq = α + εq/‖q‖.


By Lemma 5.2 and the boundedness of Φ, we have

F(q) ≥ hνq +⟨q,∫

Φdνq − α⟩+ Λνq(ϕs) ≥ 〈q, εq/‖q‖〉+ log c = ε‖q‖+ log c.

Hence F(q)→ ∞, as ‖q‖ → ∞. It follows from Lemma 5.17 that F attains a global minimum on a bounded set.

Let q(α) denote a point at which this global minimum is attained.

By Lemma 5.37, the Gibbs measure µq for ϕs · e〈q,Φ−α〉 satisfies Λµq(ϕs) > −∞. Thus, applying Lemmas

5.14 and 5.2, we obtain

F(q) = hµq +⟨q,∫

Φdµq − α⟩+ Λµq(ϕs).

Moreover, by Lemma 5.18 for each i ∈ 1, . . . , N we have

∂F(q)∂qi

∣∣∣∣q=q∗

= limqi→q∗i

P(

ϕs · e〈q∗ ,Φ−α〉 · e(qi−q∗i )(φi−αi)

)− P

(ϕs · e〈q∗ ,Φ−α〉

)qi − q∗i

=∂P((ϕs · e〈q∗ ,Φ−α〉) · eqi(φi−αi)

)∂qi

∣∣∣∣qi=0

=∫

φidµq∗ − αi.

Since F attains a minimum at q(α) it follows that∫

φidµq(α) = αi for all i ∈ 1, . . . , N. Thus, denoting

µ = µq(α), we have∫

Φdµ = α,

hµ + Λµ(ϕs) = hµ +⟨q(α),

∫Φdµ− α

⟩+ Λµ(ϕs) = P(ϕs · e〈q,Φ−α〉) ≥ 0,

and D(µ) ≥ s.

Proof of Proposition 5.35. Take α ∈ int(P(Φ)). Either dimH(

JaΦ(α)

)≤ s∞, in which case the upper bound holds,

or dimH(

JaΦ(α)

)> s∞. If dimH

(JaΦ(α)

)> s∞ then we may choose s ∈

(s∞, dimH

(JaΦ(α)

)). By Lemmas 5.36

and 5.38 there exists some µ ∈ Mσ(Σ) with D(µ) ≥ s. This completes the proof of the proposition.

This upper bound is extended to the full generality of Theorem 5.7 in Section 5.6.5. We first require some

additional lemmas.

5.6.3 Quasi upper-semicontinuity lemma

Recall that because of the non-compactness of the shift space, the space of invariant probability measures is

non-compact and the entropy is not upper semi-continuous. Nonetheless we do have the following proposi-

tion.

Proposition 5.39. If (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 1, (µn)n∈N is a sequence with µn ∈ Mσ(Σ)

for all n ∈ N and lim supn→∞ D(µn) > s∞, then there exists a sub-sequence (µnj)j∈N and a measure µ ∈ Mσ(Σ)

which is a weak∗ limit point of(

µnj

)j∈N

and satisfies D(µ) ≥ lim supn→∞ D(µn).

Before proving the proposition, we first prove a few elementary lemmas. Let P be the set of all infinite prob-

ability vectors, that is, P =(qi)i∈N ∈ [0, 1]N : ∑∞

i=1 qi = 1

, and equip it with the usual product topology. If

a = (ai)i∈N is a sequence of numbers in (0, 1) and C > 0, then we set P(a, C) = (qi)i∈N ∈ P : ∑i∈N qi log ai ≥

−C.


Lemma 5.40. If a = (ai)i∈N is a sequence of numbers in (0, 1) and C > 0, then P(a, C) is closed.

Proof. Let p = (pi)i∈N be an accumulation point of P(a, C) and let (p(n))n∈N be a sequence so that p(n) =

(pi(n))i∈N ∈ P(a, C) for all n ∈ N and limn→∞ p(n) = p. Suppose for a reductio that ∑∞i=1 pi log ai < −C.

Then there exists k ∈ N so that ∑ki=1 pi log ai < −C. Choosing now n0 ∈ N so that ∑k

i=1 pi(n) log ai < −C for

all n ≥ n0, we have arrived at a contradiction since ∑∞i=1 pi(n) log ai ≤ ∑k

i=1 pi(n) log ai < −C for n ≥ n0.

Lemma 5.41. If a = (ai)i∈N is a non-increasing sequence of numbers in (0, 1) with ∑∞i=1 ai < ∞ and C > 0, the the

function F : P(a, C)→ R, defined by F(q) = ∑∞i=1 qi log(ai/qi), is upper semi-continuous.

Proof. Let Pk =(qi)i∈N ∈ P : qi = 0 for all i ∈ k + 1, k + 2, . . .

for all k ∈N and define a map ξk : P→ Pk

by setting

ξk(q)i =

qi, if i ∈ 1, . . . , k− 1,

∑∞j=k qj, if i = k,

0, if i ∈ k + 1, k + 2, . . .,

for all q = (qi)i∈N ∈ P. Take a sequence of vectors (p(n))n∈N with each p(n) = (pi(n))i∈N ∈ P(a, C)

along with p = (pi)i∈N ∈ P such that for each i ∈ N we have limn→∞ pi(n) = pi. In particular, we have

∑∞i=1 pi(n) log ai ≥ −C. Our goal is to show that lim supn→∞ F(p(n)) ≤ F(p).

Since limn→∞ pi(n) = pi for all i ∈ 1, . . . , k − 1 and ∑∞i=k qi = 1 − ∑k−1

i=1 qi for all q ∈ P we have

limn→∞ ξk(p(n)) = ξk(p) with respect to the supremum metric. Hence limn→∞ F(ξk(p(n))) = F(ξk(p)). Simi-

larly, as in (5.4), we see that∞

∑i=k

qi

(log

aiqi− log

∞

∑j=k

aj

)≤ −

∞

∑i=k

qi log∞

∑j=k

qj,

and, consequently,

F(q) ≤k−1

∑i=1

qi logaiqi

+∞

∑i=k

qi log∑∞

j=k aj

∑∞j=k qj

= F(ξk(q))−∞

∑i=k

qi log ak +∞

∑i=k

qi log∞

∑j=k

aj

for all q ∈ P(a, C) and k ∈N. Choosing k0 ∈N so that ∑∞j=k aj < 1 for all k ≥ k0, this implies

F(ξk(p)) = limn→∞

F(ξk(p(n))) ≥ lim supn→∞

(F(p(n)) +

∞

∑i=k

pi(n) log ak)

≥ lim supn→∞

F(p(n)) + lim infn→∞

∞

∑i=k

pi(n) log ak ≥ lim supn→∞

F(p(n)) +∞

∑i=k

pi log ak

for all k ≥ k0 by Fatou’s lemma. Note that since the sequence (ai)i∈N is non-increasing we have ∑∞i=k pi log ak ≥

∑∞i=k pi log ai for all k ∈ N. Moreover, let ε > 0 and, by recalling Lemma 5.40, choose kε ∈ N so that

∑∞i=k pi log ai > −ε and ∑∞

i=k pi log ∑∞j=k pj > −ε for all k ≥ kε. Since

F(p)− F(ξk(p)) =∞

∑i=k

pi logaipi−

∞

∑i=k

pi logak

∑∞j=k pj

≥∞

∑i=k

pi log ai +∞

∑i=k

pi log∞

∑j=k

pj > −2ε

for all k ≥ kε we have

lim supn→∞

F(p(n)) ≤ lim supk→∞

(F(ξk(p))−

∞

∑i=k

pi log ak)≤ lim sup

k→∞F(ξk(p)) + ε ≤ F(p) + 3ε.

Letting ε ↓ 0 finishes the proof.


Proof of Proposition 5.39. We begin by showing that (µn)n∈N has a convergent subsequence. Let δ = lim supn→∞ D(µn)

and choose m < δ ≤ m + 1. If maxs∞, m < t0 < t1 < δ, then there exists a subsequence (nj)j∈N with

D(µnj) > t1 for all j ∈N and limj→∞ D(µnj) = δ. It follows that Λµnj(ϕt0) ≥ Λµnj

(ϕt1) > −∞ and

0 ≤ Pµnj(ϕt1) ≤ 1

k ∑ω∈Σk

µnj([ω]) logϕt1(ω)

µnj([ω])

for all k, j ∈N. Furthermore, recalling (5.4) and Lemma 5.8, we have

∑ω∈Σk

µnj([ω]) logϕt0(ω)

µnj([ω])≤ log Zk(ϕt0) ≤ k log Z1(ϕt0) < ∞

for all k, j ∈N. Since ϕt1(ω) = ϕt0(ω)γm+1(ω)t1−t0 we get

∑ω∈Σk

µnj([ω]) log γm+1(ω) ≥ − k log Z1(ϕt0)

t1 − t0(5.35)

for all k, j ∈N. Note that for every ε > 0 there is M ∈N so that ∑∞i=M γm+1(Ti)

m+1 ≤ ∑∞i=M ϕt0(Ti) < ε. Thus

for each ε > 0 there are only finitely many i’s so that log γm+1(Ti) ≥ (m + 1)−1 log ε. Therefore, (5.35) implies

that the sequence (µnj)j∈N is tight and thus has a converging subsequence. We keep denoting the subsequence

by (µnj)j∈N and let µ ∈ Mσ(Σ) be its weak∗ limit.

Let maxs∞, m < s < δ. Since ϕs(ω) ≥ γm+1(ω)m+1 for all ω ∈ Σk it follows from (5.35) that

∑ω∈Σk

µnj([ω]) log ϕs(ω) ≥ − k(m + 1) log Z1(ϕt0)

t1 − t0

for all k, j ∈ N. According to Lemma 5.40, the same estimate holds when the measure µnj is replaced by µ.

Thus Λµ(ϕs) ≥ −(m + 1)(t1 − t0)−1 log Z1(ϕt0) > −∞. Furthermore, since s < δ there is j0 ∈ N so that

D(µnj) > s for all j ≥ j0. Therefore, Lemma 5.41 implies

∑ω∈Σk

µ([ω]) logϕs(ω)

µ([ω])≥ lim sup

j→∞∑

ω∈Σk

µnj([ω]) logϕs(ω)

µnj([ω])≥ 0

and D(µ) ≥ s. The proof is finished since maxs∞, m < s < lim supn→∞ D(µn) was arbitrary.

5.6.4 Finitely many potentials lemma

In this section we prove a technical lemma which allows us to prove the upper bound in Theorem 5.7 for

boundary points of the spectrum.

Lemma 5.42. If (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 12 , the singular value function ϕs is quasi-

multiplicative for all 0 ≤ s ≤ d, a ∈ A, Φ : Σ → RN is bounded with summable variations, P(Φ) is not con-

tained within any (N − 1)-dimensional hyperplane, and α ∈ P(Φ), then for each ε > 0 there is γ ∈ int(P(Φ)) with

|α− γ| < ε and dimH(JaΦ(γ)) ≥ dimH(Ja

Φ(α))− ε for LA-almost all a ∈ A.

Proof. Fix ε > 0 and let dimH(JaΦ(α)) − ε < s < t < dimH(Ja

Φ(α)). By Lemma 5.33 and the first part of

Lemma 5.34, we may choose β ∈ int (P(Φ)) ∩ Pe(Φ, I), with respect to some finite subset I ⊂ N, satisfying

|β − α| < 1/n. Since β ∈ Pe(Φ, I) there is an ergodic invariant measure ν ∈ Mσ(Σ) with ν(IN) = 1 and


∫φidν = βi for all i ∈ 1, . . . , N. Since ν(IN) = 1 we also have Λν(ϕs) > −∞. By the sub-additive ergodic

theorem there exist τ ∈ Σ, a constant θ(τ) > 0, and L(τ) ∈N such that

|Alφi(τ)− βi| < l−1 and ϕs(τ|l) ≥ θ(τ)l

for all l ≥ L(τ) and i ∈ 1, . . . , N. Choose 0 < ρ < min1, ε/|β− α| so that

2(1−ρ)(t−s)θ(τ)ρ > 1

and let γ = ρβ + (1− ρ)α. Since β ∈ int(P(Φ)) and α ∈ P(Φ) we have γ ∈ int(P(Φ)) by the elementary

properties of convex sets in RN . Moreover, since ρ < ε/|β− α| we have |α− γ| < ε. We shall now show that

dimH(JaΦ(γ)) ≥ dimH(Ja

Φ(α))− ε.

Since t < dimH(JaΦ(γ)), it follows from Lemma 5.21 that for all l ∈N there exists q(l) ∈N such that

∑κ∈AΦ(α,l,q)

ϕt(κ) > 1.

for all q ≥ q(l). Since supi∈N ‖Ti‖ < 12 and s < t it follows that

∑κ∈AΦ(α,l,q)

ϕs(κ) > 2l(t−s).

for all q ≥ q(l). For every α, β ∈ Σ∗, according to the quasi-multiplicativity of ϕs, there exists ω ∈ Γ such that

ϕs (αωβ) ≥ cϕs(α)ϕs(β),

where c > 0 is a constant depending only on s. As in Section 5.5.2, we write α ? β for αωβ. Note that for any

given α, β ∈ Σ∗ there are at most K = max |ω| : ω ∈ Γ finite words β′ ∈ Σ∗ with α ? β′ = α ? β

Our choice of ρ implies that for each l ∈N there exists A(l) > max(1− ρ)−1q(l), ρ−1L(τ), l such that

cθ(τ)(2(1−ρ)(t−s)θ(τ)ρ

)k> 1

for all k ≥ A(l). It follows that

∑κ∈AΦ(α,l,dk(1−ρ)e)

ϕs(κ ? (τ|dkρe)) ≥ c ∑κ∈AΦ(α,l,dk(1−ρ)e)

ϕs(κ)ϕs(τ|dkρe)

> c2dk(1−ρ)e(t−s)θ(τ)dρke > cθ(τ)(2(1−ρ)(t−s)θ(τ)ρ

)k> 1

(5.36)

for all for all k ≥ A(l) and l ∈N. We shall temporarily fix l ≥ d and k ≥ A(l), and for each κ ∈ AΦ(α, l, dk(1−

ρ)e) we let r(κ) be |κ ? (τ|dkρe)| − dkρe. Since [κ] ⊂ [κ] for κ = (κ ? (τ|dkρe))|k we have

dk(1− ρ)e(αi − l−1) < Sdk(1−ρ)eφi(ω) < dk(1− ρ)e(αi + l−1)

for each i ∈ 1, . . . , N by the definition of AΦ(α, l, dk(1− ρ)e). We also have

dkρe(βi − l−1)− (K + 1)‖φi‖ < Sk−rφi(σr(ω)) < dkρe(βi + l−1) + (K + 1)‖φi‖.

Since r− dk(1− ρ)e ≤ K it follows that

dk(1− ρ)e(αi − l−1) + dkρe(βi − l−1)− (2K + 1)‖φi‖ < Skφi(ω)

< dkρe(βi + l−1) + (2K + 1)‖φi‖+ dk(1− ρ)e(αi + l−1).


Furthermore, since |αi|, |β| < ‖φi‖, γi = (1− ρ)αi + ρβi, and k ≥ l we have

γi − l−1 ((2K + 3)‖φi‖+ 1) < Akφi(ω) < γi + l−1 ((2K + 3)‖φi‖+ 1) .

Hence, if Q = ((2K + 3)maxi∈1,...,N ‖φi‖+ 1)−1, then we have κ ∈ AΦ(γ, n, k) for all n ≤ bQlc. Since κ is an

initial substring of κ ? (τ|dkρe) for any given κ ∈ AΦ(α, l, dk(1− ρ)e) it follows from (5.36) that

∑κ∈AΦ(γ,n,k)

ϕs(κ) ≥ ∑κ∈AΦ(α,l,bk(1−ρ)c)

ϕs(κ ? (τ|dkρe)) > 1

for all n ≤ bQlc and k ≥ A(l). For each n we choose l(k) ∈ N so that n ≤ bQl(n)c and let B(n) = A(l(n)). It

follows that

∑κ∈AΦ(γ,n,k)

ϕs(κ) > 1

for all n ∈N and for all k ≥ B(n). As in the proof of Proposition 5.22, we get

dimH(JaΦ(α))− ε < s ≤ lim

n→∞limk→∞

sup


Ak(φi)dµ ∈ Bn(γi) for all i ∈ 1, . . . , n

for all a ∈ A. Theorem 5.6 finishes the proof.

5.6.5 Proof of the upper bound in Theorem 5.7

Throughout this section we assume that (Ti)i∈N ∈ GLd(R)N is such that supi∈N ‖Ti‖ < 1, a ∈ A, the singular

value function ϕs is quasi-multiplicative for all 0 ≤ s ≤ d. We shall prove the following upper bound.

Proposition 5.43. Suppose that Φ : Σ→ RN is bounded with summable variations and α ∈ P(Φ). Then


d, max


∫Φdµ = α

.

The proof of Proposition 5.43 is below. We first deal with the special case in which Φ takes values in RN .

Recall that in Section 5.6.2 we proved the following proposition.

Proposition 5.35. Suppose that Φ : Σ→ RN is bounded with summable variations and α ∈ int(P(Φ)). Then


d, max


∫Φdµ = α

.

We begin by extending this theorem to the closure of the interior.

Proposition 5.44. Suppose that Φ : Σ→ RN is bounded with summable variations and α ∈ int(P(Φ)). Then


d, max


∫Φdµ = α

.

Proof. If int(P(Φ)) = ∅ then the proposition is trivial, so we may as well suppose that int(P(Φ)) 6= ∅, so

P(Φ) cannot be contained within any (N − 1) dimensional hyper plane. In addition, if dimH(JaΦ(α)) ≤ s∞

then the conclusion of the proposition holds, so we may as well assume that dimH(JaΦ(α)) > s∞.


Fix α ∈ int(P(Φ)). By Lemma 5.42, for each n ∈ N we may choose γn ∈ int(P(Φ)) with |α− γn| < n−1

and dimH(JaΦ(γn)) ≥ (Ja

Φ(α)) − n−1 for for LA-almost all a ∈ A. Since dimH(JaΦ(α)) > s∞ we must have

dimH(JaΦ(γn)) > s∞ + n−1 for all sufficiently large n. By Proposition 5.35 it follows that there exists measures

µn ∈N with∫

Φdµn = γn and

D(µn) > dimH(JaΦ(γn))−

1n

.

In particular D(µn) > s∞. Now by Proposition 5.39 this implies that the sequence (µn)n∈N has a weak ∗ limit

point µ ∈ Mσ(Σ) with D(µ) ≥ lim supn→∞ D(µn). That is, D(µ) = dimH(JaΦ(α)). Moreover, limn→∞ γn = α

so∫

Φdµ = α.

Proposition 5.45. Suppose that Φ : Σ→ RN is bounded with summable variations, and α ∈ P(Φ). Then


d, max


∫Φdµ = α

.

Proof. We let (φi)Ni=1 denote the collection of real-valued maps with Φ(ω) = (φi(ω))N

i=1 for each ω ∈ Σ. We

begin by taking the smallest possible integer M ≤ N so that there is an M-dimensional affine subspace of RN

which contains P(Φ). Then, there exists (jl)Ml=1 ⊂ 1, · · · , N along with reals (γil)

0≤l≤M1≤i≤N ∈ RMN such that

for all µ ∈ Mσ(Σ) and all i ∈ 1, · · · , N we have

∫φidµ = γi0 +

M

∑l=1

γil

∫φjl dµ.

Now let Φ : Σ→ RM denote the potential defined by

ω 7→ Φ(ω) =(φjl (ω)

)Ml=1 for ω ∈ Σ.

Given α = (αi)Ni=1 ∈ P(Φ) we let α := (αjl )

Ml=1. It follows that Ja

Φ(α) ⊂ JaΦ(α). Moreover, by our choice of M,

P(Φ) ⊂ RM cannot be contained within any proper (M− 1)-dimensional affine space. Thus, by lemma 5.33

we have P(Φ) = int(P(Φ)). Moreover, α ∈ P(Φ) so α ∈ P(Φ). Consequently, by Proposition 5.44 we have

dimH(JaΦ(α)) ≤ dimH(Ja

Φ(α))

≤ min

d, max

s∞, supD(µ) : µ ∈ Mσ(Σ) so that∫

Φdµ = α

.

Now given any µ ∈ Mσ(Σ) with∫

Φdµ = α we have∫

φjl = αjl for l = 1, · · · , M, so for each i = 1, · · · , N,

∫φidµ = γi0 +

M

∑l=1

γil

∫φjl dµ = γi0 +

M

∑l=1

γilαjl = αi.

That is,∫

Φdµ = α. Hence we have


d, max


∫Φdµ = α

.

Proof of Proposition 5.43 given Proposition 5.45. Take a bounded potential Φ : Σ → RN with summable varia-

tions and some α ∈ P(Φ). We shall apply Proposition 5.39 in a similar way to the proof of Proposition 5.44.


Again, if dimH(JaΦ(α)) ≤ s∞ then the conclusion of the proposition holds trivially, so we may as well assume

that dimH(JaΦ(α)) > s∞.

We take φi : Σ→ R and αi ∈ R so that Φ = (φi)i∈N and α = (αi)i∈N. For each n ∈N we take Φn := (φi)ni=1

and αn = (αi)ni=1. Then for each n ∈N we have Ja

Φ(α) ⊂ JaΦn(αn). Thus, by applying Proposition 5.45 we have

dimH(JaΦ(α)) ≤ dimH(Ja

Φn(αn))

≤ min

d, max

s∞, supD(µ) : µ ∈ Mσ(Σ) so that∫

Φndµ = αn

.

Since dimH(JaΦ(α)) > s∞ we see that for each n ∈N we may choose µn ∈ Mσ(Σ) so that

D(µn) > max

dimH(JaΦ(α))−

1n

, s∞

,

and∫

Φn = αn. By applying Proposition 5.39 we see that the sequence (µn)n∈N has a limit point µ ∈N with

D(µ) ≥ lim supn→∞

D(µn) ≥ dimH(JaΦ(α)).

Moreover,∫

Φndµ = αn for each n ∈N so∫

Φdµ = α.

This completes the proof of Proposition 5.43.

5.6.6 Proof of the lower bound in Theorem 5.7

In this section, we shall prove the lower bound in Theorem 5.7.


multiplicative for all 0 ≤ s ≤ d, Φ = (φi)i∈N : Σ → RN consists of bounded functions φi : Σ → R, each with

summable variations, and α ∈ RN, then

dimH(JaΦ(α)) ≥ min

d, max


∫Φdµ = α


By Theorem 5.24, to prove Proposition 5.46, it suffices to show that

limn→∞

limk→∞

sup

Dk(µ) : µ ∈ M∗σk (Σ) so that

∫Aiφidµ ∈ Bn(αi) for all i ∈ 1, . . . , n

≥ max


∫Φdµ = α

.

This inequality is shown in the following two lemmas.



summable variations, and α ∈ RN, then

D(µ) ≤ limn→∞

limk→∞

sup



for all µ ∈ Mσ(Σ) with

∫Φdµ = α.


Proof. Fix µ ∈ Mσ(Σ) with∫

Φdµ = α and let 0 ≤ s < D(µ). It follows that Pµ(ϕs) ≥ 0. Thus

∑ω∈Σk

µ([ω]) logϕs(ω)

µ([ω])≥ 0

and Dk(µ) ≥ s for all k ∈N. Moreover, since µ is σ-invariant we have

∫AkΦdµ =

∫Φdµ = α ∈ Bn(α)

for all k, n ∈N. Hence

s < limk→∞

sup

Dk(ν) : ν ∈ M∗σk (Σ) so that

∫Aiφidν ∈ Bn(αi) for all i ∈ 1, . . . , n

Letting n→ ∞ and s ↑ D(µ) completes the proof of the lemma.



summable variations, then

s∞ ≤ limn→∞

limk→∞

sup



for all α ∈ P(Φ) ⊂ RN.

Proof. It suffices to show that for any s < s∞ and n ∈ N there exists k(n) ∈ N such that for all n ≥ k(n) there

exists µ ∈ M∗σk (Σ) with Dk(µ) ≥ s and

∫Akφidµ ∈ Bn(αi) for i ∈ 1, . . . , n.

First take k0(n) so that vark (Akφi) < (4n)−1 for all i ∈ 1, . . . , n. Let Φn : Σ → Rn denote the potential

(φi)ni=1. Since α ∈ P(Φ) we have (αi)

ni=1 ∈ P(Φn) ⊂ ⋃I Pe(Φn, I) by Lemma 5.34. Thus there exists an ergodic

invariant measure ν with∫

φidν ∈ B4n(αi) for all i ∈ 1, . . . , n. Since ν is ergodic we obtain τ ∈ Σ and k(n) ≥

k0(n) such that for all k ≥ k(n) we have Akφi(τ) ∈ B4n(αi). Since k(n) ≥ k0(n) we have Akφi(κ) ∈ B2n(αi) for

all k ≥ k(n) and κ ∈ [τ|k]. Now choose ρ ∈ (0, 1) sufficiently large that

ρ(αi − 1

2n)− (1− ρ)‖φi‖ > αi − 1

n and ρ(αi +

12n)+ (1− ρ)‖φi‖ < αi +

1n .

for all i ∈ 1, . . . , n. It follows that for any k ≥ k(n), any measure µ with µ([τ|k]) = ρ will satisfy∫

Akφidµ ∈

Bn(αi) for all i ∈ 1, . . . , n.

Since s < s∞ we have ∑ω∈Σkϕs(ω) = ∞ for all k ∈ N. As such, for each k ≥ k(n) we choose a finite subset

C(k) ⊂ Σk \ τ|k with

∑ω∈C(k)

ϕs(ω) > (ϕs(τ|k))−ρ/(1−ρ).

Let µ denote the unique k-th level Bernoulli measure satisfying

µ(ω) =

(1− ρ)ϕs(ω)/ ∑κ∈C(k) ϕs(κ), if ω ∈ C(k),

ρ, if ω = τ|k,

0, if ω /∈ C(k) ∪ τ|k.

5.7. Self-affine sets of a prescribed dimension 155

Since µ([τ|k]) = ρ we have∫

Akφidµ ∈ Bn(αi) for i ∈ 1, . . . , n. Moreover,

∑ω∈Σk

µ([ω]) logϕs(ω)

µ([ω])= ρ log

ϕs(τ|k)ρ

+ ∑ω∈C(k)

(1− ρ)ϕs(ω)

∑κ∈C(k) ϕs(κ)log

∑κ∈C(k) ϕs(κ)

(1− ρ)

≥ ρ log ϕs(τ|k) + (1− ρ) log(

∑κ∈C(k)

ϕs(κ)

)> 0.

Hence Dk(µ) > s. This completes the proof of the lemma.

5.7 Self-affine sets of a prescribed dimension

Whilst Falconer’s formula holds for “typical” self-affine sets it appears to be very difficult to provide explicit

examples of finitely generated self-affine sets which satisfy Falconer’s formula. Heuter and Lalley have pro-

vided a class of examples in the plane, but all of these cases are of dimension below one [HL]. However, for

infinitely generated self-affine sets it is relatively straightforward to produce explicit examples for any given

dimension by generalising a construction due to Mauldin and Uranski (see Example 2.81).

Example 5.49. Fix d ∈N and choose any ρ ∈ (0, d]. Take any constant c ∈ (0, 1). LetD :=(n, k) ∈N2 : k ≤ 2n2−1

and for each (n, k) ∈ D we let An,k ∈ GLd(R) be a diagonal matrix with

c · 2−(n2+n+d)ρ−1 ≤ ||A−1n,k ||

−1HS < 2−(n

2+n+d)ρ−1.

Then, for any s ≥ ρ we have

∑(n,k)∈D

||An,k||sHS = ∑n∈N

2n2−1 ·(

2−(n2+n+d)ρ−1

)s(5.37)

≤ ∑n∈N

2n2−1 · 2−(n2+n+d) < 2−d.

By considering the case s = d we may choose a sequence (an,k)(n,k)∈D ⊂ Rd so that for each (n, k) ∈ D An,k

([0, 1]d

)+

an,k ⊂ [0, 1]d and the collection of orthotopesAn,k

([0, 1]d

)+ an,k

(n,k)∈D

,

are disjoint.

For each (n, k) ∈ D we shall let Sn,k : [0, 1]d → [0, 1]d by x 7→ An,k (x) + an,k. Let Λ be the self-affine limit set

associated to the iterated function system (Sn,k)(n,k)∈D . It follows that

dimH (Λ) = ρ.

To see this let ϕs denote the corresponding singular value function. Observe that for all s ∈ R we have

P(ϕs) ≤ lim supn→∞

1n

log

∑(n,k)∈D

||An,k||sHS

n

.

Consequently, it follows from equation (5.37) that for all s ≥ ρ we have P(ϕs) < 0.

Thus, by Lemma 5.21 we have

dimH (Λ) ≤ s.


Recall that Lemma 5.21 made no assumptions upon the affine maps involved.

In addition, if s < ρ then we have

∑(n,k)∈D

(||A−1

n,k ||−1HS

)s= cs · ∑

n∈N

2n2−1 ·(

2−(n2+n+d)ρ−1

)s(5.38)

≤ cs · ∑n∈N

2n2−1 · 2−(n2+n+d)(s/ρ) = ∞.

Consequently, for each s < ρ we may extract a finite subset Ds ⊂ D for which

∑(n,k)∈Ds

(||A−1

n,k ||−1HS

)s> 1.

Thus, s∞ = ρ.

Now let Λs be the limit set associated with the iterated function system (Sn,k)(n,k)∈Ds . Clearly Λs ⊂ Λ. Note also

that the family of orthotopes An,k

([0, 1]d

)+ an,k

(n,k)∈Ds

,

form a disjoint collection of subregions of [0, 1]d. Thus, it follows from Theorem [F4, Proposition 2] that dimHΛs ≥ s.

Hence, dimHΛ ≥ s. Since this holds for all s < ρ we obtain

dimHΛ = ρ.

6

Multifractal analysis on the space of lattices

6.1 Introduction

In this chapter we shall study the multifractal analysis of Birkhoff averages for the diagonal flow on the space

of unimodular lattices. In recent years this dynamical system has attracted a great deal of attention, largely

motivated by connections with the theory of Diophantine approximation (see [Kl, Mg2]).

From a dimesion theory perspective this flow is interesting in three respects. It is a partially hyperbolic flow,

on a non-compact space, lacking any discernible Markov-type structure. In spite of these challenges there has

been some progress. Much of this has focused on various forms of limsup sets discussed in [Kl], where the

lack of Markov structure proves less problematic. However, in a recent breakthrough Cheung has determined

the Hausdorff dimension of the set of divergent points for SL3(Z)\SL3(R) [Ch]. A divergent point is one such

that, for each bounded set, there is a time beyond which the point remains outside of the bounded set under

the action of the flow. Einsiedler and Kadyrov have given an alternative proof of the upper bound by giving

an upper bound on the entropy of sequences of invariant probability measures which have the zero measure

as a weak star limit [EK]. Kadyrov provides a converse by constructing sequences of compactly supported

measures with entropy arbitrarily close to this upper limit for which the weak star limit is the zero measure

[Ka].

We shall combine ideas from Kadyrov [Ka] with the techniques of multifractal analysis to determine the

Hausdorff dimension of the level sets for uniformly continuous potentials which go to infinity in the cusp.

Previous work on multifractal analysis for flows has focused on hyperbolic dynamical systems. Pesin and

Sadovskaya have determined the dimension spectrum for the local dimension of equilibrium measures on

conformal Axiom A flows [PS]. Key to their approach is the fact that Axiom A flows may be modelled by

a suspension flow over a hyperbolic Axiom A diffeomorphism, so one may employ the Markov structure of

Axiom A diffeomorphisms. Barreira and Iommi have also consider the entropy spectrum for suspension flows

over countable Markov shifts [BI].

The dimension spectrum for arbitrary suspension flows over countable Markov shifts seems more prob-

lematic. Moreover, whilst in the geodesic flow on SL2(Z)\SL2(R) may be modelled by a suspension flow over

the natural extension of the Gauss map, this is not the case in higher dimensions. Thus, rather than move to a

157

158 Chapter 6. Multifractal analysis on the space of lattices

suspension flow we instead consider the discrete time dynamical system given by the time-one map. Whilst

the time-one map lacks Markov structure and does not satisfy the specification property, a shadowing lemma

due to Kadyrov [Ka] enables us to construct points with a prescribed orbit structure. In place of the symbolic

space used for Markov systems we deploy an infinite product of homogeneous spaces, embedded into the

original homogeneous space through repeated use of topological mixing and the shadowing lemma.

6.2 Multifractal analysis for flows

Suppose we have a flow ( ft)t∈R on a space X. Given a continuous function ϕ : X → R we are interested in the

asymptotic behaviour of the Birkhoff averages of ϕ. Given α ∈ R∪ ∞ we shall study the set

Λϕ(α) :=

x ∈ X : limτ→∞

τ−1∫ τ

t=0ϕ ft(x) = α

,

along with the set of points for which the Birkhoff average does not exist Λ′ϕ := X\⋃α∈R∪∞ Λϕ(α).

Take d ≥ 2 and let G := SLd(R) and Γ := SLd(Z). Let X be the homogeneous space Γ\G. We shall consider

the flow ( ft)t∈R defined by

ft(x) = x · at,

for x ∈ X and t ∈ R, where at := diag(et, · · · , et, e−(d−1)t) ∈ SLd(R).

Let ϕ : X → R be a uniformly continuous function ϕ : X → R which tends to infinity in the cusp. That

is, the set X<m := x ∈ X : ϕ(x) < m is precompact for each m ∈ N. We let M f (X) denote the set of all

( ft)t∈R-invariant Borel probability measures on X and letMf (X) denote the set of all compactly supported

members ofM f (X). Given µ ∈ M f (X) we let hµ( f ) denote the associated measure theoretic entropy of the

time one map. We define ϕ− := inf

α ∈ R : Λϕ(α) 6= ∅

and ϕ+ := sup

α ∈ R : Λϕ(α) 6= ∅

.

Theorem 6.1. For all α ∈ (ϕ−, ϕ+) we have

dimH(Λϕ(α)

)= d(d− 1) + d−1 sup

hµ( f ) : µ ∈ M

f (X),∫

ϕdµ = α

.

In particular the multifractal spectrum α 7→ Λϕ(α) is concave and continuous.

In addition, both Λϕ(∞) and Λ′ϕ are of full Hausdorff dimension d2 − 1.

6.3 Multifractal analysis for the time one map

Take d ≥ 2 and let G denote the special linear group SLd(R) and let Γ ⊂ G be a lattice. That is, Γ is a discrete

subgroup of finite covolume with respect to the Haar measure. Let X be the homogeneous space Γ\G. Choose

d1, d2 ∈ N with d = d1 + d2 and a1 > 1 > a2 > 0 with ad11 · a

d22 = 1. We shall consider the dynamical system

T : X → X by

T(x) = x · a,

for x ∈ X where

a := diag(d1︷︸︸︷

a1, · · · , a1,d2︷︸︸︷

a2, · · · , a2).

6.3. Multifractal analysis for the time one map 159

The upper Lyapunov exponent of T is given by λ := log(a1 · a−12 ).

Take a uniformly continuous function ϕ : X → R with limx→∞ ϕ(x) = ∞, ie. for each M ∈ N, the set

X<M := x ∈ X : ϕ(x) < M is precompact.

We shall be interested in the asymptotic behaviour of the Birkhoff averages of ϕ. Given n ∈N we let

Sn(ϕ) :=n−1

∑j=0

ϕ T j.

Define the lower and upper Birkhoff averages by

ϕ∗(x) := lim infn→∞

n−1Sn(ϕ)(x),

ϕ∗(x) := lim supn→∞

n−1Sn(ϕ)(x)

Given α, β ∈ R with α ≤ β we shall study the set

Jϕ(α, β) := x ∈ X : ϕ∗(x) = α, ϕ∗(x) = β ,

and let Jϕ(α) = Jϕ(α, α). We let MT(X) denote the set of all T-invariant Borel probability measures on X

and let MT(X) denote the set of all compactly supported members of MT(X). Given µ ∈ MT(X) we let

hµ(T) denote the associated measure theoretic entropy. We define ϕ− := inf

α ∈ R : Jϕ(α) 6= ∅

and ϕ+ :=

sup

α ∈ R : Jϕ(α) 6= ∅

.

Theorem 6.2. For all α, β ∈ (ϕ−, ϕ+) with α ≤ β we have

dimH(

Jϕ(α, β))= (d2 − d1d2 − 1) + min

γ∈α,βsup

hµ(T)λ−1 : µ ∈ M

T(X),∫

ϕdµ = γ

.

In particular the function α 7→ Jϕ(α) is both concave and continuous.

Theorem 6.3. Suppose that Γ = SLd(Z) and a = (e, · · · , e, e−(d−1)). Then we have

dimH(

Jϕ(∞))= dimH

(J′ϕ)= d2 − 1.

The results for the time-one map imply the corresponding results for the flow.

Proof of Theorem 6.1 from Theorems 6.2 and 6.3. We first show that for any given x ∈ X

lim infτ→∞

τ−1∫ τ

0ϕ(x · at)dt = lim inf

n→∞n−1

∫ n

0ϕ(x · an)dt, (6.1)

lim supτ→∞

τ−1∫ τ

0ϕ(x · at)dt = lim sup

n→∞n−1

∫ n

0ϕ(x · an)dt.

Clearly,

lim infτ→∞

τ−1∫ τ

0ϕ(x · at)dt ≤ lim inf

n→∞n−1

∫ n

0ϕ(x · an)dt,

lim supτ→∞

τ−1∫ τ

0ϕ(x · at)dt ≥ lim sup

n→∞n−1

∫ n

0ϕ(x · an)dt.

Since X<0 is precompact there exists some C > 0 such that ϕ(x) > −C for all y ∈ X.


If we take any α > lim infτ→∞ τ−1∫ τ

0 ϕ(x · at)dt then there exists a sequence τr → ∞ such that

∫ τr

0ϕ(x · at)dt < α · τr

for all r ∈N. Thus, ∫ bτrc

0ϕ(x · at)dt < αbτrc+ C + |α|,

which implies that lim infn→∞ n−1∫ n

0 ϕ(x · an)dt ≤ α.

Similarly, if we take any β > lim supτ→∞ τ−1∫ τ

0 ϕ(x · at)dt then there exists a sequence τr → ∞ such that

∫ τr

0ϕ(x · at)dt > β · τr

for all r ∈N. Thus, ∫ dτre

0ϕ(x · at)dt > βdτre − C− |β|,

which implies that lim supn→∞ n−1∫ n

0 ϕ(x · an)dt ≥ β.

This completes the proof of equations (6.1). This in turn implies that

lim infτ→∞

τ−1∫ τ

0ϕ(x · at)dt = lim inf

n→∞n−1

n−1

∑i=0

(∫ 1

0ϕ((x · ai) · at)dt

),

lim supτ→∞

τ−1∫ τ

0ϕ(x · at)dt = lim sup

n→∞n−1

n−1

∑i=0

(∫ 1

0ϕ((x · ai) · at)dt

).

Thus, in order to complete the proof of Theorem 6.1 given Theorems 6.2 and 6.3 it suffices to show that for

each α ∈ R we have

sup

hµ( f ) : µ ∈ Mf (X),

∫ϕdµ = α

= sup

hµ( f1) : µ ∈ M

f1(X),

∫X

∫ 1

0ϕ(x · at)dtdµ(x) = α

.

Again, one direction is clear. Indeed if µ ∈ Mf (X) then µ ∈ M f1(X) and by definition hµ( f1) = hµ( f ).

Moreover, if∫

ϕ(x)dµ(x) = α then by f invariance we have∫

ϕ(x · at)dµ(x) = α for all t ∈ [0, 1] and so by

Fubini’s theorem we have

∫X

∫ 1

0ϕ(x · at)dtdµ(x) =

∫ 1

0

∫X

ϕ(x · at)dtdµ(x) = α.

Hence,

sup

hµ( f ) : µ ∈ Mf (X),

∫ϕdµ = α

≤ sup

hµ( f1) : µ ∈ M

f1(X),

∫X

∫ 1


.

To see the opposing inequality we take ν ∈ Mf1(X) with

∫X

∫ 10 ϕ(x · at)dtdµ(x) = α. Let µ be the measure

defined by ∫X

g(x)dµ(x) =∫

X

(∫ 1

0g(x · at)dt

)dν(x) =

∫ 1

0

(∫X

g(x · at)dν(x))

dt,

6.4. Preliminary lemmas 161

for all continuous compactly supported funtions g : X → R. It follows that∫

ϕdµ = α. Also, given any

continuous compactly supported funtions g : X → R and any τ ∈ R we have

∫X

g(x · aτ)dµ(x) =∫ 1

0

∫X

g(x · at+τ)dν(x)dt

=∫ dτe−τ

0

∫X

g(x · at+τ)dν(x)dt +∫ 1

dτe−τ

∫X

g(x · at+τ)dν(x)dt

=∫ dτe−τ

0

∫X

g(x · at+τ−dτe+1)dν(x)dt +∫ 1

dτe−τ

∫X

g(x · at+τ−dτe)dν(x)dt

=∫ 1

1−dτe+τ

∫X

g(x · at)dν(x)dt +∫ 1−dτe+τ

0

∫X

g(x · at)dν(x)dt

=∫ 1

0

∫X

g(x · at)dν(x)dt =∫

Xg(x)dµ(x),

so µ ∈ Mf (X). Finally since each map ft for t ∈ [0, 1] is an invertible map which commutes with f1 it is clear

that for each t ∈ [0, 1] the measure ν f−1t is f1 invariant and satisfies h

ν f−1t( f1) = hν( f1). Since entropy is

affine (see Theorem 2.40) it follows that hµ( f ) = hµ( f1) = hν( f1). Thus,

sup

hµ( f ) : µ ∈ Mf (X),

∫ϕdµ = α

≥ sup

hµ( f1) : µ ∈ M

f1(X),

∫X

∫ 1


.

This completes the proof.

In the remaining sections we shall prove Theorems 6.2 and 6.3. The proof consists of an upper bound, given

in Section 6.5 and a lower bound, given in Section 6.6.

6.4 Preliminary lemmas

6.4.1 Metric properties

In this section we recall some well known elementary facts regarding the metric structure of X = Γ\G. For

proofs and references we refer the reader to Section 2.6.1.

We may also choose η0 > 0 and c0 > 1 so that for any pair g1, g2 ∈ G with max dG(g1, e), dG(g2, e) < η0

or ||g1 − e||∞, ||g2 − e||∞ < η0 we have

c−10 ||g1 − g2||∞ ≤ dG(g1, g2) ≤ c0||g1 − g2||∞.

Given t = (tij)ij ∈ Md2×d1(R) we let

u(t) :=

Id1×d1 0d1×d2

t Id2×d2

.

We define a subgroup P ⊂ SL(R) by

P :=

p ∈ SLd(R) : a−n · p · an 6→ ∞ as n→ ∞

.

For every ε > 0 the set p ∈ P : ||p− e||∞ < ε is a submanifold of Hausdorff and packing dimension d2 −

d1d2 − 1. We may choose η1 > 0, c1 > 1 so that every g ∈ G with dG(g, e) < η1 there is a unique pair p ∈ P


and t ∈ Md2×d1(R) with g = p · u(t). Moreover, given g1, g2 ∈ G, p1, p2 ∈ P, t1, t2 ∈ Md2×d1(R) so that

g1 = p1 · u(t1) and g2 = p2 · u(t2) then if either maxdG(g1, e), dG(g2, e) < η1 or

max ||p1 − e||∞, ||p2 − e||∞, ||t1||∞, ||t2||∞ < η1,

we have

c−11 dG(g1, g2) ≤ max ||p1 − p2||∞, ||t1 − t2||∞ ≤ c1dG(g1, g2).

Moreover, we may also choose η1 > 0, c1 > 1 so that such that every g ∈ G with dG(g, e) < η1 there is a unique

pair p ∈ P and t ∈ Md2×d1(R) with g = u(t) · p. Moreover, given g1, g2 ∈ G, p1, p2 ∈ P, t1, t2 ∈ Md2×d1(R) so

that g1 = u(t1) · p1 and g2 = u(t2) · p2 then if either maxdG(g1, e), dG(g2, e) < η1 or

max ||p1 − e||∞, ||p2 − e||∞, ||t1||∞, ||t2||∞ < η1,

we have

c−11 dG(g1, g2) ≤ max ||p1 − p2||∞, ||t1 − t2||∞ ≤ c1dG(g1, g2).

For each z ∈ X we choose η(z) ∈ (0, minη0, η1) so that the map g 7→ z · g is an isometry from

g ∈ G : dG(g, e) < η(z) to y ∈ X : d(y, z) < η(z). Moreover, given a compact set K ⊂ X we may choose

η(K) > 0 so that η(z) > η(K) for all z ∈ K. Finally, given for any g ∈ G the map h 7→ h · g−1 is Lipschitz with

dG (h1 · g, h2 · g) ≤ ||g||HS||g−1||HS · dG(h1, h2),

for all h1, h2 ∈ G, where || · ||HS denotes the Hilbert Schmidt norm.

6.4.2 The chain lemma

The following lemma is a variant of Lemma 4.3 from [Ka].

Lemma 6.4. There exists a constant η2 > 0 and a constant c2 > 1 such that for any ε ∈ (0, η2) and any pair g−, g+ ∈ G

with dG(g−, g+) < ε there exists t ∈ Md2×d1(R) with ||t||∞ < c2ε such that h = g− · u(t) ∈ X satisfies

(i) dG

(h · al , g− · al

)< c2εelλ for all l ≤ 0,

(ii) dG

(h · al , g+ · al

)< c2ε for all l ≥ 0.

Proof. First note that given m ∈ Md(R) we have

a−n ·m · an =

a−n1 · Id1×d1 0d1×d2

0d2×d1 a−n2 · Id2×d2

m11 m12

m21 m22

an

1 · Id1×d1 0d1×d2

0d2×d1 an2 · Id2×d2

=

m11 (a1/a2)−n ·m12

(a1/a2)n ·m21 m22

=

m11 e−λn ·m12

eλn ·m21 m22

,


where

m =

m11 m12

m21 m22

∈ Md(R)

with m11 ∈ Md1×d1(R), m21 ∈ Md2×d1(R), m12 ∈ Md1×d2(R) and m22 ∈ Md2×d2(R).

Consequently if h = g− · u(t) then by Theorem 2.60 whenever ||t||∞ < η0 we have

dG

(h · al , g− · al

)= dG

(g− · u(t) · al , g− · al

)= dG

(a−lu(t)al , e

)

≤ c0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ Id1×d1 0d1×d2

eλl · t Id2×d2

− Id×d

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∞

= c0eλl ||t||∞.

Moreover, if ε < min(2c0d)−1, η0, 1

and dG(g−, g+) < ε then

||g−1+ g− − e||∞ ≤ c0dG(g−1

+ g−, e)

= c0dG(g−, g+) < min

12d

, c0ε

.

Now letting g = g−1+ g− we may write

g =

g11 g12

g21 g22

∈ Md(R)

with g11 ∈ Md1×d1(R), g21 ∈ Md2×d1(R), g12 ∈ Md1×d2(R) and g22 ∈ Md2×d2(R), ||g21||∞ < c0ε and

||g22||HC ≤ d||g22||∞ <12

,

where || · ||HS denotes the Hilbert-Schmidt operator norm. Thus g22 is invertible with

||g−122 ||∞ ≤ ||g

−122 ||HS < 2.

We let t := −g−122 · g21 ∈ Md2×d1 so g21 + g12t = 0d2×d1 , and ||t||∞ ≤ 2dc0ε so

∣∣∣∣(g11 + g12t)− Id1×d1

∣∣∣∣∞ ≤

∣∣∣∣g11 − Id1×d1

∣∣∣∣∞ + ||g12t||∞

< c0ε + d||g12||∞||t||∞ ≤ 3dc0ε.


Thus, if we let h = g− · u(t) then for all l ≥ 0 we have

dG

(g− · u(t) · al , g+ · al

)= dG

(a−l g−1

+ g−u(t)al , e)

≤ c0||a−l g−1+ g−u(t)al − e||∞

= c0||a−l (gu(t)− e) al ||∞

= c0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣a−l

g11 g12

g21 g22

Id1×d1 0d1×d2

t Id2×d2

− Id1×d1 0d1×d2

0d2×d1 Id2×d2

al

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∞

= c0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣a−l

g11 + g12t− Id1×d1 g12

g21 + g22t g22 − Id2×d2

al

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∞

= c0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ g11 + g12t− Id1×d1 e−λl g12

0d2×d1 g22 − Id2×d2

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∞

< 3dc20ε,

provided ε < η0/(3dc20). Thus, the lemma holds with η1 := η0/(3dc2

0) < η0/(2dc0) and c2 := 3dc20.

Corollary 6.5. (Shadowing lemma) There exists a constant η2 > 0 and a constant c2 > 1 such that for any ε ∈ (0, η2)

and any pair x−, x+ ∈ X with d(x−, x+) < ε there exists t ∈ Md2×d1(R) with ||t||∞ < c2ε such that y = x− · u(t) ∈

X satisfies

(i) d(

Tl(y), Tl(x−))< c2εelλ for all l ≤ 0,

(ii) d(

Tl(y), Tl(x+))< c2ε for all l ≥ 0.

Proof. Choose η2 > 0 and c2 > 1 as in the previous lemma. Take ε ∈ (0, η2) and x−, x+ ∈ X so that d(x−, x+) <

ε. It follows that there exists g−, g+ ∈ G so that x− = Γg−, x+ = Γg+ and dG(g−, g+) < ε. Thus, by the

previous lemma there exists t ∈ Md2×d1(R) with ||t||∞ < c2ε such that h = g− · u(t) ∈ X satisfies

(i) dG

(h · al , g− · al

)< c2εelλ for all l ≤ 0,

(ii) dG

(h · al , g+ · al

)< c2ε for all l ≥ 0.

Consequently, y = x− · u(t) = Γh satisfies,

(i) d(

Tl(y), Tl(x−))< c2εelλ for all l ≤ 0,

(ii) d(

Tl(y), Tl(x+))< c2ε for all l ≥ 0.

Lemma 6.6. Given a compact set K ⊂ X and a δ > 0 there exists m(K, δ) ∈N such that for all m ≥ m(K, δ) any pair

z−, z+ ∈ K there is some w ∈ X with d(z−, w) < δ and d(Tm(w), z+) < δ.


Proof. Since K is compact we may take a finite set z1, · · · , zq such that every element z ∈ K satisfies

d(z, zi) < δ/2 for some i ≤ q. Since T is topologically mixing we may find, for each pair i, j ∈ 1, · · · , q,

some mij(K, δ) ∈ N such that for all m ≥ mij(K, δ), B(zi, δ/2) ∩ T−mB(zj, δ/2) 6= ∅. Taking m(K, δ) :=

maxijmij(K, δ) completes the proof of the lemma.

Lemma 6.7. Take a compact set K ⊂ X and some δ > 0. There exists m(K, δ) ∈ N such that for all m ≥ m(K, δ),

x−, x+ ∈ X and n ∈ N with Tn(x−), x+ ∈ K there exists t ∈ Md2×d1(R) with ||t||∞ < δe−λn such that y =

x− · u(t) ∈ X satisfies

(i) d(

Tl(y), Tl(x−))< δe(l−n)λ for all l ≤ n,

(ii) d(

Tl(y), Tl−n−m(x+))< δ for all l ≥ n + m.

Proof. Fix a compact set K ⊂ X and choose some ε ∈(0, minη2, 1 · (c2(c2 + 2))−1). By Lemma 6.6 there

exists m(K, ε) ∈ N such that for all m ≥ m(K, ε), x−, x+ ∈ X and n ∈ N with Tn(x−), x+ ∈ K there exists

w ∈ X with d(Tn(x−), w), d(Tm(w), x+) < ε < η2. By the shadowing lemma we may find s1 ∈ Md2×d1(R)

with ||s1||∞ < c2ε such that z1 = Tn(x−) · u(s1) ∈ X satisfies

(i)’ d (Tq(z1), Tq+n(x−)) < c2εeqλ for all q ≤ 0,

(ii)’ d (Tq(z1), Tq(w)) < c2ε for all q ≥ 0.

In particular, d (Tm(z1), x+) < (c2 + 1)ε < η2, so by applying the shadowing lemma once more we obtain

s2 ∈ Md2×d1(R) with ||s2||∞ < c2(c2 + 1)ε such that z2 = Tm(z1) · u(s2) ∈ X satisfies

(i)” d (Tr(z2), Tr+m(z1)) < c2(c2 + 1)εerλ for all r ≤ 0,

(ii)” d (Tr(z2), Tr(x+)) < c2(c2 + 1)ε for all r ≥ 0.

Let y := T−n−m(z2). Then we have,

(ii) d(

Tl(y), Tl−n−m(x+))< c2(c2 + 1)ε for all l ≥ n + m.

By (i)” we have

d(Tl(y), Tl−n(z1)) < c2(c2 + 1)εe(l−n−m)λ

for all l ≤ n + m.

By (i)’ we have

d(Tl−n(z1), Tl(x−)) < c2εe(l−n)λ

for all l ≤ n.

Putting these two inequalities together gives

(i) d(

Tl(y), Tl(x−))< c2(c2 + 2)εe(l−n)λ for all l ≤ n.


Now by construction we have

y = z2 · a−n−m = (z1 · am · u(s2)) · a−n−m

= x− · an · u(s1) · a−n · an+m · u(s2) · a−n−m

= x− · u(

e−nλs1 + e−(n+m)λs2

).

We let t := e−nλs1 + e−(n+m)λs2 so y = x− · u(t) and ||t||∞ < c2(c2 + 1)ε.

In order to complete the proof of the lemma we note that ε < δ/(c2(c2 + 2)) so that c2(c2 + 2)ε < δ and

take m(K, δ) = m(K, ε).

We let P(X) denote the set of all compact subsets of X.

Lemma 6.8 (Chain lemma). There exists a function m : P(X)×R>0 → N such that for any sequence (δi)∞i=0 ⊂

(0, 1) with each δi ≥ δi+1, a sequence of compact sets (Ki)∞i=0 ⊂ P(X) with each Ki ⊆ Ki+1, and any pair of sequences

of natural numbers (ni)∞i=0 and (mi)

∞i=0 with each mi ≥ m(Ki, δi) there exists a pair of maps

π :∞

∏i=0

Ki ∩ T−ni (Ki)→ X

χ :∞

∏i=0

Ki ∩ T−ni (Ki)→ Md2×d1(R)

with the property that for any sequence x = (xi)∞i=0 ∈ ∏∞

i=0 Ki ∩ T−ni (Ki), y = π(x) satisfies d(

Tl(y), Tl−γi (xi))<

δi for each i ∈ N ∪ 0 and γi ≤ l < γi + ni, where γ0 := 0 and γi := ∑j<i nj + mj+1. Moreover, y = x0 · u(t) for

t = χ(x) ∈ Md2×d1(R) and ||t||∞ < δ0.

Proof. Given a compact set K ∈ P(X) we let

K := x ∈ X : ∃y ∈ K, d(x, y) ≤ 1 .

Each set K is closed and bounded and hence is compact. For each compact set K ∈ P(X) and δ > 0 we let

m(K, δ) := m(K, δ2−1(1− e−λ)

), where m(K′, δ′) is chosen as in Lemma 6.7.

Now take a sequence of reals (δi)∞i=0 with each δi ≥ δi+1 > 0, a sequence of compact sets (Ki)

∞i=0 ⊂ P(X)

with each Ki ⊆ Ki+1, and a pair of sequences of natural numbers (ni)∞i=0 and (mi)

∞i=0 with each mi ≥ m(Ki, δi)

and define (γi)∞i=0 by γ0 := 0 and γi := ∑j<i nj + mj+1.

By induction we shall show that for each i ∈ N ∪ 0 there exists yi = x0 · u(ti) ∈ X for some ti ∈

Md2×d1(R) with ||ti||∞ < δ02−1(1 − e−λ)(

∑ik=0 e−λk

)such that for each j ≤ i and γj ≤ l < γj + nj we

have d(

Tl(yi), Tl−γj(xj))< δj2−1(1− e−λ)

(∑

i−jk=0 e−λk

)and also d(yi, yi−1) < δ02−1(1− e−λ)e−λi and ||ti −

ti−1||∞ < δ02−1(1− e−λ)e−λi whenever i ≥ 1.

For i = 0 we simply take y0 = x0 and t0 = 0, which satisfies the required properties. For the inductive step

we suppose that there exists some yi and ti ∈ Md2×d1(R) with the required properties. In particular

d(Tγi+ni (yi), Tni (xi)

)< δi < 1.


Thus, Tγi+ni (yi) ∈ Ki ⊆ Ki+1. Since xi+1 ∈ Ki+1 ⊂ Ki+1 we may apply Lemma 6.7 so there ex-

ists si+1 ∈ Md2×d1(R) with ||si+1||∞ < δi+12−1(1 − e−λ)e−λ(γi+ni) ≤ δi+12−1(1 − e−λ)e−λ(i+1) such that

yi+1 = yi · u(si+1) ∈ X satisfies

(i) d(

Tl(yi+1), Tl(yi))< δi+12−1(1− e−λ)e(l−γi−ni)λ

for all l ≤ γi + ni,

(ii) d(

Tl(yi+1), Tl−γi+1(xi+1))< δi+12−1(1− e−λ)

for all l ≥ γi+1,

since γi+1 = γi + ni + mi+1. Applying (i) with l = 0 gives d(yi+1, yi) < δ02−1(1− e−λ)e−λ(i+1). Moreover, for

each j ≤ i + 1 and γj ≤ l < γj + nj we have d(

Tl(yi+1), Tl−γj(xj))< δj2−1(1− e−λ)

(∑

i+1−jk=0 e−λk

). When

j = i + 1 this is immediate from (ii). Our inductive hypothesis implies that for j ≤ i and γj ≤ l < γj + nj we

have d(

Tl(yi), Tl−γj(xj))< δj2−1(1− e−λ)

(∑

i−jk=0 e−λk

), so by (i) and δj ≥ δi+1 we have

d(


(i−j

∑k=0

e−λk + e(l−γi−ni)λ

).

Now γi = γj + ∑j≤k<i nk + mk+1 so for l < γj + nj, γi + ni − l ≥ ∑j<k≤i nk + mk+1 + 1 ≥ i− j + 1. This implies

that

d(


(i−j+1

∑k=0

e−λk

).

Finally yi+1 = yi · u(si+1) for some si+1 ∈ Md2×d1(R) with

||si+1||∞ < δi+12−1(1− e−λ)e−λ(γi+ni) ≤ δ02−1(1− e−λ)e−λ(i+1).

By hypothesis, yi = x0 · u(ti) ∈ X for some ti ∈ Md2×d1(R) with ||ti||∞ < δ02−1(1− e−λ)(

∑ik=0 e−λk

). Let

ti+1 := ti + si+1, so that ||ti+1||∞ < δ02−1(1− e−λ)(

∑i+1k=0 e−λk

)and

yi+1 = x0 · u(ti) · u(si+1) = x0 · u(ti + si+1) = x0 · u(ti+1).

This shows that for each i ∈N∪ 0 there exists yi = x0 · u(ti) ∈ X for some ti ∈ Md2×d1(R) with ||ti||∞ <

δ02−1(1− e−λ)(

∑ik=0 e−λk

)such that for each j ≤ i and γj ≤ l < γj + nj we have d

(Tl(yi), Tl−γj(xj)

)<

δj2−1(1− e−λ)(

∑i−jk=0 e−λk

)and also d(yi, yi−1) < δ02−1(1− e−λ)e−λi and ||ti − ti−1||∞ < δ02−1(1− e−λ)e−λi

whenever i ≥ 1. In particular, (yi)∞i=0 is a Cauchy sequence in X and (ti)

∞i=0 is a Cauchy sequence in Md2×d1(R),

so we may take limit points y = π(x) and t = χ(x) ∈ Md2×d1(R) which satisfy d(

Tl(y), Tl−γj(xj))≤ δj2−1 <

δj for γj ≤ l < γj + nj, ||ti||∞ ≤ δ02−1 < δ0 and y = x0 · u(t), by continuity.

6.4.3 A uniform law of large numbers

Lemma 6.9. Suppose we have a probability measure ρ on a set W along with a funtion f : W → R with∫

f dρ ∈ R.

Then for all ε > 0 there exists N(ε) ∈N such that for all n ≥ N(ε) there exists W( f , n, ε) ⊂Wn with product measure

ρn (Wn\W( f , n, ε)) < ε such that for all (yi)ni=1 ∈W( f , n, ε) and for all N(ε) ≤ l ≤ n we have

l

∑i=1

f (yi) ∈(

l∫

f dρ− lε, l∫

f dρ + lε)

.


Proof. This follows from Kolmogorov’s strong law of large numbers combined with Egorov’s theorem.

6.5 Proof of the upper bound

The proof of the upper bound proceeds as follows. We begin by showing that points with a given finite lower

or upper Birkhoff average must make regular returns to a compact region whilst simultaneously attaining a

Birkhoff average close to that value. We then proceed to relate the dimension of the level set along the unstable

manifold to the number of Bowen balls required to cover the level set. This in turn gives us an upper bound

in terms of the rate of growth of well seperated sets contained within the level set. The chain lemma is then

applied to construct compactly supported invariant measures with large entropy which integrate the potential

to within an epsilon of the appropriate Birkhoff limit. An upper-semicontinuity lemma is then required to

deduce the upper bound in Theorem 6.2.

Without loss of generality we may assume that ϕ > 0, so we make this assumption throughout the proof

of the upper bound.

6.5.1 Returns to compact regions

We begin by defining for γ ∈ R, ε > 0 and m ∈N,

Eϕ (γ, ε, m, n) :=

x ∈ X<m ∩ T−(n−1)X<m : n−1Sn(ϕ)(x) ∈ (γ− ε, γ + ε)

.

We define

Kϕ (γ, ε, m) :=⋂

l∈N

⋃n≥l

Eϕ (γ, ε, m, n) .

Lemma 6.10.

Jϕ(α, β) ⊆⋂

γ∈α,β

⋂ε>0

⋃m∈N

Kϕ (γ, ε, m) .

Proof. Take x ∈ Jϕ(α, β) and ε > 0. Take m ∈N with

m > max ϕ(x), 3 (α + ε) , 2 (β + ε) .

We shall show that x ∈ Kϕ (α, ε, m) and x ∈ Kϕ (β, ε, m).

Since lim infn→∞ n−1Sn(ϕ)(x) = α there exists some N(ε) ∈ N such that for all n ≥ N(ε), n−1Sn(ϕ)(x) >

α− ε and there exists infinitely many n ∈N for which n−1Sn(ϕ)(x) < α + ε. So take any l ∈N and take some

k ≥ max 2N(ε), 2l, 6

with k−1Sk(ϕ)(x) < α + ε. Since k ≥ 6 we may choose some j ∈N∩ [k/2, 2k/3]. Note that

k−1

∑i=j−1

ϕ(Ti(x)) ≤k−1

∑i=0

ϕ(Ti(x)) < k (α + ε) ,

so for some i ≥ j− 1 we have ϕ(Ti(x)) < 3 (α + ε) since j ≤ 2k/3. Now choose n ≤ k to be maximal so that

ϕ(Tn−1(x)) < 3 (α + ε). It follows that k/2 ≤ j ≤ n ≤ k. Since n ≥ N(ε) we have n−1Sn(ϕ)(x) > α− ε. We


also have n−1Sn(ϕ)(x) < α + ε. Indeed for n− 1 ≤ i ≤ k− 1 ϕ(Ti(x)) ≥ 3(α + ε), so if we have Sn(ϕ)(x) ≥

n(α + ε) then we would also have Sk(ϕ)(x) ≥ k(α + ε), which is a contradiction. Thus, x ∈ Eϕ (α, ε, m, n) and

n ≥ l. Since this holds for all l ∈N we have x ∈ Kϕ (α, ε, m).

To show that x ∈ Kϕ (β, ε, m) is simpler. Clearly we may assume ε ∈ (0, β). Since

lim supn→∞ n−1Sn(ϕ)(x) = β there exists some M(ε) ∈ N such that for all n ≥ M(ε), n−1Sn(ϕ)(x) < β + ε

and there exists infinitely many n ∈N for which n−1Sn(ϕ)(x) > β− ε/2. So take any l ∈N and take some

k ≥ max

M(ε), l, 2ε−1 (β− ε/2)

with k−1Sk(ϕ)(x) > β− ε/2. Since n ≥ M(ε) we have

2k−1

∑j=k

ϕ(T j(x)) ≤2k−1

∑j=0

ϕ(T j(x)) < 2k(β + ε).

Hence there exists some k < j ≤ 2k with ϕ(T j−1(x)) < 2(β + ε). Let n be the least such j. It follows that

Sn(ϕ)(x) ≥ Sn−1(ϕ)(x)

=n−2

∑i=k

ϕ(Ti(x)) + Sk(ϕ(x))

≥ (n− 1− k)2(β + ε) + k(β− ε/2)

≥ (n− 1)(β− ε/2)

> n(β− ε).

The third line holds because ϕ(Ti(x)) ≥ 2(β+ ε) for all k < i < n− 1, and the last because n > 2ε−1 (β− ε/2).

Hence, x ∈ Eϕ (β, ε, m, n) and n ≥ l. Again, since this holds for all l ∈N we have x ∈ Kϕ (β, ε, m).

6.5.2 Growth of separated sets on the unstable manifold

For each z ∈ X, p ∈ P we define,

Kz,pϕ (γ, ε, m) : =

t ∈ Md2×d1(R) : z · u(t) · p ∈ Kϕ (γ, ε, m)

.

We let P :=

p ∈ P : ||p− e||∞ < (2d)−1.

Lemma 6.11.

dimHKϕ (γ, ε, m) ≤ (d2 − d1d2 − 1) + supz∈X,p∈P

dimHKz,p

ϕ (γ, 2ε, m)

.

Proof. Since ϕ : X → R is uniformly continuous we may choose δ(ε) ∈ (0, η1) so that whenever d(x, y) < δ(ε),

|ϕ(x)− ϕ(y)| < ε. Let P(ε) :=

p ∈ P : ||p− e||∞ < min(2d)−1, δ(ε)/2c1⊂ P.

Let ηm := min

η (X≤m) , η1, δ(ε)/2c21, (2dc1)

−1. We begin by showing that given any z ∈ X≤m

the function φz : P(ε) × Md2×d1(R) → X by (p, t) 7→ z · u(t) · p maps surjectively onto a superset of

x ∈ X : d(z, x) < ηm and is Lipschitz on the preimage φ−1z x ∈ X : d(z, x) < ηm. Indeed since ηm ≤

η (X≤m) the map g 7→ z · g is an isometry from g ∈ G : d(g, e) < ηm onto x ∈ X : d(z, x) < ηm. More-

over, ηm ≤ η1 and the mapping ψ : (p, t) 7→ u(t) · p is surjective onto g ∈ G : dG(g, e) < η1 and is Lipschitz


on the preimage (under ψ) of g ∈ G : dG(g, e) < η1. Note also that if d(z · u(t) · p, z) < ηm ≤ η(X≤m) then

dG (u(t) · p, e) < ηm ≤ η1 so we must have ||p− e||∞ ≤ c1 · ηm ≤ min(2d)−1, δ(ε)/2c1. Thus, p ∈ P(ε). This

proves that the claim.

We now deduce that for each z ∈ X≤m we have

dimH(Kϕ (γ, ε, m) ∩ x ∈ X : d(x, z) < ηm

)≤ (d2 − d1d2 − 1) + sup

z∈X,p∈P

dimHKz,p

ϕ (γ, 2ε, m)

.

Since φz : (p, t) 7→ z · u(t) · p is a Lipschitz map onto x ∈ X : d(x, z) < ηm it suffices to show that

dimH(

(p, t) ∈ P(ε)×Md2×d1(R) : z · u(t) · p ∈ Kϕ (γ, ε, m))

≤ (d2 − d1d2 − 1) + supz∈X,p∈P

dimHKz,p

ϕ (γ, 2ε, m)

.

We now show that for any pair p0, p1 ∈ P(ε) we have

Kz,p0ϕ (γ, ε, m) ⊆ Kz,p1

ϕ (γ, 2ε, m) .

To see this fix p0, p1 ∈ P(ε) and choose t ∈ Kz,p0ϕ (γ, ε, m) so that z · u(t) · p0 ∈ Kϕ (γ, ε, m). Given p ∈ P(ε) we

have ||p− e|| < δ(ε)/2c1 so given any n ∈N∪0we have ||a−n pan− e|| < δ(ε)/2c1 < η1 so dG(a−n pan, e) <

δ(ε)/2. It follows that if we choose g ∈ G so that z = Γg then for any t ∈ Md2×d1(R) and any n ∈ N ∪ 0 we

have

d (Tn(z · u(t) · p0), Tn(z · u(t) · p1)) ≤ dG (g · u(t) · p0 · an, g · u(t) · p1 · an)

≤ dG(a−n p0 · an, a−n p1 · an)

≤ dG(a−n · p0 · an, e

)+ dG

(e, a−n · p1 · an) < δ(ε).

Consequently |ϕ (Tn(z · u(t) · p0))− ϕ (Tn(z · u(t) · p1)) | < ε for all n ∈N∪ 0 so

Kz,p0ϕ (γ, ε, m) ⊆ Kz,p1

ϕ (γ, 2ε, m) .

It follows that if we fix some p1 ∈ P(ε) then

(p, t) ∈ P(ε)×Md2×d1(R) : z · u(t) · p ∈ Kϕ (γ, ε, m)

⊆ P(ε)× Kz,p1

ϕ (γ, 2ε, m) .

Thus,

dimH(

(p, t) ∈ P(ε)×Md2×d1(R) : z · u(t) · p ∈ Kϕ (γ, ε, m))

≤ dimB (P(ε)) + dimH

(Kz,p1

ϕ (γ, 2ε, m))

≤ supz∈X,p∈P

dimHKz,p

ϕ (γ, 2ε, m)

,

and consequently

dimH(Kϕ (γ, ε, m) ∩ x ∈ X : d(x, z) < ηm

)≤ (d2 − d1d2 − 1) + sup

z∈X,p∈P

dimHKz,p

ϕ (γ, 2ε, m)

,

fot each x ∈ X. Clearly Kϕ (γ, ε, m) ⊆ X≤m, so by compactness Kϕ (γ, ε, m) may be covered by a finitely many

balls of radius ηm, so the lemma follows.


For each z ∈ X, p ∈ P we also define

Ez,pϕ (γ, ε, m, n) : =

t ∈ Md2×d1(R) : z · u(t) · p ∈ Eϕ (γ, ε, m, n)

,

so that

Kz,pϕ (γ, ε, m) =

⋂l∈N

⋃n≥l

Ez,pϕ (γ, ε, m, n) .

Fix z ∈ X and p ∈ P. For each n ∈N we define a metric dn on X by

dn(x, y) := max0≤i≤n−1

d(Ti(x), Ti(y))

.

For each n ∈N, x ∈ X and r > 0 we define

Bn(x, r) := y ∈ X : dn(x, y) < r .

Recall that a set A ⊂ X is said to be (n, δ)-seperated if for every distinct pair x1, x2 ∈ A we have dn(x1, x2) > δ.

A subset A ⊂ X is said to be (n, δ)-spanning if for every point y ∈ A there exists some x ∈ A with dn(x, y) ≤ δ.

For each γ ∈ R, δ > 0 and m ∈N, we take a maximal (n, δ)-seperated subset s(γ, ε, m, n, δ) of Eϕ (γ, ε, m, n).

Since X<m is precompact and T is a homeomorphism the cardinality of s(γ, ε, m, n, δ) is always finite.

Lemma 6.12. For all δ ∈ (0, min η(X≤m), η0/4) we have

dimHKz,pϕ (γ, ε, m) ≤ lim sup

n→∞(λn)−1 log #s(γ, ε, m, n, δ).

Proof. Given s ∈ Md2×d1(R) we define

Dn(s) :=

t ∈ Md2×d1(R) : d(

Tn−1(z · u(t) · p), Tn−1(z · u(s) · p))< δ

.

We begin by showing that for each s ∈ Ez,pϕ (γ, ε, m, n) and n ∈N we have diam (Dn(s)) ≤ 8c0δe−(n−1)λ.

To see this we observe that if s ∈ Ez,pϕ (γ, ε, m, n) then Tn−1(z · u(s) · p) ∈ X<m. Thus, g 7→ Tn−1(z · u(s)) · g

is an isometry from g ∈ G : dG(g, e) < η (X≤m) onto

w ∈ X : d(w, Tn−1(z · u(s) · p)) < η (X≤m)

. So given

t ∈ Dn(s) we have

d(

Tn−1(z · u(t) · p), Tn−1(z · u(s) · p))< δ < η (X≤m) .

Hence,

dG

(u(t) · p · an−1, u(s) · p · an−1

)< δ.

Now p ∈ P1 so ||p − e||∞ < (2d)−1. Thus, ||a−(n−1) · p · an−1 − e||∞ < (2d)−1, since p ∈ P, the parabolic

subgroup. Hence, ||a−(n−1) · p · an−1 − e||HS < 2−1, so ||a−(n−1) · p · an−1||HS < 2 and

||(a−(n−1) · p · an−1)−1||HS ≤∞

∑i=0||a−(n−1) · p · an−1 − e||iHS < 2.

Consequently, g 7→ g · (a−(n−1) · p · an−1)−1 is a Lipschitz map with Lipschitz constant at most 4. Thus,

dG

(u(t) · an−1, u(s) · an−1

)< 4δ < η0.


Thus,

dG

(a−(n−1)u(t− s)an−1, e

)< 4δ < η0.

Hence,

e−(n−1)λ||t− s||∞ = ||a−(n−1)u(t− s)an−1 − e||∞ < 4c0δ.

Thus, diamDn(s) ≤ 8c0δe−(n−1)λ.

Now, since s(γ, ε, m, n, δ) is a maximal (n, δ)-separated set for Eϕ (γ, ε, m, n) it must also be an (n, δ)-

spanning set for Eϕ (γ, ε, m, n). As such the collection

Dn(s) : z · u(s) · p ∈ s (γ, ε, m, n, δ) ,

forms a cover of Ez,pϕ (γ, ε, m, n). Hence, for each l ∈N, the collection

⋃n≥lDn(s) : s ∈ s (γ, ε, m, n, δ)

forms a cover of Kz,pϕ (γ, ε, m). Moreover for each s ∈ Ez,p

ϕ (γ, ε, m, n) we have diam (Dn(s)) ≤ 8c0δe−(n−1)λ.

Given any

ρ > lim supn→∞

(λn)−1 log #s(γ, ε, m, n, δ),

we have S(ρ) := ∑n∈N #s(γ, ε, m, n, δ) ·(

8c0e−(n−1)λ)ρ

< ∞. Given any ε > 0 we may choose l(ε) so that

8c0δe−(l(ε)−1)λ < ε. Hence,

Hρε

(Kz,p

ϕ (γ, ε, m))≤ ∑

n≥l(ε)∑

s∈s(γ,ε,m,n,δ)diam (Dn(s))

ρ

≤ ∑n≥l(ε)

#s(γ, ε, m, n, δ) ·(

8c0e−(n−1)λ)ρ≤ S(ρ).

Letting ε → 0 we see that Hρ(

Kz,pϕ (γ, ε, m)

)≤ S(ρ) < ∞, so dimH

(Kz,p

ϕ (γ, ε, m))≤ ρ. Since this holds for

all ρ > lim supn→∞(λn)−1 log #s(γ, ε, m, n, δ) the lemma follows.

6.5.3 Constructing a measure

Lemma 6.13. Fix m ∈ N and δ, ε > 0. There exists η = η(δ, ε) ∈ (0, δ), p = p(m, η) ∈ N and C = C(m, η, p) ≥ 0

such that for all γ ∈ R and all n ∈ N, there exists a precompact set L(n) ⊂ X such that for each q ∈ N there exists a

set Z(q) ⊂ X satisfying,

(i) Ti (Z(q)) ⊂ L(n) for all i ≤ q(n + p),

(ii) Z(q) is (q(n + p), η)-seperated set,

(iii) The cardinality of Z(q) is #s(γ, ε, m, n, δ)q,

(iv) For all x ∈ Z(q),

Sq(n+p)(ϕ)(x) ∈ (q(n(γ− 2ε)− Cp), q(n(γ + 2ε) + Cp)) .


Proof. Since ϕ is uniformly continuous we may begin by choosing η ∈ (0, δ/3) such that for all x, y ∈ X with

d(x, y) < η we have |ϕ(x)− ϕ(y)| < ε. By the Chain lemma (Lemma 6.8) we may choose p ∈ N, depeding

only upon m ∈N and η > 0, such that for any n ∈N there exists a map

πn :∞

∏i=0

X<m ∩ T−n(X<m)→ X

such that for any sequence x = (xi)∞i=0 ∈ ∏∞

i=0 X<m ∩ T−n(X<m) the point y = πn(x) satisfies

d(

Tl(y), Tl−i(n+p)(xi))< η for each i ∈N∪ 0 and i(n + p) ≤ l ≤ i(n + p) + n. We let

C := sup

|ϕ (x) | : x ∈

p⋃i=0

Ti (X<m+ε)

,

which is clearly finite and depends only upon m ∈N, η > 0 and p ∈N.

Take n ∈ N and let L(n) :=⋃p+n

i=0 Ti (X<m+ε). If s(γ, ε, m, n, δ) = ∅ then we may take Z(q) = ∅ for each

q ∈ N and conditions (i)-(iv) hold trivially, so suppose s(γ, ε, m, n, δ) 6= ∅ and take some x∗ ∈ s(γ, ε, m, n, δ)

so x∗ ∈ X<m ∩ T−nX<m. For each q ∈N we define

Z(q) := πn

(q−1

∏i=0

s(γ, ε, m, n, δ)×∞

∏i=qx∗

)

Firstly if y = πn((xi)

∞i=0)

with xi ∈ s(γ, ε, m, n, δ) for i < q and xi = x∗ for i ≥ q then for each i < q, xi ∈ X<m

and d(

Ti(n+p)(y), xi

)< η which implies Ti(n+p)(y) ∈ X<m+ε. Hence Ti(n+p)+l(y) ∈ L(n) for 0 ≤ l ≤ n + p,

so T j(y) ∈ L(n) for all j ≤ q(n + p), so (i) holds.

Also note that Tn(xi) ∈ X<m and d(Ti(n+p)+n(y), Tn(xi)) < η, so Ti(n+p)+n(y) ∈ X<m+ε for i < q and

hence∣∣ϕ (Ti(n+p)+n+l(y)

) ∣∣ ≤ C for l < p, so for each i < q,

|Sp(ϕ)(

Ti(n+p)+n(y))| ≤ Cp. (6.2)

On the other hand, since xi ∈ s(γ, ε, m, n, δ), for i < q, we have Sn(ϕ)(xi) ∈ (nγ − ε, nγ + ε). Since

d(

Ti(n+p)+l(y), Tl(xi))< η for l ≤ n it follows that for each i < q we have

Sn(ϕ)(

Ti(n+p)(y))∈ (nγ− 2ε, nγ + 2ε). (6.3)

By combining (6.3) with (6.2) we see that for each i < q,

Sn+p(ϕ)(

Ti(n+p)(y))∈ (nγ− 2ε− Cp, nγ + 2ε + Cp) ,

so (iv) holds.

To see (ii) and (iii) we shall choose ya, yb ∈ Z(q) so that ya = πn(xa), yb = πn(xb) and xa 6= xb and show

that d(

Tl(ya), Tl(yb))> η for some l < q(n + p). Write xa = (xa

i )∞i=0 and xb = (xb

i )∞i=0. Since xi = x∗ for all

i ≥ q, there must be some i′ < q with xai′ 6= xb

i′ . Since xai′ , xb

i′ ∈ s(γ, ε, m, n, δ) are distinct, there must exist some

0 ≤ j < n with d(

T j(xai′), T j(xb

i′))

> δ. Now, by construction we also have d(

Ti′(n+q)+j(ya), T j(xai′))

< η

and d(

Ti′(n+q)+j(yb), T j(xbi′))< η. Since η < δ/3 we have d

(Ti′(n+q)+j(ya), Ti′(n+q)+j(yb)

)> δ/3 > η. This

completes the proof of (ii) and (iii).


Given a measure ν on X we shall define for n ∈N an n-th level average measure by

An(ν) :=1n

n−1

∑i=0

ν T−i.

Lemma 6.14. Let µ be a Borel probability measure on X and let ξ be a finite Borel partition of X. Then for each pair

n, l ∈N with n ≥ 2l we have

1n

Hν

(n−1∨i=0

T−iξ

)≤ 1

lHAn(ν)

(l−1∨i=0

T−iξ

)+

2ln

log #ξ.

Proof. See [CFH, Lemma 2.4].

Lemma 6.15. Fix m ∈ N and δ, ε > 0. There exists p = p(m, δ, ε) ∈ N, C = C(m, δ, ε) ≥ 0 such that for all γ ∈ R

and all n ∈N there exists a measure µ ∈ MT(X) such that

(i) hµ(T) ≥ 1n+p log #s (γ, ε, m, n, δ),

(ii)∫

ϕdµ ∈[

n(γ−2ε)−Cpn+p , n(γ+2ε)+Cp

n+p

].

Proof. We begin by choosing η > 0, p ∈ N and C ≥ 0 as in Lemma 6.13. Now choose γ ∈ R and n ∈ N. It

follows from Lemma 6.13 that there exists a precompact set L(n) ⊂ X such that for each q ∈ N there exists a

set Z(q) ⊂ X satisfying,

(i) Ti (Z(q)) ⊂ L(n) for all i ≤ q(n + p),

(ii) Z(q) is (q(n + p), η)-seperated set,

(iii) The cardinality of Z(q) is #s(γ, ε, m, n, δ)q,

(iv) For all x ∈ Z(q),


Since L(n) is precompact we may choose a finite partition ξ of L(n) into finitely many Borel sets diameter

strictly less than η. Let ξ be the finite partition ξ ∪ X\L(n). For each q ∈N we define

νq :=1

#Z(q) ∑x∈Z(q)

δx

and let µq := Aq(n+p)(νq). Consider any given q ∈ N. Since Z(q) is a (q(n + p), η)-seperated set, Ti (Z(q)) ⊂

L(n) for all i ≤ q(n + p), and ξ partitions L(n) into sets of diameter less than η, each member of Z(q) must

belong to a different members of the partition∨q(n+p)−1

i=0 T−iξ. It follows from the definition of νq that

Hνq

q(n+p)−1∨i=0

T−iξ

≥ log #Z(q)

= q log #s(γ, ε, m, n, δ).


Hence, by Lemma 6.14 for all l ≤ b2−1q(n + p)c we have

1l

Hµq

(l−1∨i=0

T−iξ

)≥ 1

n + plog #s(γ, ε, m, n, δ)− 2l

q(n + p)log #ξ.

Now since Ti (Z(q)) ⊂ L(n) for all i ≤ q(n + p) and νq is supported on Z(q), the measure µq = Aq(n+p)(νq) is

supported on L(n) for all q ∈ N. Since L(n) is precompact this implies that the sequence (µq)q∈N has a weak

∗ limit point µ. Clearly µ is a T-invariant measure supported on L(n), so µ ∈ MT(X). Moreover, by taking

limits, the above inequality shows that for each l ∈N we have

1l

Hµ

(l−1∨i=0

T−iξ

)≥ 1

n + plog #s(γ, ε, m, n, δ).

Also, for each x ∈ Z(q),


Hence, for each q ∈N we have∫Sq(n+p)(ϕ)dνq ∈ (q(n(γ− 2ε)− Cp), q(n(γ + 2ε) + Cp)) .

Thus, as µq := Aq(n+p)(νq) we have,∫ϕdµq ∈

(n(γ− 2ε)− Cp

n + p,

n(γ + 2ε) + Cpn + p

).

Taking limits we obtain ∫ϕdµ ∈

[n(γ− 2ε)− Cp

n + p,

n(γ + 2ε) + Cpn + p

].

Corollary 6.16. For all m ∈N, γ ∈ R and δ, ε > 0 we have

lim supn→∞

n−1 log #s(γ, ε, m, n, δ)

≤ sup

hµ(T) : µ ∈ MT(X),

∫ϕdµ ∈ (γ− 3ε, γ + 3ε)

.

Proof. This follows from Lemma 6.15 by taking sufficiently large n.

Proposition 6.17. Given any α, β ∈ R we have

dimH(

Jϕ(α, β))≤ d2 − d1d2 − 1 + min

γ∈α,β

limε→0

sup

hµ(T)λ−1 : µ ∈ MT(X),

∫ϕdµ ∈ (γ− ε, γ + ε)

.

Proof. Take any γ ∈ α, β and ε > 0. By Lemma 6.10 we have Jϕ(α, β) ⊆ ⋃m∈N Kϕ(γ, ε, m), so it suffices to

show that for each m ∈N we have

dimH(Kϕ(γ, ε, m)

)≤ d2 − d1d2 − 1 + sup

hµ(T)λ−1 : µ ∈ M

T(X),∫

ϕdµ ∈ (γ− 3ε, γ + 3ε)

.

By combining Lemmas 6.11 and 6.12 along with Corollary 6.16 the proposition holds.

The proof of the upper bound is now almost complete. To remove the ε from Proposition 6.17 we must

show that the entropy-theoretic formula given by Theorem 6.2 is upper-semicontinuous.


6.5.4 The entropy function is concave and continuous

Recall that we defined ϕ− and ϕ+ by

ϕ− := inf

α ∈ R : Jϕ(α) 6= ∅

ϕ+ := sup

α ∈ R : Jϕ(α) 6= ∅

.

Lemma 6.18. We have

ϕ− = inf∫

ϕdµ : µ ∈ MT(X)

ϕ+ = sup

∫ϕdµ : µ ∈ M

T(X)

.

Proof. Take β > ϕ−. Then there exists some α ∈ [ϕ−, β) with Jϕ(α) 6= ∅. By Proposition 6.17 there must exist

some µ ∈ MT(X) with

∫ϕ ∈ (ϕ−, β). Hence, β > inf

∫ϕdµ : µ ∈ M

T(X)

. Since this holds for all β > ϕ−

we have

ϕ− ≥ inf∫


.

The proof of

ϕ+ ≤ sup∫


is almost identical.

The opposite inequalites follow from Birkhoff’s ergodic theorem.

Note that∫


is an interval sinceM

T(X) is closed under convex combinations.

We define a function Hϕ : (ϕ−, ϕ+)→ R by

Hϕ(γ) = sup

hµ(T) : µ ∈ MT(X),

∫ϕdµ = γ

.

Proposition 6.19. The map Hϕ is concave and continuous.

The proof of Proposition 6.19 follows the next two lemmas.

Lemma 6.20. The function Hϕ : (ϕ−, ϕ+)→ R is concave.

Proof. Take γ1, γ2 ∈ R so that ϕ− < γ1 < γ2 < ϕ+ and ρ ∈ (0, 1). Choose µ1, µ2 ∈ MT(X) so that

∫ϕdµ1 =

γ1,∫

ϕdµ2 = γ2 and

hµ1(T) > sup

hµ(T) : µ ∈ MT(X),

∫ϕdµ = γ1

− ε

hµ2(T) > sup

hµ(T) : µ ∈ MT(X),

∫ϕdµ = γ2

− ε.

Note that ∫ϕd(ρµ1 + (1− ρ)µ2) = ρ

∫ϕdµ1 + (1− ρ)

∫ϕdµ2 = ργ1 + (1− ρ)γ2.


Hence,

Hϕ(ργ1 + (1− ρ)γ2) ≥ hρµ1+(1−ρ)µ2(T)

= ρhµ1(T) + (1− ρ)hµ2(T)

> ρHϕ(γ1) + (1− ρ)Hϕ(γ2)− ε.

Letting ε→ 0 proves that Hϕ is concave.

Lemma 6.21. The function Hϕ : (ϕ−, ϕ+)→ R is upper-semicontinuous.

Proof. We must show that for each γ ∈ (ϕ−, ϕ+) we have For all γ ∈ (ϕ−, ϕ+) we have

limε→0

sup

hµ(T) : µ ∈ MT(X),

∫ϕdµ ∈ (γ− ε, γ + ε)

≤ sup

hµ(T) : µ ∈ M

T(X),∫

ϕdµ = γ

.

We first choose µ+, µ− ∈ MT(X) so that ∫

ϕdµ+ ∈ (γ, ϕ+) ,∫ϕdµ− ∈ (ϕ−, γ) .

Now choose a sequence of measures (µj)j∈N ⊂MT(X) with each∫

ϕdµj ∈(

max

γ− j−1,∫

ϕdµ−

, min

γ + j−1,

∫ϕdµ+

),

and so that

limj→∞

hµj(T) = limε→0

sup

hµ(T) : µ ∈ MT(X),

∫ϕdµ ∈ (γ− ε, γ + ε)

.

Note that the sequence on the right is non-increasing and bounded below, so the limit certainly exists.

For each j ∈N we define

ρj :=

γ−∫

ϕdµj∫ϕdµ+−

∫ϕdµj

if∫

ϕdµj ≤ γ,∫ϕdµj−γ∫

ϕdµj−∫

ϕdµ−if∫

ϕdµj > γ.

We also define

νj :=

(1− ρj)µj + ρjµ+ if

∫ϕdµj ≤ γ,

(1− ρj)µj + ρjµ− if∫

ϕdµj > γ.

In each case we see that νj ∈ MT(X) and

∫ϕdνj = γ. Moreover, limj→∞ ρ(j) = 0. Thus,

limj→∞

hνj(T) ≥ limj→∞

(1− ρj)hµj(T)

≥ limj→∞

hµj(T)

= limε→0

sup

hµ(T) : µ ∈ MT(X),

∫ϕdµ ∈ (γ− ε, γ + ε)

.

This completes the proof of the lemma.


Proof of Proposition 6.19. This follows immediately from lemmas 6.20 and 6.21.

To complete the proof of the upper bound in Theorem 6.2 we combine Lemma 6.21 with Proposition 6.17.

Note that the upper bound in Theorem 6.3 is trivial.

6.6 Proof of the lower bound

We begin by constructing certain measures which will be required for the proof of the lower bound in Theorem

6.3 before introducing the key proposition (Proposition 6.26) in the proof of the lower bound.

6.6.1 Compactly supported measures with high entropy

As an application of Proposition 6.17 we shall prove the existence of compactly supported measures with

entropy arbitrarily close to the maximal entropy of the system. This result was previously proved by Kleinbock

and Margulis using an alternative method [KM].

Proposition 6.22 (Kleinbock, Margulis). For each ε > 0 there exists µ ∈ MT(X) with hµ(T) > d1d2λ− ε.

Lemma 6.23. There exists a uniformly continuous potential ϕ : X → R which is non-negative, integrable with respect

to the Haar measure and tends to infinity in the cusp.

Proof. Let m be the Haar measure on X. Fix a point x0 ∈ X. Then we have

∑n∈N

m (x ∈ X : n− 1 ≤ d(x, x0) < n) = m(X) = 1.

Thus, we may define a strictly increasing sequence of natural numbers (N(l))l∈N by taking N(1) ∈N so that

∑n≥N(1)

µ (x ∈ X : n− 1 ≤ d(x, x0) < n) < 12

,

and for each l ≥ 2 we choose N(l) > N(l − 1) so that N(l) ∈N and

∑n≥N(1)

µ (x ∈ X : n− 1 ≤ d(x, x0) < n) < 12l .

We define a function g : [0, ∞)→ [0, ∞) by

g(y) :=

0 for 0 ≤ y ≤ N(1)

y + l − 1− N(l) for N(l) ≤ y ≤ N(l) + 1, l ∈N

l for N(l) + 1 ≤ y ≤ N(l + 1), l ∈N.

Then g is uniformly continuous. Hence, ϕ : X → R defined by ϕ(x) = f (d(x, x0)) is uniformly continuous.

Moreover, it is clear that ϕ tends to infinity in the cusp. Finally ϕ(x) ≤ l− 1 for all x ∈ X with d(x, x0) ≤ N(l),


so ∫ϕdm ≤ ∑

n∈N

m (x ∈ X : n− 1 ≤ d(x, x0) < n) · sup ϕ(x) : n− 1 ≤ d(x, x0) < n

≤ ∑l∈N

∑n≥N(l)

m (x ∈ X : n− 1 ≤ d(x, x0) < n) · l

≤ ∑l∈N

l2l < ∞.

Proof of Proposition 6.22. Take the potential ϕ constructed in Lemma 6.23. Since∫

ϕdm < ∞ it follows from the

Birkhoff ergodic theorem applied to the Haar measure m that

m(

Jϕ

(∫ϕdm

))= 1.

Hence, dimH(

Jϕ (∫

ϕdm))= d2 − 1. On the other hand, by Proposition 6.17 we have

dimH

(Jϕ

(∫ϕdm

))≤ d2− d1d2− 1+ lim

ε→0sup

hµ(T)λ−1 : µ ∈ M

T(X),∫

ϕdµ ∈(∫

ϕdm− ε,∫

ϕdm + ε

).

Thus,

sup

hµ(T) : µ ∈ MT(X),

∫ϕdµ ∈ (γ− ε, γ + ε)

= λ · d1d2.

Kadyrov has proved the existence of compactly supported measures arbitrarily high in the cusp [Ka].

Theorem 6.24 (Kadyrov). Suppose that Γ = SLd(Z) and a = (e, · · · , e, e−(d−1)). There exists a sequence νnn∈N ⊂

MT(X) of compactly supported T-invariant measures such that for any compact set K ⊂ X we have limn→∞ νn(K) = 0.

Lemma 6.25. Suppose that Γ = SLd(Z) and a = (e, · · · , e, e−(d−1)) and let ϕ be a uniformly continuous potential

which goes to infinity in the cusp. Then for any ε > 0 there exists a sequence of compactly supported invariant measures

µnn∈N ⊂MT(X) with hµn(T) > λ · d1d2 − ε for all n ∈N and limn→∞

∫ϕdµn = ∞.

Proof. Fix ε > 0. By Proposition 6.22 we may choose there exists µ0 ∈ MT(X) with hµ0(T) > d1d2λ − ε.

Moreover, by Theorem 6.24 there exists a sequence νnn∈N ⊂ MT(X) of compactly supported T-invariant

measures with limn→∞ ϕdνn = ∞, since for each m ∈ N the set x ∈ X : ϕ(x) < m is precompact. Now, for

each n ∈N we choose q(n) ∈N so that∫

ϕdνq(n) > n2.

For each n ∈N we let µn :=(

1− 1n

)µ0 +

1n νq(n). Then, for each n ∈N we have

∫ϕdµn ≥

(1− 1

n

) ∫ϕdµ0 + n,

so limn→∞∫

ϕdµn = ∞ and for each n ∈N we have

hµn(T) >(

1− 1n

)(λ · d1d2 − ε) ,

so lim infn→∞ hµn(T) > λ · d1d2 − ε. Thus, by moving to a tail sequence if necessary, we obtain a sequence of

compactly supported measures with the required properties.


6.6.2 Central lemma in the proof of the lower bound

The key proposition for the proof of the lower bound is as follows.

Proposition 6.26. Suppose we have a sequence of compactly supported invariant measures

νq

q∈N⊆M

T(X). Then

dimH Jϕ

(lim inf

q→∞

∫ϕdνq, lim sup

q→∞

∫ϕdνq

)≥ (d2 − d1d2 − 1) + λ−1 inf

q∈Nhνq(T).

Using the results of the previous section we shall show that Proposition 6.26 implies the lower bounds in

Theorems 6.2 and 6.3.

Proof of the lower bounds in Theorems 6.2 and 6.3 given Proposition 6.26. To prove the lower bound in Theorem 6.2

we take any α, β ∈ R so that ϕ− < α ≤ β < ϕ+. Given ε > 0 we choose any µα, µβ so that∫

ϕdµα = α,∫ϕdµα = β so that

hµα(T) > sup

hµ(T) : µ ∈ MT(X),

∫ϕdµ = α

− ε,

hµβ(T) > sup

hµ(T) : µ ∈ M

T(X),∫

ϕdµ = β

− ε.

Let νqq∈N be the sequence defined by

νq :=

µα if q ∈ 2N− 1

µβ if q ∈ 2N.

It follows that lim infq→∞∫

ϕdνq = α and lim supq→∞∫

ϕdνq = β. By Proposition 6.26 we have

dimH(

Jϕ(α, β))> (d2 − d1d2 − 1) + λ−1 min

hµα(T), hµβ

(T)

> (d2 − d1d2 − 1) + minγ∈α,β

sup

hµ(T) : µ ∈ MT(X),

∫ϕdµ = γ

− ε.

Letting ε→ 0 completes the proof of the lower bound in Theorem 6.2.

We now deduce Theorem 6.3 from Proposition 6.26. So suppose that Γ = SLd(Z) and a =(e, · · · , e, e−(d−1)

). To prove that dimH

(J′ϕ)≥ d2 − 1 we take ε > 0 and let µnn∈N be the corresponding

sequence given by Lemma 6.25 which satisfies hµn(T) > λd1d2 − ε for each n ∈ N and limn→∞∫

ϕdµn = ∞.

It follows that we may choose n1, n2 ∈N so that∫

ϕdµn1 <∫

ϕdµn2 . Let νqq∈N be the sequence defined by

νq :=

µn1 if q ∈ 2N− 1

µn2 if q ∈ 2N.

Then lim infq→∞ ϕdνq =∫

ϕdµn1 and lim supq→∞ ϕdνq =∫

ϕdµn2 , so by Proposition 6.26 we have

dimH

(Jϕ

(∫ϕdµn1 ,

∫ϕdµn2

))> (d2 − d1d2 − 1) + λ−1 (λd1d2 − ε) .

Since∫

ϕdµn1 6=∫

ϕdµn2 we gave Jϕ (∫

ϕdµn1 ,∫

ϕdµn2) ⊆ J′ϕ so

dimH

(J′ϕ)> (d2 − d1d2 − 1) + λ−1 (λd1d2 − ε) .

Letting ε → 0 gives dimH

(J′ϕ)≥ d2 − 1. To prove that dimH

(Jϕ(∞)

)≥ d2 − 1 we simply apply Proposition

6.26, taking νqq∈N to be the sequence given by Lemma 6.25 for a given ε, before letting ε tend to zero.


We now proceed to prove Proposition 6.26. The proof of Proposition 6.26 proceeds as follows. We begin by

reducing the question to a lower bound along local unstable manifolds. We then relate the existence of high

entropy invariant measures which integrate the integral to a given value to the existence of large (n, δ) sets

with a given average n-th level Birkhoff sum. The chain lemma is then used to sew these well-seperated sets

together in such a way as to construct a Cantor set of a high dimension, contained within the intersection of

the level set and the local unstable manifold.

6.6.3 Reduction to unstable manifolds

In order to prove Proposition 6.26 we will first reduce the question to studying individual unstable manifolds.

Proposition 6.27. Suppose we have a sequence of compactly supported invariant measures

νq

q∈N⊆ M

T(X), a

point z ∈ X and some η∗ > 0. Then there exists a set T ⊂ Md2×d1(R) such that for all t ∈ T ||t||∞ < η∗ the point

x = z · u(t) satisfies

ϕ∗(x) = lim infq→∞

∫ϕdνq, ϕ∗(x) = lim sup

q→∞

∫ϕdνq,

and

dimHT ≥ λ−1 infq∈N

hνq(T).

Proof of Proposition 6.26 from Proposition 6.27. We shall make use of the observations from Section 6.4.1. Fix a

sequence of compactly supported invariant measures

νq

q∈N⊆ M

T(X). Take some z0 ∈ X and let η∗ :=

min

η1, c−11 η(z0)

. We let P(η∗) := p ∈ P : ||p− e||∞ < η∗. By Proposition 6.27, for each p ∈ P(η∗) we

may choose Tp with

dimHTp ≥ λ−1 infq∈N

hνq(T)

and such that for all t ∈ Tp ||t||∞ < η∗ the point x = z0 · p · u(t) satisfies

ϕ∗(x) = lim infq→∞

∫ϕdνq, ϕ∗(x) = lim sup

q→∞

∫ϕdνq.

It follows that ⋃p∈P(η∗)

⋃t∈Tp

z0 · p · u(t) ⊆ Jϕ

(lim inf

q→∞

∫ϕdνq, lim sup

q→∞

∫ϕdνq

).

Since dimH P(η∗) ≥ d2 − d1d2 − 1 it follows from Marstrand’s fibre lemma (Proposition 2.10) that

dimH

⋃p∈P(η∗)

p × Tp

≥ (d2 − d1d2 − 1) + λ−1 infq∈N

hνq(T).

Moreover, for each p ∈ P(η∗) and t ∈ Tp we have ||p− e||∞ < η1 and ||t||∞ < η1 and so

dG (p · u(t), e) ≤ c1 max ||p− e||∞, ||t||∞ < η(z0),

and by the local equivalence of metrics we have

dimH

⋃p∈P(η∗)

⋃t∈Tp

p · u(t)

≥ (d2 − d1d2 − 1) + λ−1 infq∈N

hνq(T).


Since each p ∈ P(η∗) and t ∈ Tp satisfies dG (p · u(t), e) < η(z0) it follows that the map from⋃

p∈P(η∗)⋃

t∈Tpp ·

u(t) to⋃

p∈P(η∗)⋃

t∈Tpz0 · pu(t) by g 7→ z0 · g is an isometry and thus,

dimH

⋃p∈P(η∗)

⋃t∈Tp

z0 · pu(t)

≥ (d2 − d1d2 − 1) + λ−1 infq∈N

hνq(T),

which implies

dimH Jϕ

(lim inf

q→∞

∫ϕdνq, lim sup

q→∞

∫ϕdνq

)≥ (d2 − d1d2 − 1) + λ−1 inf

q∈Nhνq(T).

6.6.4 Constructing a partition

Given a set A we let ∂A denote the boundary A∩X\A. Given a collection of setsAwe let ∂A denote the union

of the boundaries of elements in A.

Lemma 6.28. Let ν ∈ MT(X) be a compactly supported invariant measure. For each ε > 0 there exists a finite partition

ξ along with a distinguished element γ ∈ ξ such that,

(i) hν(T, ξ) > hν(T)− ε,

(ii) ν(∂ξ) = 0,

(iii) X\γ is precompact,

(iv) ν(γ) = 0.

Proof. First note that X =⋃

m∈N X<m. Since each ν is compactly supported we may choose M so that for all

m ≥ M ν (X<m) = 1 for all l ≤ p.

We shall begin by showing that for each m ≥ M we may choose a partitionAm of X into finitely many Borel

sets with a distinguieshed element Am0 ∈ Am such that Am

0 ⊂ X\X<m and diamA < 1/m for all A ∈ Am\Am0

with the additional property that ν (∂Am) = 0.

Indeed for each z ∈ X the set of r > 0 for which ν (y : d(z, y) = r) > 0 is at most countable, so there

exists some η(z) ∈(

0, 12m

)with ν

(∂Bη(z)(z)

)= 0.

By compactness there exists a finite set z1, · · · , zk ⊂ X≤m for which X<m ⊂⋃k

i=1 Br(zi)(zi). For each

i ∈ 1, · · · , k we let Ami := Br(zi)

(zi)\⋃

j<i Br(zj)(zj) and let Am

0 := X\⋃j≤k Br(zj)(zj). We define a partition by

Am :=

Am0 , Am

1 , · · · , Amk

.

Note that for any pair of sets A, B ⊂ X we have ∂(A ∪ B) ⊆ ∂A ∪ ∂B and by taking complements we also

have ∂(A ∩ B) ⊆ ∂A ∪ ∂B. It follows that ν(∂Am) = 0. We also have diamAmi < 1

m for all i = 1, · · · , k and

Am0 ⊂ X\X<m. In particular ν(Am

0 ) = 0.

Given a finite partition C we let C denote the algebra of sets generated by C. Since for each m ∈N we have

Am0 ⊂ X\X<m and diamAm

i < 1m for all i 6= m the collection

⋃m≥M Am generates the Borel sigma algebra.


Indeed given x0 ∈ X and m0 ∈ N we may choose m ≥ max3m0, M so that x ∈ X<m. Since X<m is open we

may take ρ ∈(

0, 13m0

)so that Bρ(x0) ⊂ X<m. By taking N(x0, m0) :=

⋃ Am

i : Ami ∩ Bρ(x0) 6= ∅

we obtain a

neighbourhood of x0 of diameter no more than 1/m0. As⋃

m≥M Am is countable it is clear that every open set

in X is a countable union of members of⋃

m≥M Am so the collection generates the Borel sigma algebra.

For each m ≥ M we take Bm :=∨m

j=MAj. Using the fact that ∂(A ∩ B) ⊆ ∂A ∪ ∂B once more, we see that

ν(∂Bm) = 0. Moreover, for each m ∈N we have Am ⊂ Bm and so⋃

m≥M Bm generates the Borel sigma algebra.

Since Bm ⊆ Bm+1 for each m ∈N. Thus, by [W, Theorem 4.22] there exists some m∗ ≥ M with

hν(T,Bm∗) > hν(T)− ε.

Taking ξ = Bm∗ we see that (i) and (ii) hold. We let γ :=⋂m∗

j=M Aj0 ∈ Bm∗ . Every other element of Bm∗

is contained within Ami for some i > 0 and m ≤ m∗ so is bounded and hence precompact. Since there are

only finitely many such elements it follows that X\γ =⋃Bm∗\γ is precompact which implies (iii). Also

ν(Am∗0 ) = 0 so ν(γ) = 0, so (iv) also holds.

6.6.5 Lower bound construction

In order to prove Proposition 6.27 we shall make use the chain lemma (Lemma 6.8) to construct a set with the

correct limiting behaviour and large dimension along the unstable manifold.

Proof of Proposition 6.27. Choose a sequence of compactly supported invariant measures

νq

q∈N⊆ M

T(X)

and a point z ∈ X. We let ν0 := δz. Choose some h < infq∈N hνq(T).

Given a partition ξ and a natural number n ∈N we let ξn denote the partition

ξn :=n−1∨j=0

T−jξ.

By Lemma 6.28 for each q ∈ N there exists a finite partition ξq along with a distinguished element γq ∈ ξq

such that,

(i) hνq(T, ξq) > h,

(ii) νq(∂ξq) = 0,

(iii) X\γq is precompact,

(iv) νq(γq) = 0.

Let K0 := x ∈ X : d(x, z) ≤ δ0 and for each q ∈N we define Kq := Kq−1 ∪ X\γq. By (iii) each Kq is compact.

As before we define

Kq :=

x ∈ X : ∃y ∈ X\γq, d(x, y) ≤ 1

and for each q ∈N we let

Φ(q) := sup

|ϕ(x)| : x ∈m(q+2)⋃

i=0

Ti (Kq)+ 1.


Since T and ϕ are continuous this implies that Φ(q) < ∞ for all q ∈ N. We also define for compact set

K ∈ P0(X) and δ > 0 the set q ∈N

Vδ(ϕ, K) := sup |ϕ(x)− ϕ(y)| : x, y ∈ K, d(x, y) < δ .

Again, since ϕ is continuous we have limδ→0 Vδ(ϕ, K) = 0 for all K ∈ P0(X).

Let δ0 := min η(z), η0, η1, η∗, 1. We shall define constants (δq)q∈N ⊂ (0, 1) and thickenings of the bound-

ary (D(q))q∈N inductively. Suppose that δq−1 ∈ (0, 1) and D(q − 1) have been defined. For each δ > 0 we

let

Dq(δ) :=

x ∈ X : ∃y ∈ ∂ξq, d(x, y) ≤ δ

.

Note that⋂

δ>0 Dq(δ) = ξq. Thus, by (i) we may choose δq ∈(0, δq−1q−1) so that

Vδq(ϕ, Kq) < q−1 and νq(

Dq(3δq))< (2q log #ξq)

−1

and let D(q) := Dq(3δq).

For each q ∈N∪ 0 let

m(q) := max m (Kl , δl) : l ≤ q + 1 ,

where m : P(X)×R>0 is the function given by the chain lemma (Lemma 6.8).

We let choose k(0) = 1 and for q ∈N define k(q) := qm(q)Φ(q).

Since νq is T-invariant and h < hνq(T, ξ) we have∫log νq

(ξk(q)(x)

)dνq(x) < −k(q)h, (vii’)∫

Sk(q)

(log #ξq · 1D(q)

)(x)dνq(x) < k(q)(2q)−1 (viii’)∫

Sk(q)(ϕ)(x)dνq(x) = k(q)∫

ϕdνq. (ix’)

Thus, by Lemma 6.9, for each q ∈ N there exists ι(q) ∈ N such that for all τ ≥ ι(q) there exists Wτ(q) ⊂ Kτq

such that

ντq (Wτ(q)) > e−q−2

, (vi)

and for all ι(q) ≤ l ≤ τ we have,

l

∑i=1

log νq

(ξk(q)(xi)

)< −lk(q)h (vii)

l

∑i=1

Sk(q)

(log #ξq · 1D(q)

)(xi) < lk(q)q−1 (viii)

l

∑i=1

Sk(q)(ϕ)(xi) ∈ B(

lk(q)∫

ϕdνq, lk(q)q−1)

. (ix)

We shall recursively define a pair of sequences (τ(q))∞q=0 and (φ(q))∞

q=0 by τ(0) = 1, φ(0) = 1 + m(0) and

for q ∈N,

τ(q) := q(ι(q + 1)(k(q + 1) + m(q + 1)) + φ(q− 1)− log δq+1

)Φ(q) + ι(q) (x)

φ(q) := φ(q− 1) + (m(q) + k(q))τ(q). (xi)


We let W(0) := z and for each q ∈N we let W(q) := Wτ(q)(q).

We let Λ := ∏∞q=0 Kτ(q)

q and define a set W ⊂ Λ by W := ∏∞q=0 W(q).

We shall now use the chain lemma (Lemma 6.8) to define a suitable projection π from the infinite product

space Λ to the underlying homogeneous space X along with a projection χ from Λ to the unstable manifold

coordinates Md2×d1(R).

Let γ(0, 1) := 0 and for q ∈N and 1 ≤ l ≤ τ(q) we define

γ(q, l) := φ(q− 1) + (l − 1) (m(q) + k(q)) .

By Lemma 6.8 we may define a projection π : Λ → X such that for each x = (xq)∞q=0 ∈ Λ with each xq =

(xqi)τ(q)i=1 ∈ Λτ(q) the point y = π (x) satisfies

d(

Tn(y), Tn−γ(q,l)(xql))< δq < 1/q, (xii)

for each q ∈N, 1 ≤ l ≤ τ(q) and γ(q, l) ≤ n < γ(q, l) + k(q). Moreover, y = z · u(t) for t = χ(x) ∈ Md1×d1(R)

with ||t||∞ = χ(x) < δ0 < η0.

Proposition 6.29. For all y ∈ π(W) we have

ϕ∗(y) = lim infq→∞

∫ϕdνq and ϕ∗(y) = lim sup

q→∞

∫ϕdνq.

Proof. The proof of Proposition 6.29 is broken down into several lemmas.

Lemma 6.30. For all y ∈ π (Λ) we have |ϕ(Tl(y))| ≤ Φ(q− 2) for all l ≤ φ(q).

Proof. By (xii) combined with the fact that δr < 1 and Kr ⊂ Kr+1 for all r ∈N we see that Tl(y) ∈ Kq whenever

l ∈q⋃

r=0

τ(r)⋃i=1γ(r, i) ≤ j < γ(r, i) + k(r) .

Since m(r) ≤ m(q) for all r ≤ q it follows that y ∈ ⋃m(q)i=0 Ti (Kq

)for all

l ∈q⋃

r=0

τ(r)⋃i=1γ(r, i) + k(r) ≤ j < γ(r, i) + k(r) + m(r) .

Thus, |ϕ(Tl(y))| ≤ Φ(q− 1) for all l ≤ φ(q), which implies the claim.

Lemma 6.31. For all q ∈N,

0 ≤ φ(q)− k(q)τ(q) ≤ (φ(q)− k(q)τ(q))Φ(q) ≤ 2φ(q)q−1.

Proof. The first inequality follows from the definition of φ(q). The second follows from the fact that Φ(q) ≥ 1.

Finally, since τ(q) ≥ qφ(q− 1)Φ(q) and k(q) ≥ qm(q)Φ(q) we have

(φ(q)− k(q)τ(q))Φ(q) = φ(q− 1)Φ(q) + m(q)Φ(q)τ(q)

≤ τ(q)q−1 + q−1k(q)τ(q)

≤ 2k(q)τ(q)q−1

≤ 2φ(q)q−1.


Lemma 6.32. For all y ∈ π (W), ϕ∗(y) ≤ lim infq→∞∫

ϕdνq and ϕ∗(y) ≥ lim supq→∞∫

ϕdνq.

Proof. Fix some y = π(x) where x =(

xq)∞

q=0 ∈ W ⊂ Λ with xq = (xq,i)τ(q)i=1 ∈ W(q) ⊂ Λτ(q). Take sequences

(qαl )l∈N and (qβ

l )l∈N so that

liml→∞

∫ϕdνqα

l= lim inf

q→∞

∫ϕdνq,

liml→∞

∫ϕdν

qβl= lim sup

q→∞

∫ϕdνq.

It suffices to show that

liml→∞

φ(qαl )−1Sφ(qα

l )(ϕ)(y) ≤ lim inf

q→∞

∫ϕdνq (α)

liml→∞

φ(qβl )−1S

φ(qβl )(ϕ)(y) ≥ lim sup

q→∞

∫ϕdνq. (β)

We shall prove (β). The proof of (α) is almost identical.

Take q ∈ qβl l∈N. By (ix) we have

τ(q)

∑i=1

k(q)−1

∑j=0

ϕ(T j(xq,i)) > τ(q)k(q)(∫

ϕdνq − q−1)

.

By (xii) and Vδq(ϕ, Kq) < q−1 this implies that

τ(q)

∑i=1

k(q)−1

∑j=0

ϕ(Tγ(q,i)+j(y)) > τ(q)k(q)(∫

ϕdνq − 2q−1)

.

By Lemma 6.30 we have ϕ(Tn(y)) ≥ −Φ(q− 1) for all n ≤ φ(q). Hence,

φ(q)−1

∑n=0

ϕ(Tn(y)) > τ(q)k(q)(∫

ϕdνq − 2q−1)− (φ(q)− τ(q)k(q))Φ(q− 1).

As such, it suffices to show that

limq→∞

(φ(q)− τ(q)k(q))Φ(q− 1)φ(q)−1 = limq→∞

(φ(q)− τ(q)k(q))φ(q)−1 = 0.

But this follows immediately from Lemma 6.31 as Φ(q) ≥ Φ(q− 1).

Given n ≤ φ(0) we let q(n) = l(n) = a(n) = 0 and for n > φ(0) we define q(n), l(n) ∈N by

q(n) := min q ∈N : n ≤ φ (q) + k(q + 1) + m(q + 1) ,

l(n) := max l ≤ τ(q(n)) : φ(q(n)− 1) + l (k(q(n)) + m(q(n))) ≤ n .

For each n ∈N, we define A(n) by

A(n) :=

⋃τ(q(n)−1)i=1 γ(q(n)− 1, i) ≤ j < γ(q(n)− 1, i) + k(q(n)− 1)

if l(n) < ι(q(n))

⋃τ(q(n)−1)i=1 γ(q(n)− 1, i) ≤ j < γ(q(n)− 1, i) + k(q(n)− 1)

∪⋃l(n)i=1 γ(q(n), i) ≤ j < γ(q(n), i) + k(q(n))

if l(n) ≥ ι(q(n)).


The set A(n) is a subset of natural numbers j ≤ n for which the j-th iterate T j(y) of point y = π(x) ∈ π(Λ)

may be controlled by components of x. For each n ∈N we let a(n) := #A(n).

Equivalently,

a(n) :=

τ(q(n)− 1) · k(q(n)− 1) if l(n) < ι(q(n))

τ(q(n)− 1) · k(q(n)− 1) + l(n) · k(q(n)) if l(n) ≥ ι(q(n)).

Lemma 6.33. For all n ∈N we have

0 ≤ n− a(n) ≤ Φ(q(n)− 1) (n− a(n)) ≤ 4n (q(n)− 1)−1 .

Proof. The first inequality follows from the fact that τ(q(n) − 1) · k(q(n) − 1) ≤ φ (q(n)− 1) and from the

definitions of q(n) and l(n) that φ(q(n)− 1) + l(n) (k(q(n)) + m(q(n))) ≤ n, so a(n) ≤ n for all n ∈ N. The

second inequality follows from the fact that Φ(q(n)− 1) ≥ 1.

To see the third inequality we first recall from Lemma 6.31 that we have φ(q)− k(q)τ(q) ≤ 2φ(q) (Φ(q)q)−1

for all q ∈N.

Given n ∈ N there are three cases. Either l(n) < ι(q(n)), ι(q(n)) ≤ l(n) < τ(q(n)) or l(n) = τ(q(n)). In

the first two cases, it follows from the definition of l(n) that

n < φ(q(n)− 1) + (l(n) + 1) (k(q(n)) + m(q(n))) .

Thus, in the first case where l(n) < ι(q(n)) we have

n− a(n) < φ(q(n)− 1) + (l(n) + 1) (k(q(n)) + m(q(n)))− a(n)

= (φ(q(n)− 1)− τ(q(n)− 1) · k(q(n)− 1))

+ (l(n) + 1) (k(q(n)) + m(q(n)))

≤ 2φ(q(n)− 1)Φ(q(n)− 1)−1 (q(n)− 1)−1

+ ι(q(n)) (k(q(n)) + m(q(n)))

≤ 2φ(q(n)− 1)Φ(q(n)− 1)−1 (q(n)− 1)−1

+ ι(q(n)) (k(q(n)) + m(q(n))) + l(n)m(q(n)).

In the second case where ι(q(n)) ≤ l(n) < τ(q(n)) we again have

n− a(n) < φ(q(n)− 1) + (l(n) + 1) (k(q(n)) + m(q(n)))− a(n)

= (φ(q(n)− 1)− τ(q(n)− 1) · k(q(n)− 1))

+ (k(q(n)) + m(q(n))) + l(n)m(q(n))

≤ 2φ(q(n)− 1) (q(n)− 1)−1

+ ι(q(n)) (k(q(n)) + m(q(n))) + l(n)m(q(n)),


since ι(q(n)) ≥ 1. Thus, in either of the first two cases, so whenever l(n) < τ(q(n)), we have

Φ(q(n)− 1)(n− a(n)) ≤ 2φ(q(n)− 1) (q(n)− 1)−1

+ ι(q(n)) (k(q(n)) + m(q(n)))Φ(q(n)− 1)

+ l(n)m(q(n))Φ(q(n))

≤ 2φ(q(n)− 1) (q(n)− 1)−1

+ τ(q(n)− 1) (q(n)− 1)−1

+ l(n)k(q(n))q(n)−1

≤ 3φ(q(n)− 1) (q(n)− 1)−1 + n(q(n)− 1)−1

≤ 4n (q(n)− 1)−1 ,

where we used the definitions of τ, φ and q(n).

Now suppose l(n) = τ(q(n)). Then, by the definition of q(n) we have

Φ(q(n)− 1)(n− a(n)) ≤ Φ(q(n)− 1)(φ(q(n)) + k(q(n) + 1) + m(q(n) + 1))

−Φ(q(n)− 1)τ(q(n))k(q(n))

≤ 2φ(q(n))q(n)−1

+ (k(q(n) + 1) + m(q(n) + 1))Φ(q(n)− 1)

≤ 2φ(q(n))q(n)−1 + τ(q(n))q(n)−1

≤ 4φ(q(n))q(n)−1.

Moreover,

φ(q(n)) = φ(q(n)− 1) + τ(q(n)) (k(q(n)) + m(q(n))) ≤ n.

Hence, in all three cases we have

Φ(q(n)− 1)(n− a(n)) ≤ 4n (q(n)− 1)−1 .

Lemma 6.34. For all y ∈ π (W), ϕ∗(y) ≥ lim infq→∞∫

ϕdνq and ϕ∗(y) ≤ lim supq→∞∫

ϕdνq.

Proof. Fix some y = π(x) where x =(

xq)∞

q=0 ∈W ⊂ Λ with xq = (xq,i)τ(q)i=1 ∈W(q) ⊂ Λτ(q).

It suffices to show that

lim infn→∞

n−1Sn(ϕ)(y) ≥ lim infq→∞

∫ϕdνq (α)

lim supn→∞

n−1Sn(ϕ)(y) ≤ lim supq→∞

∫ϕdνq. (β)

We shall prove (α). The proof of (β) is similar.


By (vii) for all q ∈N and all l ≥ ι(q) we have

l

∑i=1

Sk(q)(ϕ)(xiq) ≥ lk(q)(∫

ϕdνq − q−1)

.

Hence, for all n ∈N τ(q(n)− 1) ≥ ι(q(n)− 1) so we have

τ(q(n)−1)

∑i=1

k(q(n)−1)−1

∑j=0

ϕ(T j(xi,q(n)−1))

> τ(q(n)− 1)k(q(n)− 1)(∫

ϕdνq(n)−1 − (q(n)− 1)−1)

,

and if l(n) ≥ ι(q(n)) then we also have

l(n)

∑i=1

k(q(n))−1

∑j=0

ϕ(T j(xi,q(n)))

> l(n)k(q(n))(∫

ϕdνq(n) − q(n)−1)

.

By (xii) and Vδq(ϕ, Kq) < q−1 it follows that

τ(q(n)−1)

∑i=1

k(q(n)−1)−1

∑j=0

ϕ(Tγ(i,q(n)−1)+j(y))

> τ(q(n)− 1)k(q(n)− 1)(∫

ϕdνq(n)−1 − 2(q(n)− 1)−1)

,

for all n ∈N and if l(n) ≥ ι(q(n)) then we also have

l(n)

∑i=1

k(q(n))−1

∑j=0

ϕ(Tγ(i,q(n))+j(y))

> l(n)k(q(n))(∫

ϕdνq(n) − 2q(n)−1)

.

Hence,

∑j∈A(n)

ϕ(T j(y)) > a(n)(

infq≥q(n)−1

∫ϕdνq

− 2(q(n)− 1)−1

).

Moreover, by Lemma 6.30 we have |ϕ(T j(y))| ≤ −Φ(q(n)− 1) for all j ≤ nφ(q(n) + 1) so

∑j∈1,··· ,n\A(n)

ϕ(T j(y)) > −(n− a(n))Φ(q(n)− 1).

Putting these two inequalities together gives Sn(ϕ)(y)

> a(n)(

infq≥q(n)−1

∫ϕdνq

− 2(q(n)− 1)−1

)− (n− a(n))Φ(q(n)− 1).

Hence it suffices to show that

limn→∞

a(n)n−1 = 1

limn→∞

Φ(q(n)− 1)(n− a(n))n−1 = 0.

and these limits follow immediately from Lemma 6.33, so the proof is complete.


This completes the proof of Proposition 6.29.

Lemma 6.35. For all x ∈W we have

limn→∞

1n

(τ(q(n)−1)

∑j=1

Sk(q(n)−1)(log #ξq(n)−1 · 1D(q(n)−1))(xq(n)−1,j)+

l(n)

∑j=1

Sk(q(n))(log #ξq(n) · 1D(q(n)))(xq(n),j)

)= 0.

Proof. The proof is essentially the same as the proof of Lemma 6.34 using (viii) in place of (ix).

The set Λ supports the product probability measure ν := ∏∞q=0 ν

τ(q)q .

Lemma 6.36. ν∞ (W) > 0.

Proof. ν∞ (W) = ∏∞q=0 ν

τ(q)q (W(q)) ≥ ∏∞

q=0 e−q2> 0.

This allows us to define probability measure ν supported on W defined by

ν(A) =ν (A ∩W)

ν(W).

For each q ∈N and 1 ≤ l ≤ τ(q) we define

D(q, l) :=τ(q−1)

∏j=1

ξk(q−1)q−1 ×

l

∏j=1

ξk(q)q .

Thus, every A ∈ D(q, l) is of the form

A :=((

Aq−1,1, · · · , Aq−1,τ(q−1)

),(

Aq,1, · · · , Aq,l

))where each Ai,j ∈ ξ

k(i)i . Recall that every x ∈ Λ is of the form x = (xi)

∞i=0 where xi = (xi,j)

τ(i)j=1 ∈ Λτ(i).

For each A ∈ D(q, l) we define a set A ⊂ Λ by

A :=τ(q−1)⋂

j=1

x ∈ Λ : xq−1,j ∈ Aq−1,j

∩

l⋂j=1

x ∈ Λ : xq,j ∈ Aq,j

.

Lemma 6.37. For all n ∈N we have

sup

log ν(

A)

: A ∈ D(q(n), l(n))≤ −nh + o(n).

Proof. Take n ∈ N and A ∈ D(q(n), l(n)). Now either A ∩W = ∅, in which case ν(A) = 0, or we may find

some x ∈W such that

A =τ(q(n)−1)⋂

j=1

w ∈ Λ : wq(n)−1,j ∈ ξ

k(q(n)−1)q(n)−1 (xq(n)−1,j)

∩l(n)⋂j=1

w ∈ Λ : wq(n),j ∈ ξ

k(q(n))q(n) (xq(n),j)

.


It follows from the construction of ν that

ν(

A)= ν

(A)

/ν(W)

≤τ(q(n)−1)

∏l=1

νq(n)−1

(ξ

k(q(n)−1)q(n)−1 (xl,q(n)−1)

)·

l(n)

∏l=1

νq(n)

(ξ

k(q(n))q(n) (xl,q(n))

)· ν(W)−1.

By (vii) along with the fact that τ(q(n)− 1) ≥ ι(q(n)− 1) we see that

log ν(

A)≤ −τ(q(n)− 1)k(q(n)− 1)h− log ν(W),

and if l(n) ≥ ι(q(n)) then

log ν(

A)≤ − (τ(q(n)− 1)k(q(n)− 1) + l(n)k(q(n))) h− log ν(W).

Thus, for all n ∈N we have

sup

log ν(

A)

: A ∈ D(q(n), l(n))≤ −a(n)h− log ν(W).

Moreover, Lemma 6.33 implies a(n) ≥ n− o(n). This completes the proof of the lemma.

Lemma 6.38. Take y = π(x) with x ∈W. For all n ∈N,

#

A ∈ D (q(n), l(n)) : π(A) ∩ Bn

(y, δq(n)

)6= ∅

< exp(o(n)).

Proof. Take y = π(x) with x ∈W. By Lemma 6.35 it suffices to show that for any given n ∈N we have

#

A ∈ D (q(n), l(n)) : π(A) ∩ Bn

(y, δq(n)

)6= ∅

≤τ(q(n)−1)

∏j=1

#ξSk(q(n)−1)(1D(q(n)−1))(xq(n)−1,j)

q(n)−1 ·l(n)

∏j=1

#ξSk(q(n))(1D(q(n)))(xq(n),j)

q(n) .

Note that any A ∈ D(q(n), l(n)) may be written in the form

A =((

Aq(n)−1,1, · · · , Aq(n)−1,τ(q(n)−1)

),(

Aq(n),1, · · · , Aq(n),l(n)

))where each Ar,s =

⋂k(r)j=0 T−jEA

j (r, s) and EAj (r, s) ∈ ξ. To prove the lemma it suffices to prove the claim that for

any A ∈ D(q(n), l(n)) with

π(A) ∩ Bn

(y, δq(n)

)6= ∅ (*)

we have EAj (r, s) = ξ(Tγ(r,s)+j(y)) whenever T j(xr,s)(y) /∈ D(r). Here (r, s, j) is a triple of integers, either

satisfying r = q(n)− 1, 1 ≤ s ≤ τ(q(n)− 1) and 0 ≤ j < k(q(n)− 1) or satisfying r = q(n), 1 ≤ s ≤ l(n) and

0 ≤ j < k(q(n)).

To prove the claim we assume (*) and choose w = (wr)∞r=0 ∈ Λ with each wr = (wr,s)

τ(r)s=1 ∈ Λτ(r) so

that w ∈ A and π(w) ∈ Bn

(y, δq(n)

). Now take r, s, j so that either r = q(n)− 1, 1 ≤ s ≤ τ(q(n)− 1) and

0 ≤ j < k(q(n)− 1) or r = q(n), 1 ≤ s ≤ l(n) and 0 ≤ j < k(q(n)) and so that T j(xr,s) /∈ D(r). This implies


that d(T j(xr,s), ∂ξr

)> 3δr and by (xii) we have d

(Tγ(r,s)+j(π(w)), T j(wr,s)

)< δr so d

(Tγ(r,s)+j(π(w)), ∂ξr

)>

2δr. Now γ(r, s) + j ≤ n and π(w) ∈ Bn

(y, δq(n)

)so d

(Tγ(r,s)+j(π(w)), Tγ(r,s)+j(y)

)< δq(n) ≤

δr. Since Tγ(r,s)+j(y) ∈ ξ(Tγ(r,s)+j(y)) and d(

Tγ(r,s)+j(y), ∂ξr(Tγ(r,s)+j(y)))

> 2δr, it follows that

Tγ(r,s)+j(π(w)) ∈ ξ(Tγ(r,s)+j(y)) and d(

Tγ(r,s)+j(π(w)), ∂ξr(Tγ(r,s)+j(y)))

> δr ≥ δr. By (xii) we have

d(

Tγ(r,s)+j(π(w)), T j(wr,s))< δr, so indeed

EAj (r, s) = ξ

(T j(wr,s)

)= ξ(Tγ(r,s)+j(y)).


Lemma 6.39. Take y = π(x) with x ∈W. For all n ∈N,

ν π−1(

Bn

(y, δq(n)

))< exp(−nh + o(n)).

Proof. Combine Lemmas 6.37 and 6.38.

We define a measure µ on Md2×d1(R) by µ := ν χ−1. Also, given t ∈ Md2×d1(R) and r > 0 we define

B(t, r) :=

s ∈ Md2×d1(R) : ||s− t||∞ < r

.

Lemma 6.40. Take t = χ(x) with x ∈W. For all n ∈N,

µ(

B(

t, δq(n)c−10 e−λn

))< exp(−nh + o(n)).

Proof. By Lemma 6.39 it suffices to show that whenever t′ = χ(x′) ∈ Md2×d1(R) for some x′ ∈ W satisfies

||t′ − t||∞ < δq(n)c−10 e−λn then y′ = π(x′) ∈ X satisfies d(Tl(y), Tl(y′)) < δq(n) for all l ≤ n.

So take t′ ∈ Md2×d1(R) with ||t′ − t||∞ < δq(n)c−10 e−λn. It follows that for any l ≤ n we have

||a−lu(t)−1u(t′)al − e||∞ = ||a−lu(t′ − t)al − e||∞

= ||a−l (u(t′ − t)− e)

al ||∞

= ||t′ − t||∞ · elλ

< δq(n)c−10 e−λn · elλ ≤ δq(n)c

−10 .

Since δq(n)c−10 < δ0 < η0 it follows that

dG(a−lu(t)−1u(t′)al , e) ≤ c0||a−lu(t)−1u(t)al − e||∞ < δq(n).

Now take g ∈ G so that z = Γg. We have y = Γg · u(t) and y′ = Γg · u(t′). Hence, for each l ≤ n we have

d(Tl(y′), Tl(y)) = d(

Γg · u(t′)al , Γg · u(t)al)

≤ dG

(g · u(t′)al , g · u(t)al

)= dG

(a−lu(t)−1u(t′)al , e

)< δq(n).



Lemma 6.41. Take t = χ(x) with x ∈W. Then

limr→0

log µ (B (t, r))log r

≥ hλ

.

Proof. Given r > 0 we choose define nr ∈N by

nr := min

n ∈N : δq(n)c−10 e−λn < r

.

Then r ≤ δq(nr−1)c−10 e−λ(nr−1) so B(t, r) ⊆ B

(t, δq(nr−1)c

−10 e−λ(nr−1)

). Thus, by Lemma 6.40

− log µ (B(t, r)) ≥ (nr − 1)h− o(nr).

Moreover, by the defitions of q(n), τ(q) and φ(q) we have

nr > φ(q(nr)− 1) > τ(q(nr)− 1) ≥ −(q(nr)− 1) log δq(nr).

Now δq(nr)c−10 e−λnr < r so

− log r < λnr + log c0 − log δq(nr)

< nr(λ + (q(nr)− 1)−1 + n−1r log c0).

Hence,

log µ (B (t, r))log r

≥ (nr − 1)h− o(nr)

nr(λ + (q(nr)− 1)−1 + n−1r log c0)

.

Letting r → 0 so nr → ∞ completes the proof of the lemma.

By Proposition 2.3 this completes the proof of Proposition 6.27 and in turn Proposition 6.26.

Recall that in Section 6.6.2 we showed that Proposition 6.26 implies the lower bounds in both Theorem 6.2

and Theorem 6.3. The proof of the upper bounds for both theorems was given in Section 6.5 and the proof of

the regularity properties of the spectrum in Theorem 6.2 follow from Proposition 6.19.


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