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The C1 Stability Conjecture*
and
Structural Stability of C1 Diffeomorphisms
*Commentary on the proof of C1 Stability Conjecture by Ricardo Mañe, first published in
Publications Mathématiques de l’Institut des Hautes Études Scientifiques 66 (1988)
MSci Project Advisor: Prof. Jeroen Lamb
CRISTINA SARGENT (CID 00424479)
Mathematics Department, Imperial College, London
17 June 2008
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CONTENTS
1 Introduction and overview of the content ........... ...................................................................3
2 Structural Stability ............................... .....................................................................................5
2.1 Motivation....................................................................................................................................... 5
2.2 What is structural stability?.......................................................................................................... 8
2.3 The C1 Stability Conjecture ....................................................................................................... 13
3 The Proof of the C1 Stability Conjecture........... ...................................................................15
3.1 The space / problem setting:..................................................................................................... 16
3.2 Proof – Outline ............................................................................................................................ 19
3.3 The contradiction ........................................................................................................................ 24
4 Main theorems – Outline of proofs .................. .....................................................................31
4.1 Outline of proof Theorem I.1 ............................................................................................31
4.2 Outline of proof of Theorem I.2. .......................................................................................32
4.3 Outline of proof of Theorem I.3 ........................................................................................37
4.4 Outline of proof of Lemma I.5...........................................................................................44
4.5 Comment on proof of Theorem I.7 ...................................................................................45
5 Examples of Axiom A Diffeomorphisms ................ ..............................................................47
5.1 Hyperbolic sets............................................................................................................................ 47
5.1.1 Smale’s Horseshoe ..........................................................................................................48
6 Some key concepts and theorems..................... ...................................................................56
6.1 Stable and unstable manifolds for points of a hyperbolic invariant set............................... 56
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1 Introduction and overview of the content
The subject of this project is the proof of the C1 Stability Conjecture as given by Ricardo
Mañe, first published in 1988. Although Mañe’s proof established it as a theorem its
statement became famous under the name of C1 Stability Conjecture, hence I kept the term
conjecture rather than theorem.
Chapter Two starts with a general presentation of the concept of structural stability and an
overview of the formal mathematical language developed to describe it. I introduce the C1
Stability Conjecture and its important role in unifying many results that attempted to
describe and classify dynamical systems.
The proof of the C1 Stability Conjecture is an elaborate and complex construction based on
six main theorems and four supporting lemmas. The objective of my work was first to
collect and familiarise myself with all the key concepts and ingredients necessary for
understanding the proof. Mañe’s proof uses results developed by other famous
mathematicians who made sustained efforts towards solving it. Some of the main papers
that supported Mañe’s proof and had been essential to prepare this project are:
R. Mañe – An ergodic closing lemma [18]
S. Smale Differentiable dynamical systems[3]
J. Frank – Necessary conditions for stability of diffeomorphisms [14]
Chapter Three contains an outline of the proof where I present for each theorem and
lemma, the scope and the rationale behind introducing those particular results. It also
identifies the steps of the proof and how the requirement of the proof is being reduced from
one statement to another. I have also enclosed a flowchart of the proof, to help the reader
follow the argument and overall strategy.
In Chapter Four I present and discuss the proofs for a selection of the supporting theorems.
I have omitted the proof of some theorems as their length and level of technicality is
beyond the scope of this project.
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Chapter Five follows with a detailed presentation of a prototypical structurally stable
diffeomorphism, one the most famous and useful for understanding the dynamic behaviour
of such systems, the Smale horseshoe.
The project ends with a presentation of some key concepts and theorems which play an
essential role in the understanding of behaviour of dynamical systems, for example the
Stable Manifold Theorem.
I would like express my gratitude and give special thanks to my advisor, Professor Jeroen
Lamb who patiently, over countless hours, helped me unravel and understand this
wonderful piece of mathematics.
I would also like to thank Dr. Stefano Luzzatto who kindly provided me with the paper of
Enrique Pujals and Martin Sambarino.
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2 Structural Stability
2.1 Motivation The notion of structural stability is something that preoccupied natural philosophers and
later scientists alike. For example the question of structural stability of the solar system
appears in Newton’s Principia Mathematica where he noted that the solar system is not
perfectly stable - he actually believed that God must intervene at various intervals to re-set
the orbits of the planets to preserve the structure of the universe. 1
Later Pierre-Simon Laplace tried to solve the problem of the solar system’s stability by
making some simplifying assumptions about the nature of the gravitational interactions of
the planets. Laplace showed that this simplified system was integrable and that there were
periodicities in the movements of the planets – he thought that he found an analytical
solution to this problem. However, exactly the terms that he had neglected were those
which could provide the sources of chaos.
At the end of the 19th century Henri Poincaré (April 29, 1854 – July 17, 1912) tackled the
problem of orbits of three celestial bodies experiencing mutual gravitational attraction in
depth and he proved that a simple, general solution does not exist. Poincaré’s was the first
to appreciate the complicated behaviour that could result from the gravitational interaction
of just three bodies. Poincaré was the first to appreciate the true source of the problem: the
difficulty lay not necessarily in formulating the equations, but rather in specifying the
initial conditions.
By considering the behaviour or orbits arising from sets of initial points (rather than
focusing on individual orbits) Poincaré was able to show that very complicated (now called
chaotic) orbits were possible.
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At the beginning of this century, in his essay Science and Method, Poincaré wrote:
“A very small cause which escapes our notice determines a considerable
effect that we cannot fail to see, and then we say that that effect is due to chance..
[….]; it may happen that small differences in the initial conditions produce very
great ones in the final phenomena. A small error in the former will produce an
enormous error in the latter. Prediction becomes impossible, and we have the
fortuitous phenomenon”.
Poincaré's discovery of sensitive dependence on initial conditions in what are now termed
chaotic dynamical systems shifted the emphasis from a local solution - knowing the exact
motion of an individual trajectory - to a global solution - knowing the qualitative behaviour
of all possible trajectories for a given class of systems.
Dynamical systems theory and nonlinear dynamics grew out of the qualitative study of
differential equations, which in turn began as an attempt to understand and predict the
motions that surround us. The most successful class of mathematical forms for describing
natural phenomena so far are differential equations. All the major theories of physics are
stated in terms of differential equations. The same phenomena and problems of the
qualitative theory of ordinary differential equations are present in their simplest form in the
diffeomorphism problem2. Birkhoff, at the beginning of the 20th century, adopted and
furthered Poincaré's point of view, and realised the importance of the study of
diffeomorphisms of surfaces as a means of understanding the more difficult dynamics
arising from differential equations 3.
The 20th century saw the emergence and consolidation of the field of dynamical systems as
a distinct branch of mathematics, with the important work of Adronov and Pontryagin in
the 1920’s , M.L. Cartwright and J.E Littlewood in the 1940’s, and in 1960’s S. Smale and
Moser in US, Peixoto in Brasil and from Russian mathematicians, notably A.N.
Kolmogorov and his co-workers.
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Furthermore, the notion of structural stability has become such a well established concept
that aptly illustrates the interplay between the concepts of stability and a certain structure,
and essentialy characterises the phase portrait, subject to topological equivalence. Besides,
the very concepts of "structure" and "stability" are fundamental for so many human
endeavors, inside and outside mathematics, being widely used in numerous fields that
include civil engineering, soil science, thermodynamics, human physiology, electrical
engineering, cosmology, and psychology, to name just a few.
Page 8 of 61
2.2 What is structural stability? The study of dynamical systems was accompanied by continuous efforts to categorise and
classify the systems of interest and their evolution. The strategy was to find formal
equivalence relations on Diff(M) as these relations are expected to preserve the orbit
structures in some sense. Associated to each equivalence relation is the notion of stability.
More precisely if the equivalence relation on
Diff(M) is called E, ( )f Diff M∈ is called E-
stable if there is a neighbourhood N(f) of f in Diff
(M) such that if ( )g Diff M∈ (i.e. g approximates
f sufficiently) then f and g are in the same E
equivalence class.
A reasonable question is whether there is a dense
open set ( )U Diff M⊂ such that the equivalence
classes could be distinguished by numerical and algebraic invariants and that the E-stable
diffeomorphisms are dense in Diff(M)4.
To achieve a certain degree of classification, the
notion of conjugacy is employed. The notion of
conjugacy associated to topological conjugacy is
called structural stability.
Structural stability - robustness of the system as a whole, i.e. the property of a dynamical
system to remain unchanged qualitatively when it is subjected to small perturbations,
although an easy concept to grasp intuitively, it needs a lot of tools from analysis and
topology to make the statement precise. Mathematically, the concept of structural stability
of a Cr diffemorphisms on a closed manifold was introduced in the thirties by Andronov
and Pontrjagin in the context of flows on the two dimensional disk.
Definition 2.1 - A map f is a homeomorphism if it is bijective and both f and f-1 are continuous. Definition 2.2 A differentiable map is Ck if derivatives up to order k exist and are continuous. Definition 2.3 f is a Ck diffeomorphisms if it bijective and f and f-1 are Ck.
Definition 2.4. Two continuous maps, f1: M→M and f2: N→N are topologically equivalent (conjugate) if there is a homeomorphism h: M →N such that h-1 f2 h = f1.
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An influential article by Smale [3] in 1967 set out the framework for the modern global
qualitative theory of dynamical systems which has underpinned much of the progress in
understanding nonlinear dynamics over the next forty years. It drew attention to the
important role played in the dynamics of flows or iterated maps by the nonwandering set
Ω (the set of points all of whose neighbourhoods eventually return to themselves – see
exact definition in the next section) and to the significance of uniform hyperbolic structure
of exponential contraction / expansion on Ω . This structure provides a way into
addressing the question of structural stability.
The precise mathematical statement for structural stability is:
Definition 2.5. Cr diffeomorphism f on a closed manifold M is structurally stable if it has
a Cr neighbourhood U such that everyg U∈ is topologically equivalent to f.
In particular if ( )xξ is a smooth function of x, then there exists some positive number 0ε
such that the perturbed map ( ) ( )f x xεξ+ can be transformed to the original map ( )f x by
a one to one change of variables for all ε satisfying 0ε ε< . In particular in the range
0ε ε< the perturbed map and the original map have the same number of periodic orbits
for any period and have the same symbolic dynamics. The non-stability is associated with
the apparition of bifurcations, which can be viewed as the topological transition between
stable structures.
For some time it was thought that structurally stable diffeomorphisms might be dense in
Diff(M), i.e. the set of all diffeomorphisms f defined on manifold M. Smale showed5 that
structurally stable systems are not dense in the space of all systems on a given manifold.
However, for every manifold M structurally stable diffeomorphisms form non-empty open
subsets of Diff(M)[26].
In the context of flows, Diff(M) is replaced by, the set of smooth vector fields on M, ℵ say,
equipped with the C1 topology, and the flow generated by each Χ ∈ℵ is structurally stable
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if and only if for each Y in some neighborhood of X in ℵ there is a homeomorphism
that sends the orbits of X to the orbits of Y, preserving the orientation of the
orbits.
An important results concerning structural stability of flows is Peixoto’s Theorem which
establishes the conditions for structural stability of flows on 2-dimensional plane:
Theorem 2.1 (Peixoto): A smooth vector field on a 2D compact planar domain of R2 is
stable if and only if the number of critical points and closed orbits are finite and hyperbolic
and there are no integral curves connecting the saddle points.
Here we note that non-hyperbolic critical points or closed orbits are structurally unstable
because arbitrarily small perturbations can make them hyperbolic. Saddle connections can
also be broken by small perturbations as well.
Figure 2.1: Structural stability implies that the two phase portraits are homeomorphic.
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Figure 2.2: The connection between the saddle points breaks. The phase portraits are not homeomorphic.
More progress was made on the problem of
classifying dynamical when other types of
equivalence classes were considered, for
example if the relation on Diff(M) of topological
conjugacy is relaxed then stability can be
studied on Ω - the set of non-wandering points.
The corresponding stability is the Ω - stability.
Definition 2.6. A diffeomorphism f: M →M of a compact manifold is Ω -stable if
i) there is a neighbourhood N(f) of f in the C1 topology such that g∈ N(f) implies that
there is a homeomorphism h from the nonwandering set of f, Ω(f) to the nonwandering
set of g, which satisfies g h = h f and
ii) if p is a periodic point of f then dim ( , )sW p f = dim ( ( ), )sW h p g
Structural stability implies Ω - stability, and it was though that although structural is not
generic (i.e. residual), perhaps Ω - stability is. Ralph Abraham and Steve Smale
demonstrated that this is not true6.
Definition 2.5. For a map f: M → M, a
point p is called a nonwandering point provided for every neighbourhood U of p there is an integer n>0 such that fn(U) ∩ U ≠ ∅ . Thus there is a point q ∈U such with fn(q) ∈U. The set of all nonwandering points for f is called the nonwandering set and is denoted Ω(f).
Page 12 of 61
Other types of structural stability type in dynamics are (the examples below are from E.
Pujal’s website).
• Finite stability. Jorge Sotomajor proposed that a diffeomorphism f be called finitely
structurally stable if there exist a 1C neighborhood U of f and a 1C embedded n -
disc D N⊂ such that f D∈ and each g N∈ is conjugate to some member of D .
Robinson and Williams show that finite structural stability is not generic (Robinson
and Williams 1973).
• Future stability. If the decomposition of the phase space into sets of points with
mutually asymptotic orbits is stable under perturbation, the system is said to be
future stable. Mike Shub and Bob Williams show that future stability is not generic
(Shub and Williams 1969).
• Prevalence stability. Jim Yorke and others proposed an idea to speak of "almost
every" in function spaces (Hunt et al. 1992, Kaloshin 1997). Then a system is
prevalence stable if almost every element of its neighborhood is conjugate to it.
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2.3 The C1 Stability Conjecture
The definition of structural stability does not give a direct, practical way to establish
whether a diffeomorphisms or flow is stable or not. Various conjectures were proposed by
mathematicians working in this field to try to characterise the structurally stable systems.
The discussions focused a set of key conditions which included the no-cycles condition, the
Transversality condition (see definition included in the next section), hyperbolicity and the
Ω -stability.
Trying to unify Morse-Smale systems and Anosov's
globally hyperbolic systems7 Smale devised Axiom A
[3] (whose definitions are included in the next section).
If f satisfies Axiom A and in addition, the stable and
unstable manifolds of all points of Ω meet one another
transversally (the Strong Transversality Condition) then
it is an AS systems, an abbreviation that can also be
read as "Anosov-Smale." It is known that MS AS⊂
and Palis and Smale formally made the conjecture that
that a vector field is structurally stable if and only if it
satisfies the two conditions known as Axiom A and the
Strong Transversality Condition. Their sufficiency was proved in the papers of Joel
Robbin8 (for 2r ≥ ) and Robinson9 (for r=1). The question of their necessity was reduced
to prove that Cr structural stability implies Axiom A10. This problem became known as the
1C Structural Stability Conjecture and after long efforts by many mathematicians R. Mañe
in solved this problem for the discrete case in 198811 i.e. for diffeomorphisms of nM .
Definition 2.6 A Morse-Smale dynamical system has only trivial recurrence, only finitely many periodic orbits, all of which are hyperbolic, and the invariant manifolds of the periodic orbits meet transversally. Definition 2.7 A diffeomorphism is called an Anosov diffeomorphism provided f has a hyperbolic structure on all of M. It is called a toral Anosov diffeomorphism provided in
addition M is a torus, Tn.
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The corresponding issue for C2-structural stability (defined analogously) is still open.
Later S. Hu and S. Hayashi completed the picture for flows:
• In 1994 Sen Hu established the similar to the C1 stability conjecture for three-
dimensional flows.12, i.e. gave the proof that there exists a hyperbolic structure over the
Ω set for the structurally stable three-dimensional flows.
• In 1997 Hayashi S. provided the solution of the C1-stability and Ω-stability conjectures
for flows on invariant manifolds13
- thus yielding the satisfying characterization of structural stability in all dimensions: SS =
AS.
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3 The Proof of the C1 Stability Conjecture In his remarkable paper of 1988 in Mañe proved that 1C structural stability implies Axiom
A. Mañe used, innovatively, results and techniques from ergodic and measure theory that
helped him to translate the dynamics of orbits into some more quantitative result.
The proof is a complex construction based on six main theorems and four lemmas.
Mañe systematically reduces the requirements of Axiom A, in particular the uniform
hyperbolicity of the set Ω (f), to a condition on the measure of the closure of sets of
periodic points, chosen in a specific way. He then uses the induction method to prove that
the required condition holds for all the sets of periodic points (their closure) and finally
derives a contradiction that concludes the proof.
As the proof is very elaborate, I will first present an outline of each theorem and lemma,
the scope and the rationale behind introducing those particular results and how they are
used for the purpose of the proof. I also highlight the steps of the proof and how the
requirement of the proof is being reduced from one statement to another. I have also
enclosed at the end of this chapter a flowchart of the proof, which contains minimum
details to help the reader follow the argument but not so much as to detract from
understanding the overall strategy.
Some of the proofs are too lengthy and technical for the scope of this project, therefore I
present and discuss only a selection of them in the next section.
Supporting concepts and definitions are introduced where necessary; however some of the
key concepts and theorems that are relevant for this proof but require more space and
detail, for example The Stable Manifold Theorem, are included in a following chapter.
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3.1 The space / problem setting:
Define ( )rF M as the set of diffeomorphisms f : M → M having a Cr neighbourhood U
such that all the periodic points of every g∈U are hyperbolic. Cr stable or Ω -stable
diffeomorphism belong to FFFF r(M)14 . Most of the steps in Mañe’s proof use only the weaker
assumption that such diffeomorphisms belong to FFFF r(M).
M is a closed manifold, f : M → M is a diffeomorphism in ( )rF M ) defined as above, Per(f)
the set of periodic points of f, and if ( )x Per f∈ let ( )sE x and ( )uE x be the stable and
unstable subspaces of x in xT M i.e. the subspaces associated with the eigenvalues of
:nx xDf T M T M→ where n is the period of x, that have respectively modulus< 1 and >1.
Denote ( )Per f the closure of Per(f).
C1 Stability Conjecture. Every 1C structurally stable diffeomorphism of a closed manifold satisfies Axiom A.
So what are the properties satisfied by f in virtue of
the fact that it is structurally stable? Nowadays we
are aware of many such results, however at the time
of Mañe proving that structural stability implies
Axiom A, some of the key results that were certain to
be true are as follows:
If diffeomorphism f: M→ M is structurally stable
then:
a. f structurally stable implies Ω - stable [18].
b. All periodic points of f: M→ M are hyperbolic and they form a hyperbolic set [18].– see
definition of hyperbolic set below.
Definition 3.1 A basic set Λ of f ∈ FFFF 1(M) is a hyperbolic set if
is transitive (i.e. there exists x∈ Λ such that its orbit is dense in Λ ) and isolated, i.e. it has a compact neighbourhood U satisfying: ( )n
n
f U = Λ∩
Page 17 of 61
c. If basic sets 1Λ and 2Λ are attached then
1 2dim ( ) dim ( )s sE EΛ = Λ [14].
d. If 1p∈ Λ and 2q∈ Λ then ( )sW p and ( )uW q
always intersect transversely. This condition
means that whenever ( ) ( )s ux W p W q∈ ∩ :
( ) ( )s ux x xT W x T W x T M+ = (3.1)
e. Moreover there is constant 0α > such that for any ( )sW p and 2q∈ Λ and any point of
intersection of ( )sW p and ( )uW q the angle between ( )sW p and ( )uW q is greater than
α .[14]
One of the most important characteristic of transversality is that it persists under small
perturbations whereas tangencies can be easily destroyed by small perturbations hence
altering qualitative features of the phase space.
f. If f: M→ M is Ω - stable and p and q are periodic points with their orbits attached then
dim ( ) dim ( )s uW p W q= . Also (Orbit ) and (Orbit )s uW p W q always intersect
transversely.[14]
There are obviously more implications of structural stability, the above are only a selection
of such consequences of structural stability which will play a key role in the proof of the C1
Stability conjecture. An Axiom A diffeomorphism that satisfies the transverse intersection condition above for
every x M∈ is said to satisfy the Strong Transversality Condition (AS system).
So what are the requirements for a structurally stable diffeomorphism to be an Axiom A diffeomorphism? Axiom A. A diffeomorphism f: M→ M satisfies Axiom A provided: i) Ω (f) is a hyperbolic set ii) The set of periodic points of f, Per(f) is dense in Ω (f), i.e. ( )Per f = Ω (f). See overleaf for the definition of uniform hyperbolicity.
Definition 3.2. Two basic sets 1Λ
and 2Λ are attached if there are
periodic points 1, 1 1p q ∈ Λ and
2, 2 2p q ∈ Λ such that
1 2( ) ( ) 0s uW p W p ≠∩ and
1 2( ) ( ) 0u sW q W q ≠∩
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Definition 3.3. A compact invariant set Λ has a uniform hyperbolic structure for a diffeomorphism f on M provided: i) at each point p in Λ the tangent space to M splits as the direct sum of the
eigenspaces of p, upE and
spE , TpM =
upE ⊕ s
pE
ii) The splitting is invariant under the action of the derivative of the map f in the sense
that Dfp(upE ) =
upE and Dfp (
spE ) =
spE
iii) There exist 0 < λ < 1 and C≥ 1 independent of p such that for all n ≥ 0,
C for n n sp pDf v v v Eλ≤ ∈ (3.2)
and
C for v En n up pDf v vλ− ≤ ∈ (3.3)
Remark 2.2 If m is a positive integers such that ρ = C mλ < 1, then
for m sp pDf v v v Eρ≤ ∈ (3.4)
for m up pDf v v v Eρ− ≤ ∈ (3.5)
so D m
pf | spE is a contraction and D m
pf − | upE is an expansion. Thus the constant C
determines how many iterates of f are necessary before the vectors get contracted,
(respectively expanded) in the subbundle spE (respectively
upE ).
So, in summary, the aim is to prove that:
AI . C1 Structural Stability
AII. Axiom A
A C1 diffeomorphism f on a closed manifold M is structurally stable if it has a C1 neighbourhood U such that everyg U∈ is topologically equivalent to f.
i. ( )fΩ is a uniformly hyperbolic set i.e. it is:
- Compact - Invariant - There is continuous splitting TxM =
( )uE x ⊕ ( )sE x for all ( )x f∈Ω ,
this splitting is invariant with respect to nDf . Furthermore, under this linear
map uE expands uniformly and sE contracts uniformly
ii. ( )Per f = ( )fΩ
Page 19 of 61
3.2 Proof – Outline The first theorem is due to Mañe, published in 197815 and establishes that part (ii) of AII above is satisfied by any f ∈ F 1(M): Theorem I.1 If f ∈ F 1(M) then ( )Per f = ( )fΩ . So the rest of the proof is concerned with proving that:
1. ( )Per f is a hyperbolic set. Let Pi(f) be the set of periodic points such that dim( )sE x = i. By Theorem I.1 we have that
when f ∈ FFFF 1(M):
( )fΩ = dim
0( )
M
ii
P f=U (3.6)
Then if f ∈ FFFF 1(M) it is sufficient to show that dim
0( )
M
ii
P f=U is hyperbolic, that is:
2. ( )iP f is hyperbolic for each 0 dimi M≤ ≤ .
The next theorem says that f ∈ FFFF 1(M) implies that the sets of sinks and sources (i.e. periodic points whose stable manifolds have respectively dimensions dim M and 0) are finite.
Theorem I.2 If f ∈ F 1(M) then for the cases i=0 and i=dim M the sets Pi(f) are finite This conjecture was first advanced by Smale16 and its proof is due to Pliss17 . The actual
proof is quite technical but I will present an outline in the next section.
As M is a compact manifold, a finite subset is a closed set, hence 0 0( ) ( )P f P f= and
dim dim( ) ( )M MP f P f= . 0( )P f and
dim ( )MP f consist of hyperbolic periodic points, hence
they are hyperbolic sets.
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Note1.2.1: From above, if dim( ) 2M = , ( )iP f are hyperbolic hence f satisfies the
conditions of Axiom A. The proof now is reduced to show that:
3. ( )iP f is hyperbolic for 1 dimi M≤ < .
To prove the uniform hyperbolicity of the sets ( )iP f for 1 dimi M≤ < Mañe starts with the
splitting TxM = ( )uE x ⊕ ( )sE x that applies when ( )ix P x∈ and shows that this splitting
TM | Per(f) extents to a continuous invariant splitting TM | ( )Per f , hence satisfying the condition of hyperbolicity. :
4. Induction method: If f ∈ F F F F 1(M), then 0( )P f is hyperbolic as per previous statement.
Suppose ( )iP f is hyperbolic for all i such that 0 i≤ ≤ j . The requirement is to show that
1( )jP f+ is also hyperbolic
To complete the induction step, Mañe assumes 1( )jP f+ is not hyperbolic. Using properties
that are direct consequences of the fact thatf ∈ FFFF 1(M) and the structural stability of f he
derives a contradiction.
The periodic points are hyperbolic so for each periodic point ( )ix P f∈ there is a natural
hyperbolic decomposition of the tangent bundle over the periodic orbit in two
complementary directions s uE E⊕ . So the question is whether the tangent bundle
decomposition over the periodic points can be extended to the closure of these sets, which
by Theorem I.1 means that it would extend to the non-wandering set.
The extension of splitting of the tangent manifold to ( )Per f is proved using the concept of
dominated splitting which is a decomposition of the tangent bundle TM, with respect to
invariant sets, into two invariant subbundles. Dominated splitting is a weaker form of
hyperbolic splitting. Formal definition and details with respect to dominated splitting are
included in the next section. With the next theorem Mañe establishes that iff ∈FFFF1(M)
then TM | ( )iP f has the dominated splitting property.
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Theorem I.3 If f ∈ F F F F 1(M) there exist constants C > 0 and 0< λ < 1, m>0 and a C1
neighbourhood U of f such for all g∈U and 0 < i < dim M there exists a dominated
splitting TM | ( )iP f = s ui iE E⊕ satisfying:
a) / ( ) . / ( ( )) for all ( )s um m mi i iDf E x Df E f x x P fλ− < ∈ (3.7)
b) s si iE E= and
u ui iE E= when ( )x Per f∈
c) If ( )ix P g∈ and has period n m> then
[ / ] 1
[ / ]
0
/ ( ( ))n m
m s mj n m
j
Dg E g x Cλ−
=
≤∏ and [ / ] 1
[ / ]
0
/ ( ( ))n m
m u mj n m
j
Dg E g x Cλ−
−
=
≤∏
d) For all ( )ix P g∈
1
0
1log / ( ( )) loglim
nm s mj
n j
Dg E g xn
λ−
→+∞ =
≤∑
1
0
1log / ( ( )) loglim
nm u mj
n j
Dg E g xn
λ−
−
→+∞ =
≤∑
This theorem was independently proved in papers of V.A. Pliss, S.T. Liao and R. Mañe18.
What we should note from theorem I.3 is, first of all, that all the hyperbolic periodic point
have the property of dominated splitting and secondly that for an f-invariant set exhibiting
dominated splitting of index i (i.e. the dimension of the subspace s
E of the tangent bundle),
its closure also exhibits a dominated splitting of index i, i.e. we established that the
property of dominated splitting applies to all points ( )ix P f∈ , i.e. including the non-
periodic points on the boundary of the set.
The main idea is that if the angles of the eigenspaces of hyperbolic periodic points are
bounded away from zero in a robust way then it guarantees the existence of dominated
splitting.
The aim is now to show that this dominated splitting is in fact exactly a uniform hyperbolic
splitting, so the argument of the proof is reduced to:
Page 22 of 61
5. The splitting TM | ( )iP f = s ui iE E⊕ is hyperbolic for all 1 i≤ < dim M .
Next theorem, still assuming that f ∈ F F F F 1 , simplifies the requirements for uniform
hyperbolicity to showing that Df contracts the subbundle siE .
Theorem I.4 If f ∈ F F F F 1(M), 0< i < dim M. and siE is contracting then
uiE is expanding.
This theorem is due to Mañe and it is proved in the C1
Stability conjecture paper (1988) [11]. Now the problem is reduced to show that if f ∈ FFFF1(M) then:
6. siE is contracting for all 0< i < dim M
So far we have the property of dominated splitting, which gives results with of a ‘quantitative’ nature with respect to the dynamics of periodic orbits so Mañe uses very ingeniously elements of ergodic theory to establish a connection between the contracting /
expanding behaviour of tangent subspaces siE and the measure of the sets of periodic
points. Lemma I.5 Let Λ be a compact invariant subset of 1( )f Diff M∈ and E TM⊂ Λ a
continuous invariant subbundle and if there exists m > 0 such that
log /mDf E dµ < 0∫ (3.8)
for every ergodic µ ∈ M M M M(f m | Λ ) then E is contracting. This theorem is due to Mañe [11]–the proof is presented in the next section.
Note: In the context of our proof, the above reads ( )iP f instead of Λ and siE for E.
Definition 3.3 A subbundle
/E TM⊂ Λ is contracting if it is
continuous, invariant and there
exist 0C < and 0 1λ< < such
that / ( )n nDf E x Cλ≤
Page 23 of 61
So, the claim is that if f ∈ FFFF 1(M), then 0( )P f is hyperbolic. Suppose( )iP f is hyperbolic
for all i such that 0 i≤ ≤ j . Then the aim is to show that 1( )jP f+ is also hyperbolic and this
has been reduced to prove that the above property (3.8) holds for every ergodic µ ∈ MMMM(f m
| 1( )jP f+ ), i.e. the next step in the argument is that:
7. 1log m sjD f dµ+
Λ∫ Ê < 0 for every ergodic µ ∈ M M M M(f m | 1( )jP f+ )
Using again elements of measure and ergodic theory Mañe formulates theorem I.6 which has the most interesting and unexpected statement, a testimony to what a brilliant and creative mathematician he was. Theorem I.6 will give the condition sufficient for proving
the hyperbolicity of 1( )jP f+ :
Theorem I.6 If f ∈ F F F F 1(M) and 0m> is as given by Theorem I.3 there exists 0 < 0λ < 1
such that if ( )iP f is hyperbolic for all 0 i≤ <j and µ ∈ M M M M (f m | ( )jP f ) satisfies
0log / log
smiDf E dµ λ≥∫ (3.9)
then
0
( ( )) 0i
i j
P fµ≤ <
>∪ (3.10)
This theorem is due to Mañe and it is proved in [11] To complete the induction step, it is necessary to shows that:
8. For all µ ∈ M M M M (f m | 1( )jP f+ we have that
0
( ( )) 0ii j
P fµ≤ ≤
=∪ (3.11)
This would imply that there are no measures µ ∈ MMMM (f m | 1( )jP f+ satisfying I.6.(i). Hence
0log / log 0
smiDf E dµ λ< <∫ (3.12)
for all µ ∈ MMMM (f m | 1( )jP f+ and then step 6 is completed, i.e. siE is contracting and then
step 5 is completed, i.e. 1( )jP f+ is hyperbolic, thus completing the induction step and also
the proof of Axiom A property for f.
Page 24 of 61
3.3 The contradiction From this point onwards Mañe sets up the context which he will use to derive two
contradictory results. One result rests on the assumption of structural stability of f and the
other on the assumption that there exists 10 ( / ( ))mjM f P fµ +∈ such that 0
0
( ( )) 0ii j
P fµ≤ ≤
>∪
A basic set Λ of f ∈ FFFF 1(M) is a hyperbolic set if is transitive (i.e. there exists x∈ Λ such that its orbit is dense in Λ ) and isolated, i.e. it has a compact neighbourhood U satisfying: ( )n
n
f U = Λ∩ (3.13)
Theorem I.7 Let Λ be a compact invariant set of ( )f Diff M∈ such that ( )fΩ Λ = Λ and having a dominated splitting TM E FΛ = ⊕ . Suppose there exist basic sets 1 2,...., sΛ Λ Λ of f and constants 0m> , 0c > and 0 1λ< < satisfying: I) Ind( ) dim ( )i E xΛ < for all 1 i s≤ ≤ and x∈ Λ II) There exists a 1C neighbourhood U of f such that if g U∈ coincides with f in a
neighbourhood of 1
s
kΛ∪ then
( ) ( )s ug i g i iW WΛ Λ = Λ∩
for all 1 i s≤ ≤ .
III) If µ ∈ M M M M(f m | Λ ) satisfies
log /mDf E d cµ ≥ −∫ (3.14)
then
1
( ) 0s
kµ Λ >∪ (3.15)
IV) / ( ) . / ( ( )) for all s um m mi iDf E x Df E f x xλ− < ∈ Λ
Then, given 1 i s≤ ≤ such that iΛ − Λ is not closed, there exists 1( )g Diff M∈ arbitrarily near f, coinciding with f in a neighbourhood of
1
s
kΛ∪ and 1 r s≤ ≤ , i r≠ such that rΛ − Λ is not closed and ( ) ( )s u
g i g rW WΛ Λ ≠ ∅∩ (3.16)
This theorem is due to Mañe and it is proved in the C1 Stability conjecture paper (1988)- [11]
Page 25 of 61
Suppose there exists 10 ( / ( ))m
jM f P fµ +∈ which does not satisfy (3.11) i.e.
(3.17) 00
( ( )) 0ii j
P fµ≤ ≤
>∪
From Smale’s Spectral Decomposition
Theorem, the hyperbolic set 0
( )ii j
P f≤ ≤∪ can be
uniquely decomposed into basic disjoint sets:
10
( ) ...i si j
P f≤ ≤
= Λ Λ∪ ∪∪
The transitivity of the basic sets implies that the dimension of the fibers of the stable
subspace of the hyperbolic splitting / iTM Λ is constant and it is called index of iΛ
(denoted Ind( iΛ )). This observation will be important in the course of deriving the
contradiction. The stable and unstable sets of Λ are defined by:
( ) lim ( ( ), ) 0s nf
nW y d f y
→∞Λ = Λ = (3.18)
( ) lim ( ( ), ) 0u nf
nW y d f y−
→∞Λ = Λ = (3.19)
The next two results ensure that there exist sets iΛ such that 1( )j iP f+ − Λ is not closed
which is a requirement of theorem I.7 above. Otherwise we would get a contradiction with
our assumption that there exists 10 ( / ( ))mjM f P fµ +∈ that satisfies 0
0
( ( )) 0ii j
P fµ≤ ≤
>∪
and
contradicting the existence of 0µ with such property would in fact prove Axiom A.
Lemma I.9 If 1( )j iP f+ Λ ≠ ∅∩ then 1( )j iP f+ − Λ is not closed.
Still assuming that (3.17) holds we have that:
Corollary I.10.
There exist values of i such that 1( )j iP f+ − Λ is not closed.
Theorem 2.7 -Spectral Decomposition of Diffeomorphisms Suppose f : M
→ M is a C1 diffeomorphism on compact manifold M. Then if f satisfies Axiom A there is a unique way of writing Ω (f) as a finite union of disjoint, closed, invariant indecomposable subsets on each of which f is topologically transitive.
1 2 kΩ = Ω Ω Ω U . . . . .U
Page 26 of 61
If 1( )j iP f+ − Λ is closed for all 1 i s≤ ≤ then by Lemma I.9 the intersections 1( )j iP f+ Λ∩
are empty for all 1 i s≤ ≤ . But then
0 00 1
( ( )) ( ) 0s
i ii j
P fµ µ≤ ≤
= Λ =∪ ∪ (3.20)
because the support of 0µ is contained in 1( )jP f+ thus contradicting (3.17).
Now with the set 1( )jP f+Λ = , the dominated splitting 1 11/ ( ) j j
s ujTM P f E E+ ++ = ⊕ , the
basic sets 1 2,...., sΛ Λ Λ ,
0m> , 0< λ <1 as given by theorem I.3 and 0logc λ= − given by
theorem I.6 all the hypotheses of theorem I.7 are satisfied.
Therefore we take iΛ such that 1( )j iP f+ − Λ is not closed then conform I.7 there exists
rΛ such that 1( )j rP f+ − Λ is not closed, i r≠ and there is diffeomorphism g arbitrarily
1C close to f coinciding with f in a neighbourhood of 1
s
kΛ∪ such that
( ) ( )s ug i g rW WΛ Λ ≠ ∅∩ (3.21)
Up to this point, we have used only the assumption that f ∈ F 1(M). In the remainder of
the argument Mañe will use structural stability and its properties.
Let t be the minimum of the indexes of the sets kΛ such that 1( )j kP f+ − Λ is not closed.
Take iΛ such that ( )iInd tΛ = and 1( )j iP f+ − Λ is not closed and there do not exist sets kΛ
with k i≠ such that 1( )j kP f+ − Λ is not closed and
( ) ( )s uf i f rW WΛ Λ ≠ ∅∩ (3.22)
Such a iΛ exists because otherwise there is a family of different basic sets 1,...
pi iΛ Λ such
that their indexes are all t and
1( ) ( )
n n
s uf i f iW W
+Λ Λ ≠ ∅∩ (3.23)
for 1 n p≤ ≤ and
Page 27 of 61
1
( ) ( )p
s uf i f iW WΛ Λ ≠ ∅∩ (3.24)
i.e. we have a circular intersection, or a cycle. All these intersections are transversal
because of the structural stability of f . Let 1 2
( ) ( )s uf i f iz W W∈ Λ Λ∩ . Here Mañe makes the
comment that it is ‘well known’ that ( )tz P f∈ . The explanation for this statement is
based on the robustness of the condition of transversality and the fact that periodic
solutions lay dense in a cycle. If the stable and unstable manifold intersect transversally
once then they must intersect infinitely many times (heteroclinic tangle – an explanation of
homoclinic tangle, which is the result of the intersection of the stable and unstable manifold
of the same periodic point is included in the next chapter – the principle of why this leads
to an infinite number of intersections is similar). Hence we can construct periodic solutions
in a sufficiently small neighbourhood because we can easily find a sequence of periodic
solutions that accumulates to z and z would belong to the closure of a set of periodic
points, in this case with index t hence it must be that ( )tz P f∈ .
So if we have that ( )tz P f∈ then it belongs to some set qΛ and the orbit is contained in
qΛ as qΛ it is a basic set, therefore invariant. This implies that qΛ intersects 1i
Λ and
2iΛ thus implying that 1 2q i i= = . This contradicts the fact that all the sets 1 2,...., sΛ Λ Λ
are
completely different hence there exists iΛ such that ( )iInd tΛ = and 1( )j iP f+ − Λ is not
closed and there do not exist sets kΛ with k i≠ such that 1( )j kP f+ − Λ is not closed and
( ) ( )s u
f i f rW WΛ Λ ≠ ∅∩
Define ( )T f as the set of pairs ( , )n q such that n q≠ , ( ) ( )n qInd Ind tΛ = Λ = and
( ) ( )s qf n f qW WΛ Λ ≠ ∅∩
( )T f = ( , )n q | n q≠ , ( ) ( )n qInd Ind tΛ = Λ = and ( ) ( )s q
f n f qW WΛ Λ ≠ ∅∩ (3.25)
By above we established that there are no elements in T(f) with i in the first entry, i.e.
( , ) ( )i r T f∉ , ,0 dimr r M∀ ≤ ≤ Now we follow the conclusion of Theorem I.7 i.e. take g U∈ coinciding with f in a
neighbourhood of 1
s
kΛ∪ with r i≠ satisfying
( ) ( )s ug i g rW WΛ Λ ≠ ∅∩ (3.26)
Page 28 of 61
and such that 1( )j rP f+ − Λ is not closed. The intersection in (3.26) is transversal because
the structural stability of f implies structural stability for g, then ( ) ( )i rInd Ind tΛ ≥ Λ = .
But since ( )iInd tΛ = , t being the minimum of indexes such the 1j iP + − Λ is not closed it
follows that ( ) ( )r iInd Ind tΛ = Λ = . We can assume without loss of generality that g is
topologically conjugate to f. Let :h M M→ be a homeomorphism such that gh hf= .
Conjugacy preserves periodic points therefore ( ( )) ( )i ih P f P g= for all 0 dimi M≤ ≤ . Hence:
1 1 1
( ) ( ( )) ( )s
k kkk j k j
h h P f P g≤ ≤ ≤ ≤
Λ = =∪ ∪ ∪ (3.27)
Besides this theorem, the following lemmas are needed. If g ∈ FFFF 1(M) denote by N(i, n, g) the number of periodic points of gn contained in Pi(g). Lemma I.8. If f ∈ F F F F 1(M) there exists a C1 neighbourhood U of f such that:
i. N(i, n, g1) = N(i, n, g2) for all 1 2, g U g U∈ ∈ , n > 0 and 0 dimi M≤ ≤
ii. If g U∈ and g coincides with f in a neighbourhood of ( )iP f for some
0 dimi M≤ ≤ then ( )iP g = ( )iP f
By (ii) of Lemma I.8, as f and g coincide on a neighbourhood of 1
( )k
k j
P f≤ ≤∪ we have
( ) ( )k kP f P g= for all 0 k j≤ ≤ . Then (3.27) implies that
1 1
( )s s
k kh Λ = Λ∪ ∪ (3.28)
and this means that for all 1 k s≤ ≤ , ( )kh Λ is another set of the family 1 2,...., sΛ Λ Λ .
Define ( )T g in the same way as for f , i.e. as the set of pairs ( , )n q such that n q≠ ,
( ) ( )n qInd Ind tΛ = Λ = and
( ) ( )s u
g i g rW WΛ Λ ≠ ∅∩ (3.29)
by replacing f with g. From the fact that h maps every set of the family 1 2,...., sΛ Λ Λ onto another set of the family
with the same index it follows that
Page 29 of 61
( ) ( )T g T f# = # (3.30)
All the intersections in (3.29) are transversal by the structural stability of f. Hence when g
is sufficiently close to f, if (3.29) holds for certain values n and q , it holds replacing f by
g. Then ( ) ( )T g T f⊃ and by (3.30) this implies that:
( ) ( )T g T f= (3.31)
But we have seen that ( , ) ( )i r T f∉ because we chose i, no pair with i in the first entry belongs to ( )T f . But on the other hand ( , ) ( )i r T g∈ because of theorem I.7. and
( )rInd tΛ = . Hence we should have ( ) ( )T g T f≠ . So we found a contradiction.
Now to see how this contradiction leads to the proof of the C1 Stability conjecture we have
to carefully look back at our assumptions.
Theorem I.7 hypotheses I, II and IV are consequences of the fact that f ∈ FFFF 1(M) which
must hold as all structurally stable diffeomorphism belong to FFFF 1(M). The extra
assumption we made in order to be able to apply Theorem I.7 is that there exists
10 ( / ( ))mjM f P fµ +∈ that satisfies 0
0
( ( )) 0ii j
P fµ≤ ≤
>∪ . This yields a set T(g) as defined
above which contains pairs with i as the first entry, i.e. ( , ) ( )i r T g∈ .
By structural stability of f we find a set T(f) such that (i,r) ∉T(f) . By structural stability of
f , and conjugacy with g, we find that ( ) ( )T g T f= . Both can not hold hence we must
discard the assumption that that there exists 10 ( / ( ))mjM f P fµ +∈ that satisfies
00
( ( )) 0ii j
P fµ≤ ≤
>∪ which is necessary for hypothesis (III) of Theorem I.7 hence we proved
Step 8, i.e. that there are no measures µ ∈ MMMM (f m | 1( )jP f+ satisfying I.6.(i). Hence
0log / log 0
smiDf E dµ λ< <∫ (3.32)
for all µ ∈ MMMM (f m | 1( )jP f+ and then step 6 is completed, i.e. siE is contracting and then
step 5 is completed, i.e. 1( )jP f+ is hyperbolic, thus completing the induction step and also
the proof of Axiom A property for f.
Page 30 of 61
Page 31 of 61
4 Main theorems – Outline of proofs
Theorem I.1 If f ∈ F 1(M) then ( )Per f = ( )fΩ .
This first theorem, was proved by Mane in his paper ‘Contributions to the Stability
Conjecture’, (Lemma 3.1.), published in 1978 [15] and is a corollary to Pugh Closing
Lemma.
Pugh’s Closing Lemma is an important theorem, part of a suite of theorems and results on
perturbations. It has a very intuitive statement, that any orbit going back arbitrarily close to
itself can be closed by a 1C perturbation. For 1r > the problem of the closing lemma
remains open. I have come across comments in the literature19 about how despite having
such simple statement, Pugh’s closing lemma has very complex and elaborate proof. The
proof has two main arguments that have been generalised and led to new important
perturbation lemmas.
The first ingredient of the proof is to estimate the effect of small perturbations of the
diffeomorphism on a small neighbourhood of a finite segment of orbit. The second one is
to select two returns of the orbit close to itself, which are in some sense closer to one
another than to any intermediary returns. Mañe used each of these arguments to prove
important results connected with structural stability, for example the Ergodic Closing
Lemma [18] and Theorem I.1 of the 1C Stability Conjecture.
4.1 Outline of proof Theorem I.1 Suppose f ∈ F 1(M), ( ) ( )x f Per f∈Ω − . Let U be a connected neighbourhood of f
contained in F 1(M). By the proof of the closing lemma20 there exist 0n > such that for
all family jU of neighbourhoods of ( )jf x , j n≤ there exists g U∈ such that
( ) ( )g y f y= if n
jn
y U−
∉∪ and 0( )Per g U ≠ ∅∩ . Take neighbourhoods jU satisfying
Page 32 of 61
( )jU Per f = ∅∩ for all j n≤ and let 0 ( )q U Per g∈ ∩ . Then the number of fixed points
of ( , )g qf π is less than the number of fixed points of ( , )g qgπ . On the other hand for all
0m> the number of fixed points of mf is constant on each connected component of f ∈ F
1(M). Theorem I.2 - If f ∈ F F F F 1(M) then for the cases i=0 and i=dim M the sets Pi(f) are finite, As stated in the previous section, Smale had advanced the hypothesis that in the ‘coarse
case’ a system of differential equations can have only a finite number of stable periodic
motions. The term ‘coarse’ was introduced in the foundation paper of Andronov and
Pontrjagin21 in 1937 and later was replaced by the term structural stability; the expression
used was in fact ‘grossier’ – as the paper was initially published in French. Pliss proved
[17] that if a given periodic system of differential equations and all systems close to the
original system have no periodic solutions with zero characteristic indices then the number
of stable periodic solutions is finite.
Below, I outline the main parts of the proof as given by Pliss in [17].
4.2 Outline of proof of Theorem I.2.
Consider the n-dimensional system of differential equations
( , )dy
Y y tdt
= (4.1)
where y and Y are n-vectors, the vector valued function Y is continuous and continuously
differentiable with respect to y for all y and t and we assume that Y is periodic in t:
( , ) ( , )Y y t Y y tω+ = (4.2)
We assume that (4.1) satisfies the following condition: Fundamental condition: There is a positive number ε such that for every vector-valued function R(y,t), ω - periodic in t, continuous and continuously differentiable with respect to y , and satisfying
( , )R y t ε< , R
yε∂ <
∂ (4.3)
the system
Page 33 of 61
( , ) ( , )dy
Y y t R y tdt
= + (4.4)
has no periodic solutions with zero characteristic indices.
Under these conditions Pliss proves the following: for each bounded domain space there
exists only a finite number of stable periodic solutions of (4.1). He proves this result by
assuming that there exists an infinite number of periodic solutions 1 2( ), ( ), . . .t tϕ ϕ of
period mω in which lim mm
ω→∞
= ∞ . He then linearises the system (4.1) – about each
( )my tϕ= to obtain the sequence of systems
( )m
dxP t x
dt= , ( ) ( )m m mP t P tω+ = , ( 1,2...)m= (4.5)
For (4.5) Pliss finds that all the solutions ( )mx t satisfy
0 0 0 0( , , mam mx t t x x eωω+ ≤ (4.6)
for sufficiently small a. Pliss then uses this fact and other estimates to obtain a
contradiction:
Suppose that the theorem as stated does not hold, i.e. the system (4.1) has a sequence of stable periodic solutions 1 2( ), ( ), . . .t tϕ ϕ , satisfying
( )i t cϕ < , 1,2,...i < (4.7)
By linearising the system in (4.1) in the neighbourhood of the solution ( )my tϕ= we
obtain:
( )m
dxP t x
dt= ( 1,2...)m= (4.8)
where x is an n-vector, the ( )mP t are continuous square matrices of order n, periodic with
period mω :
( ) ( )m m mP t P tω+ = (4.9)
the periods mω of the ( )m tϕ are multiples of ω and are unbounded while the ( )mP t are
uniformly bounded:
( ) , 1mP t M M≤ ≥ (4.10)
Page 34 of 61
where ( )mP t is the Euclidean norm.
We write 0 0( , , )mx t t x for the solution of (4.8) with initial conditions 0 0, t t x x= = .
Also, let θ be any positive number and put
0
01
( , , ) ( , ( 1), )max mx
r k m x k k xθ θ θ=
= − (4.11)
1,2,... mkωθ
=
Theorem I.2 (1) If the system (4.8) is uniformly coarse then for arbitrary (0, )σ λ∈ there
is a number 0θ such that for all 0θ θ> there is ( )m θ such that
1
ln ( , , )
m
mk
r k m
ωθ
θ σω
=
<∑ (4.12)
if ( )m mθ> .
I will omit the proof of this theorem. Details can be found in [17]. Choose a multipleθ of ω so that (4.12) holds for sufficiently large m. Let 0 0( , , )y t t y and
0 0( , , )x t t x be solutions of (4.1) and (4.8) respectively. Assume that (4.8) is such that there
are numbers 0l and 1 0l l> such that
0
01
ln ( , , ) ( )2
l
k l
r k m l lσθ θ
= +< − −∑ (4.13)
holds for all positive integers 0 1( , )l l l∈
Since (4.8) was obtained by linearization of (4.1) in the neighbourhood of ( )my tϕ= there
is a number 0γ > independent of m, 0l and 1l so small that, for any two vectors ξ and η
satisfying ξ γ< and η γ< we have
0( )
30 0 0 0( , , ( ) ( , , ( )
l l
m my l l l y l l l eσ θ
θ θ ϕ θ ξ θ θ ϕ θ η ξ η− −
+ − + ≤ − (4.14)
for 0 1l l l< <
Let K be a positive integer such that the set of vectors 1,... ky y satisfying iy c< , 1,2...i = ,
contains at least two vectors iy and jy ( i j≠ ) for which
Page 35 of 61
2i jy yγ− < (4.15)
Inequality (4.10) implies
ln ( , , )r k m Mθ θ< (4.16)
Choose a number m of the solution ( )my tϕ= so large that
12
m KM
ω σθ
> + (4.17)
By virtue of (4.16), (4.17) and (4.12) there is a sequence of positive integers
1 2 . . . kν ν ν< < < for which
1
ln ( , , ) ( )2
j
jk
r k mν
ν
σθ θ ν ν= +
< − −∑ (4.18)
for j kν ν ν< ≤ . It follows from the choice of K that there are two points iθν and jθν
( i j> ) such that
( ) ( )2m j m i
γϕ θν ϕ θν− < (4.19)
and
( )i j mν ν θ ω− < (4.20)
Let T denote the transformation 0 0( , , )i jTy y yθν θν= and let Γ denote the ball
( )m jy ϕ θν γ− < then (4.19) and (4.20) imply that
TΓ ⊂ Γ (4.21) Moreover (4.14) implies that T is a contraction transformation in Γ . Hence a solution ( )tψ of (4.1) with period ( )i jν ν θ− passes through the region Γ for
jt θν= and any solution 0( , , )jy t yθν with 0y ∈Γ tends to ( )tψ for t → ∞ . In particular
the solution ( )my tϕ= tends to ( )tψ as t → ∞ . This contradicts (4.20) and the fact that
( )m tϕ has the smallest period mω and the result of the theorem follows.
An alternative idea for the proof of this theorem: Suppose that the diffeomorphism
:f M M→ has an infinity of stable periodic points with dim dimsE M= . Consider one
of these sinks, say 0x , and take a small disk 0( )sW xε of radius 0ε > , centered at 0x . By
Page 36 of 61
definition of the stable manifold we have that for any two points 1 2 0, ( )sx x W xε∈ ,
0 1( ( ), ( )) 0n nd f x f x → and 0 2( ( ), ( )) 0n nd f x f x → as n → ∞ hence
1 2( ( ), ( )) 0n nd f x f x → as n → ∞ . If we assume that the compact manifold M is a subspace
of a metric space, say nℝ then it is also complete hence we can apply the Contraction
Mapping Theorem and conclude that there is a unique fixed point (which in our case
conform the Stable Manifold Theorem is 0x ) and under the action of iterates of f all points
converge exponentially to it. By density of periodic points, however small 0ε > we still
must have another 0 0' ( )sx W xε∈ to which the Contraction Mapping Theorem equally must
apply. Hence we get a contradiction with the uniqueness of the fixed point on 0( )sW xε ,
hence on a finite compact manifold we can not have an infinity of periodic points with
dim dimsE M= .
Page 37 of 61
4.3 Theorem I.3 If f ∈ F F F F 1(M) there exist constants C > 0 and 0< λ < 1, m>0 and a C1 neighbourhood U
of f such for all g∈U and 0 < i < dim M there exists a dominated splitting TM | ( )iP f =
s ui iE E⊕ satisfying:
a) / ( ) . / ( ( )) for all ( )s um m mi i iDf E x Df E f x x P fλ− < ∈
b) s si iE E= and
u ui iE E= when ( )x Per f∈
c) If ( )ix P g∈ and has period n m> then
[ / ] 1
[ / ]
0
/ ( ( ))n m
m s mj n m
j
Dg E g x Cλ−
=
≤∏ and [ / ] 1
[ / ]
0
/ ( ( ))n m
m u mj n m
j
Dg E g x Cλ−
−
=
≤∏
d) For all ( )ix P g∈
1
0
1log / ( ( )) loglim
nm s mj
n j
Dg E g xn
λ−
→+∞ =
≤∑
1
0
1log / ( ( )) loglim
nm u mj
n j
Dg E g xn
λ−
−
→+∞ =
≤∑
This theorem was independently proved in papers of V.A. Pliss, S.T. Liao and R. Mañe22. Theorem I.3 introduces the property of dominated splitting which is very useful for
understanding how dynamics of the tangent map Df controls or determines the underlying
dynamics of f, even when hyperbolicity can not be established for the whole set of points.
Definition I.3 (1) Let f : M → M a diffeomorphism on a closed manifold M and Λ any f-
invariant subset. Then Λ has dominated splitting if there is a decomposition over the
tangent bundle into in two invariant subbundles: TM | Λ = E F⊕ . Furthermore the
splitting components are continuous and invariant under the derivative Df i.e. there are
constants C >0 and 0< λ <1 such that:
( ) . ( ( ))n n n nD f E x D f F f x Cλ− ≤ (4.22)
Page 38 of 61
for all x∈ Λ and 0n ≥ . C and λ can be interpreted as measures of the strength of the
domination. The constant λ is called the constant of domination. The definition above
says that for n large, the “greatest expansion” of Dfn on E is less than the “greatest
contraction of Dfn on F and by a factor that becomes exponentially small with n. As a
result every direction not belonging to E must converge exponentially fast under iterations
of Df to the direction F23.
Dominated splitting has many important and useful properties, for example it can not be destroyed by small perturbations.
4.3.1 Theorem I.3 – Proof outline
The proof of theorem I.3 is based on two main lemmas. The first lemma is due to Franks
([14] Lemma I.1) and it allows with a slight perturbation of the diffeomorphism along a
periodic orbit to achieve a desired derivative at a finite number of points.
Lemma I.3.(1a) Let θ be a finite set of points in M and let x xQ TMθ∈= Θ and let
( )' x f xQ TMθ∈= Θ . IF 0ε > is sufficiently small and : 'G Q Q→ is an isomorphism such that
/10G df ε− < then there exist a diffeomorphism :g M M→ , ε -close to f in the 1C topology,
such that /x xdg G TM= for any x θ∈ . Moreover if R is a compact subset of M disjoint from θ
we can require f(x)=g(x) for x R∈ .
The version that Mañe uses is a more general statement.
Lemma I.3.(1b) If f ∈ Diff (M) for any neighbourhood U of f there exists ε > 0 and a neighbourhood U0 ⊂ U of f such that given
0g U∈ , a finite set 1,...., Nx x and linear maps
( ):i ii x g xL T M T→ such that
ii xL D g ε− ≤ for all
1 i N≤ ≤ , then there exists g U∈ such that
( ) ( )g x g x= if 1,...., ( )Nx x x M U∈ ∪ − and
ix iD g L= for all 1 i N≤ ≤ .
Definition 4.1. Two groups (G,) and (H,*) are isomorphic if there exists a function :G Hφ → such that both of the following statements hold: a. φ is bijective
b. for all 1g and 2g G∈ ,
1(gφ 2)g = 1( )gφ * 2( )gφ
Such a function is called an isomorphism.
Page 39 of 61
So using this lemma we can perturb g to obtain a diffeomorphism that coincides with g on
M-U and on the g-orbit of p but such that its derivative at points gi(p) is essentially a
translation of ( )if p
D f .
The second supporting lemma refers to a result for families of periodic sequences of linear maps. Let GL(N) be a group of linear isomorphisms of RN . Let ξ :ℤ →GL(N) a sequence of
isomorphisms of RN . The sequence is periodic if there exists 0 1n ≥ such that 0j n jξ ξ+ = for
all j. Denote ( )sjE ξ ,
( )u
jE ξ respectively the space of vectors v∈ RN such that
10
sup ( ) , 0n
ji
v nξ +=
≥ < ∞
∏ (4.23)
respectively
11
0
sup ( ) , 0n
j ii
v nξ −− −
=
≥ < ∞
∏ (4.24)
We say that the sequence ϕ is hyperbolic if ( )s
jE ξ ⊕ ( )ujE ξ ≡ RN for all j ∈Z.
The hyperbolicity of the sequence is equivalent to the hyperbolicity of the linear map 0 1
0
n
jj
ξ−
=∏ .
We say that the sequence ξ is contracting if ( )sjE ξ = RN for every j ∈Z.
Let ( )αξ α ∈D be a family of periodic sequences of linear maps. We say that it is
hyperbolic (respective contracting) if every sequence of the family is hyperbolic
(respectively contracting) and ( )sup , ,n nαξ α ∈ ∈ < ∞ℤD . If, ( )αξ α ∈D,
( )αη α ∈D are families of periodic sequences of linear maps of RN we define
( ) ( )( , ) sup n nd nα αξ η ξ η α= − ∈ ∈ℤD, (4.25)
and we say that two families are periodically equivalent if for every α ∈D the minimum
periods of ( )αξ and ( )αη coincide. Finally we say that a hyperbolic family ( )αξ α ∈D is
Page 40 of 61
uniformly hyperbolic if there exists 0ε > such that any periodically equivalent family
( )αη α ∈D satisfying ( , )d ξ η is also hyperbolic (respectively contracting.
The key property of uniformly hyperbolic families is the following
Lemma I.3.(2). If ( )αξ α ∈D is a uniformly hyperbolic family of periodic sequences of
isomorphisms of Nℝ , then there exist constants 0K > , m∈ℤ , and 0 1λ< < such that:
a) If α ∈D and ( )αξ has minimum period n m≥ , then:
1 1
( ) ( )
0 0
( ) / ( )k m
s kmj i mj
j i
E Kα αξ ξ λ− −
+= =
≤∏ ∏ (4.26)
and
1 1
( ) 1 ( )( 1)
0 0
( ) / ( )k m
u kmj i m j
j i
E Kα αξ ξ λ− −
−+ +
= =
≤∏ ∏ (4.27)
where [ ]/k n m= .
b) For all α ∈D , j ∈ℤ :
1 1
( ) ( ) ( ) 1 ( )
0 0
( ) / ( ) ( ) / ( )m m
s umj i mj mj i j m
i i
E Eα α α αξ ξ ξ ξ λ− −
−+ + +
= =
≤∏ ∏˙ (4.28)
c) For all α ∈D
11
( ) ( )
0 0
1limsup log ( ) / ( ) 0
mns
mj i mjn
j i
En
α αξ ξ−−
+→∞ = =
<∑ ∏ (4.29)
and
11
( ) 1 ( )( 1)
0 0
1limsup log ( ) / ( ) 0
mnu
mj i m jn
j i
En
α αξ ξ−−
−+ +→∞ = =
<∑ ∏ (4.30)
Although the expressions above look very complicated, the manner in Mañe then applies
them to prove Theorem I.3 and the previous lemma, gives an indication of how they should
be interpreted. The sequences of isomorphisms can be viewed as the derivatives of the
diffeormorphism f, ( )jf x
D f computed at the periodic points ( )jf x . Equations (4.26) and
(4.27) indicate contraction, respectively expansion effect of the derivative when sampling
the periodic orbit at different rates.
Page 41 of 61
To prove Theorem I.3 from this lemma take the neighbourhood 0U of f and the constant
0ε > given by Lemma I.3 (1) when applied to f and U =F (M) . The claim is that the
family of periodic sequences ( , )0, ( )x g g U x Per gξ ∈ ∈ where ( , )x gξ is the matrix
( )jg xD g
with respect to orthonormal basis of ( )jg x
T M and 1( )jg xT M+ is uniformly hyperbolic. If it
was not, a family ( , )0, ( )x g g U x Per gη ∈ ∈ could be found such that ( , )d ξ η ε≤ and such
that for some 0g U∈ , ( )x Per g∈ the sequences ( , )x gη and ( , )x gξ have the same minimum
period and ( , )x gη is not hyperbolic. Then by I.3.(1) could find 1( ( )) ( )j jg g x g x+= for all j
and ( , )
( )j
x gjg x
D g η= . Hence x would be a non-hyperbolic periodic point of g , contradicting
g ∈ F (M).
Now applying Lemma I.3 (2) to the family ( , )0, ( )x g g U x Per gξ ∈ ∈ we obtain property
(d) of Theorem I.3 as a corollary of (c) of Lemma I.3 (2) and property (c) of Theorem I.3 as a corollary of (a) of Lemma I.3 (2). To prove (a) and (b) of Theorem I.3 – which is the part used directly in the proof of the 1C
Stability Conjecture, we define the splitting / ( )s ux x iE E TM P f⊕ = . We also define
s sx xE E= and
u ux xE E= if ( )x Per f∈ and we know thatdim ( )s
xE f i= . To extend the
definition to points of ( ) ( )i iP f P f− we fix, for each orbit ( )nf x n∈ℤ in ( )iP f such
that x is not periodic, with dim ( )sxE f i= , a sequence ( )nx Per f∈ such that the limnx x=
and then lim ( ) ( )k knf x f x= for all k. For all k the subspaces
( )( )k
n
s
f xE f ,
( )( )k
n
u
f xE f converge to subspaces of
( )kf xT M which we define as ( )k
sf xE , ( )k
uf xE . Now we
have attached to every ( )ix P f∈ two subspaces sxE ,
uxE with the following properties:
dim dim dims ux xE E M+ = (4.31)
and
( ) and ( )s s u ux x x xx xD f E E D f E E= = (4.32)
for all ( )ix P f∈ and from property (b) of Lemma I.3.(2) we have that:
/ ( ) . / ( ( )) s um m mi iDf E x Df E f x λ− < (4.33)
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for all for all ( )x Per f∈ and dim ( )sxE f i= . Since these points are dense in ( )iP f it
follows that (4.33) holds for every ( )ix P f∈ . From (4.33) it follows that 0s ux xE E =∩
which together with (4.31) implies that:
s ux x xE E T M⊕ = (4.34)
for all ( )ix P f∈ . By Proposition 1.3 of [24]24 properties (4.32) - (4.34) imply that the
subspaces sxE ,
uxE depend continuously on x. By (4.33) these subspaces satisfy the
inequality in Theorem I.3 (a) and by definition s sx xE E= and
u ux xE E= if ( )x Per f∈ and
dim ( )sxE f i= , thus completing the proof of Theorem I.3.
For the proof of the C1 Stability Conjecture, Mañe used only properties (a) and (b) of
theorem I.3. The proof as given by Mañe raised a few questions for me, for example I
could not understand based on the reasoning he gave in his proof why can we deduce that
due to the density of ( )iP f the property (4.33) holds for the non-periodic points as well.
Pujals and Sambarino [ 23]offer a simplified version of the theorem which asserts the
presence of dominated splitting at hyperbolic periodic points. Following I will present the
main ideas in Pujals and Sambarino’s proof as I found it gave me a good understanding of
the relation between hyperbolicity and dominated splitting.
Theorem I.3.(3) Assume that there is a neighbourhood U(f) such that
∡ ( ( , ), ( , )) 0s uE p g E p g α> > (4.35)
for every ( )g U f∈ and ( )ip P f∈ . Then there is a dominated splitting of index i over
( )iP f .
Outline of proof: The condition stated in the hypothesis of Theorem I.3 (3) means that
there are no tangencies allowed, only transversal intersections and this means that the
angles between eigenspaces are bounded away from zero. The proof will show that the
absence of dominated splitting forces the presence of arbitrarily small angles.
The first step is to prove the domination at the end of the period of each periodic point,
meaning the following: there exists 0 1λ< < such that if p is a hyperbolic periodic point of
index i of f then:
Page 43 of 61
/ ( ) / ( )np s np u npDf E p Df E p λ− ≤ (4.36)
where pn is the period of p. If not, it means that we do not have uniform contraction at the
end of the period neither on / ( )np sDf E p nor on / ( )np uDf E p− . Thus we can perturb, to
obtain a hyperbolic periodic point gp of the diffeomorphism g of index i close to f having
one eigenvalue less but close to one and another eigenvalue greater but also close to one.
From this it is possible to break down the angle between spaces sE and uE . For the second
and last step, assume that we do not have a dominated splitting over ( )iP f . Thus it can be
proved that there is a periodic point p of f whose period is arbitrarily large (say k) and for
some large m we have
/ ( ) / ( ( ) 1/ 2j s j u jDf E p Df E f p− ≥ , 1 j m≤ ≤ (4.37)
In some sense this means that the action of Df along the piece of orbit , ( ),..., ( )mp f p f p is
neutral, although a the end of the period we have domination, that we may assume is due to
contraction at the end of the period on sE .
Next define ( ) ( )
: i ii f p f pT T M T M→ such that ( ( )) (1 )s i
iT E f p Idε= + and
( ( ))u iiT E f p Id= and a map i iL T= 1( )if p
Df − . Here, the number ε is small enough in
order to still have a contraction on ( )sE p at the end of the period, i.e. under the action of
1nL − … 0L .
Let us see the effect of this perturbation: take a space S close to ( )uE p then due to the
expansion added in sE along , ( ),..., ( )mp f p f p , we have that mL … 0( )L S will move
(‘fall down’) into the direction of sE , meaning that the angle between mL … 0( )L S and
( ( ))s mE f p becomes small. On the other hand, due to domination at the end of the period,
we will have that nL … 0( )L S is ‘up again’ meaning that nL … 0( )L S and ( )uE p are
close again. Thus, adding another perturbation on nL so that nL … 0( )L S =S and using
Frank’s Lemma (See I.3.(1)) we find g near f so that p is hyperbolic periodic point of g and
( , ) ( , )s sE p g E p f= and ( , ) ( , )u uE p g E p f= but we have ‘destroyed the angle’ at
( ) ( )m mg p f p= , i.e. it is less than α .
So, we can see that dominated splitting prevents the presence of tangencies. The tool of
dominated splitting has been used by many mathematicians to formulate a wealth of
theorems and establish many useful results in the area of dynamical systems. I will only
Page 44 of 61
mention one result, due to Mañe, pertinent to the topic of the C1 Stability conjecture, which
answers the question whether a set having dominated splitting is hyperbolic. Two
necessary conditions follow [Error! Bookmark not defined. ]: all the periodic points in the
set must be hyperbolic and no attracting (repelling) closed curves supporting an irrational
rotation are in the set. Also, Mañe stated and proved a theorem25 which establishes that
these two conditions are also sufficient if the diffeomorphism is smooth enough
( 2 2( )f Diff M∈ . Theorem I.4 – I will omit the proof as its length and degree of technicality is beyond the scope of this project. Details can be found in [11]
4.4 Lemma I.5 Lemma I.5 Let Λ be a compact invariant subset of 1( )f Diff M∈ and E TM⊂ Λ a
continuous invariant subbundle and if there exists m > 0 such that
log /mDf E dµ < 0∫ (4.38)
for every ergodic µ ∈ M M M M(f m | Λ ) then E is contracting.
4.4.1 Lemma I.5 – Proof Outline If for each x∈ Λ there exists n> 0 satisfying
1
0
( )n
n
j
D f E x−
=∏ < 1
Suppose this property is false. Then there exists x∈ Λ such that
1
0
( ( )) 1n
m mj
j
D f E f x−
=
≥∏
for all n. Hence, for all n
1
0
1log ( ( )) 0
nm mj
j
D f E f xn
−
=
≥∑
Define a probability nµ by
Page 45 of 61
1
( )0
1mj
n
n f xjn
µ δ−
=
= ∑
and let 0kn kµ ≥ be a convergent subsequence. Its limit 0µ belongs to MMMM(f m/ Λ ) and
0log lim log k
m mn
kD f E d D f E dµ µ
→∞Λ Λ
=∫ ∫
= 1
0
1lim log ( ( )) 0
knm mj
kjk
D f E f xn
−
→∞ =
≥∑
But if 0log 0mD f E dµΛ
≥∫ then by the Ergodic Decomposition Theorem there exists an
ergodic µ ∈ MMMM (f m/ Λ ) with the same property and the lemma is proved.
4.5 Theorem I.7 Theorem I.7 treats the problem of creation of homoclinic points i.e. aims to create a linking between transitive hyperbolic sets that are bound together by orbits that accumulate in all of them. The problem is, using this loose linking, to create a real linking, meaning by this an intersection between a stable and an unstable manifold of these sets. In a simplified manner, the theorem’s underlying aim to find a solution to the following problem:
Suppose that ( )rf Diff M∈ has a hyperbolic
point p such that there exists ( ( ) )sq W p p∈ −
whose α − limit set satisfies
( ) ( ( ) )uq W p pα ∩ − ≠ ∅ . Then, the question is
whether it is possible to find a diffeomorphism g, Ck near f such that it coincides with f in a neighbourhood of p and satisfies
( ) ( )s ug gq W p W p∈ ∩ ?
Mañe presents two approaches to this question: one local and one global, as follows:
Definition 4.2 Let :f M M→ be a continuous map on a compact manifold M. If f is invertible then α - limit set is defined as:
0
( ) ( ( ) )n
N n N
x cl f xα≤ <
= ∩ ∪
Page 46 of 61
The local method consists in taking a 0n ≥ such that the point ( )nf q− is very close to a
point ( )uz W p p∈ − and trying to find a diffeomorphism ϕ , Ck near to the identity such
that ( ( ))nf q zϕ − = and is the identity outside the a ball ( )rB z that does not contain p. Then
if ( )rB z does not contain points of the form ( )jf q for ( 1) 0n j− − ≤ ≤ the sequence
( ) ( 1) 0 ( ) 0j jf q n j f z j− − ≤ ≤ ∪ ≥ is an orbit of the diffeomorphism
1 1( )g fϕ − −= contained in ( ) ( )s ug gW p W p∩ .
The global approach consists in taking a small neighbourhood U of q and a diffeomorphism
ϕ that is the identity outside U and then defining the diffeomorphism g fϕ= with the
hope of finding ϕ near to the identity and also satisfying the relation ( ) ( )n ugg q W q− ∈ for
some n>0. The intention is to exploit the dynamics of f in such a way that the small
perturbations introduced by ϕ will be amplified under iterations in such a way that the orbit
of q under f will move towards ( )uW p and hit it. To accomplish this we need
supplementary hypotheses that grant certain expanding behaviour to 1f − through which the
amplification of small perturbation can be obtained.
The strategy of the proof of theorem I.7 is that either the local method works or there are
enough expanding dynamics in f to make the global method work.
I will omit the actual proof as it is 22 pages long.
Page 47 of 61
Definition .5.1 Suppose :f X X→ . A compact set A X⊂ is an attractor for f if there is a neighbourhood U such that
( )f U U⊂ and ( )n
n
f U A∈
=∩N
.
Definition 5.2 The basin of attraction B of an attracting periodic orbit
10 0 0, ( ),... ( )mx f x f x−=O is the set of x
such that 0( ) ( ) 0n n kf x f x+− → as
n → ∞ for some k. Definition 5.3 Given an interval S, let
0( , )S xµ denote the fraction of time an
orbit originating from initial condition 0x
in B spends in the interval S in the limit that the orbit length goes to infinity. If
0( , )S xµ is the same for every 0x in the
basin of attraction except for a set of 0x -
values of Lebesgue measure zero, then we say that 0( , )S xµ is the natural
measure of S.
5 Examples of Axiom A Diffeomorphisms
5.1 Hyperbolic sets
Hyperbolic invariant sets are mainly of interest because the property of hyperbolicity
allows many rigorous results to be obtained, for example:
• Stable and unstable manifolds at x, ( )sW x
and ( )uW x can be defined (See Stable
Manifold Theorem). Two points on the same
stable manifold for example approach each
other exponentially in time as illustrated in
Figure 6.1.
• The dynamics of the invariant set can be
represented via symbolic dynamics as a full
shift or a shift of finite type on a bi-infinite
symbol sequence
• If the invariant hyperbolic set is an
attractor, then a natural measure exists.
• The invariant set and its dynamics are
structurally stable.
The simplest example of a hyperbolic set is a
hyperbolic orbit. One of the best examples of a non-trivial hyperbolic set is Smale’s
original horseshoe which is an invertible map with infinitely many periodic points. It is
created on a piece of Euclidean space and so it can occur on any manifold. Another
example with infinitely many periodic points which is more global in nature as the
dynamics occur on the whole manifold is the Tn torus.
Page 48 of 61
Figure 5.1 – Construction of a horseshoe map
5.1.1 Smale’s Horseshoe
The horseshoe map is an example of a
diffeomorphism 2 2:hM S S→ that has an invariant
set which is a Cantor set.
The map hM is specified geometrically in figure
4.1. The map takes the square [0,1] [0,1]S = ×
,(Figure 5.1 (a)) and uniformly stretches it
vertically by a factor greater than 2 (say 2λ > )
and uniformly compresses it horizontally by a
factor less than 1
2 (say 1/ 2µ < ) (Figure 5.1.b)
Then the long strip is bent into a horseshoe shape
with all the bending deformation taking place in
the cross-hatched region of Figures 4.1.(b) and
(c).Then the horseshoe is placed on top of the
original square as shown in Figure 4.1. (d). If the
initial conditions are spread over the square with a
uniform distribution, then the fraction of initial
conditions that generate orbits that do not leave S
during the application of the map is just nf .
Since 0nf → as n → ∞ almost every initial
condition with respect to Lebesgue measure eventually leaves the square. Thus there is no
attractor contained in the square.
We are interested in characterising the invariant set Λ (which is of Lebesgue measure zero)
of points which never leave the square. Furthermore we will investigate the orbits followed
by points in Λ . Firstly we note that the intersection of the horseshoe with the square
represents the regions that points in the square map to if they return to the square on one
iterate. These regions are the two cross-hatched vertical strips labelled 0V and 1V in Figure
5.2 (a).
S
(a) (b)
Page 49 of 61
To see where these strips came from we look at the horseshoe construction backward in
time (i.e. from (d) to (c) to (b) to (a) in Figure 4.1).
To make the statements more precise, let jH for j=1,2, be the two horizontal strips
1 2( , ) : 0 1, j jjH x y x y y y= ≤ ≤ ≤ ≤ with 1 1 2 2
1 2 1 20 1y y y y≤ ≤ ≤ ≤ ≤ . Similarly let jV for
j=1,2 be two disjoint vertical strips 1 2( , ) : ,0 1j jjV x y x x x y= ≤ ≤ ≤ ≤ with
1 1 2 21 2 1 20 1x x x x≤ ≤ ≤ ≤ ≤ . hM is a
diffeomorphism such that ( )h j jM H V= for
j=1,2, 11 2hS M H H− =∩ ∪ and for
1 2p H H∈ ∪
0
( )0
p
hp
aDM p
b
=
(4.39)
with 1/ 2pa µ= < and 2pb λ= > . As
neither of the eigenvalues are the unit, all points 1 2p H H∈ ∪ are hyperbolic.
Thus we find that the two vertical strips 0V and 1V are the images of two horizontal strips 1
0 0( )hH M V−= and 11 1( )hH M V−= as shown in Figure 5.2.(b). Thus taking the intersection
of 0( )hM V and 1( )hM V with S (Figure 5.2. (c)) we see that the points originating in the
square which remain in the square for two iterates are mapped to four vertical strips
00 01 10 11, , ,V V V V . (Figure 5.2.(c)) and Figure 5.2.(d) shows that the four vertical strips ijV
( ( , 0,1)i j = came from two previous iterates 2( )ij h ijH M V−= .
Page 50 of 61
Denote ( )n
n jm
j m
S M S=
=∩ . By this description 10S is the union of the two vertical strips 0V
and 1V of width µ .and for n=2:
2 1 10 0 1 0 2[ ( ] [ ( ]S f S V f S V= ∩ ∪ ∩ (4.40)
See (Figure 5.2 (a)) - 20S is the union of 22 vertical strips of width 2µ . By induction
1 10 0 0 0 1( ) ( )n n n
h hS M S H M S H− −= ∩ ∪ ∩ (4.41)
is the union of 2n vertical strips of width nµ . Taking the infinite intersection
0 10
[0,1]n n
n
S S C∞
−∞=
= = ×∩ (4.42)
is a Cantor set of vertical lines. We then consider the sets: 01 0 1S H H− = ∪ is the union of
horizontal strips of height 1λ− , 02S− is the union of horizontal strips of height 2λ− .
Continuing by induction 0mS− is the union of 2m horizontal strips of height mλ− and
Figure 5.2 – Vertical and horizontal strips Vi, Vij, Hi, Hij for the horseshoe map
Page 51 of 61
0 02
0
[0,1]mm
S S C∞
−∞ −=
= = ×∩ (4.43)
is a Cantor set of horizontal line segments. Since points in the invariant set Λ never leave
S, all iterates of Λ must be in the square. Hence Λ is contained in 0 1H H∪ and is also
contained in 0 1V V∪ . Thus Λ is contained in the intersection
11S− = 0 1 0 1( ) ( )V V H H∪ ∩ ∪ (4.44)
This intersection consists of four squares as shown in Figure 4.3.(a). Similarly Λ must also lie in the intersection
22 00 01 10 11 00 01 10 11( ) ( )S H H H H V V V V− = ∪ ∪ ∪ ∩ ∪ ∪ ∪ (4.45)
shown in Figure 5.3.(b).This intersection consists of 16 squares, four of which are contained in each of the four squares of figure 5.3.(a). Proceeding in stages of this type, at each successive stage, each square is replaced by four squares that it contains. Thus Λ is found by intersecting the two sets 0S∞ and 0S−∞ and get that
0
0 1 2S S S C C∞ ∞−∞ −∞Λ = = = ×∩ (4.46)
is the product of two Cantor sets. Λ has the properties of a Cantor set on a line: it is perfect as both the sets 1C and 2C are Cantor sets, i.e. perfect and its connected components are
points as the set nnS it is the union of 2n rectangles of dimensions nµ and nλ − .
Let us look at the stable and unstable manifolds of the invariant set. We first consider the
set:
Figure 5.3
Page 52 of 61
0 02
0
[0,1]mm
S S C∞
−∞ −=
= = ×∩ (4.47)
A point q is in 0S−∞ if and only if is in ( )j
hM S for all 0j ≤ , if and only if ( )jhM q S∈ for all
0j ≥ . Thus such a point is in the stable manifold of the whole set Λ . If 2[0,1]q C∈ × and
1 2p C C∈ × = Λ have the same y coordinate, then they are on the same horizontal Cantor
line segment, so ( ) ( )j j jh hM q M p µ− ≤ and ( )sq W p S∈ ∩ . Thus, these horizontal line
segments are the local stable manifolds of points in Λ . Similarly for p∈ Λ , ( )uW p S∩ is the vertical line segment through p. To prove that a horseshoe has infinitely many periodic points it is shown that it is conjugate
to the subshift of finite type. Smale’s horseshoe is in fact an example of perfect coding.
The idea is to follow the orbit of a point p∈ Λ and see in which of the horizontal boxes
iH it lies for each iterate. The itinerary map φ is the map which assigns to the point
p∈ Λ the binary sequence a which labels the box ajH in which the point ( )jhM p lies.
Let x be a point in the invariant set Λ . Let 2Σ be the full two-sided two shift space where
we can specify bi-infinite symbol sequences a, 3 2 1 0 1 2... ...a a a a a a a− − −= · (4.48)
and each symbol ia is a function of x specified by
0
1
0 if ( )
1 if ( )
ih
i ih
M p Ha
M p H
∈= ∈
(4.49)
In Figure 6.3.(b) the 16 rectangles are labelled by the symbols 2 1 0 1a a a a− − · that correspond
to the four middle symbols in Figure 6.1. The correspondence given by (4.48) - (4.52).
From eqn. (4.49) we have
1
01
1
0 if ( )'
1 if ( )
ih
i ih
M p Ha
M p H
+
+
∈= ∈
(4.50)
On 2Σ we have the shift map
( ) 'a aσ = (4.51) where '
1i ia a+= .
Page 53 of 61
The above represents a correspondence between bi-infinite symbol sequences a and points
x in Λ . The map
2:φ Λ → Σ , ( )p aφ = (4.52)
is bijective and continuous hence φ is a topological conjugacy from hM Λ to σ on 2Σ .
5.1.2 Proof that φ is continuous
Let ( )p aφ = . A neighbourhood of a is given by
0 0: for j jU t t a n j n= = − ≤ ≤ (4.53)
With 0n fixed, the continuity of hM ensures that there is a 0δ > such that for p∈ Λ and
p q δ− ≤ , ( )jh a jM p H∈ for 0 0n j n− ≤ ≤ . Thus if ( )t pφ= and p q δ− ≤ then t U∈ .
This proves the continuity of φ .
5.1.3 Proof that φ is bijective
To check that φ is surjective, we apply induction on n to show that 1
( )n
jh s j
j
M H −=∩ is a
vertical strip of width nµ for all strings of symbols 2a∈Σ . Let 2a∈Σ . For n=1 this set is
just 1 1( )h s sM H V− −= which is a vertical strip of width µ . Then,
11
1 2
( ) ( ( )) ( )n n
j jh s j h h s j h s
j j
M H M M H M H−− − −
= =
= ∩∩ ∩ (4.54)
is a strip of width nµ since 2
( )n
jh s j
j
M H −=∩ is a strip of width 1nµ − . Letting n go to infinity,
2
( )jh s j
j
M H∞
−=∩ is a vertical line segment.
0
( )jh s j
j
M H −=−∞∩ is a horizontal line segment and
( )jh s j
j
M H∞
−=−∞∩ is a single point q. So ( )q aφ = and φ is surjective.
Page 54 of 61
Figure 5.4 A transverse homoclinic point and the associated homoclinic tangle.
To prove that φ is injective, we assume that ( ) ( )p q aφ φ= = . Then for all j , both ( )jhM p−
and ( )jhM q− are in s jH − so , ( )j
h s jp q M H −∈ . Letting j run from 1 to ∞ we see that
, ( )jh s j
j
p q M H∞
−=−∞
∈ ∩ so they are in the same horizontal line. Using all j from -∞ to ∞ we
see that p q= . This completes the proof of conjugacy and this proves that the shift on the bi-infinite symbol space is equivalent to the horseshoe map applied to the invariant set Λ .
Fixed points of nσ are mapped by 1φ − to fixed points of nhM . Since the former are just
sequences that repeat after n shifts and since there are 2n ways of choosing a sequence of n
zeros and ones, we see that there are 2n fixed points of nhM in Λ . This implies that the set
of points on periodic orbits is dense in the invariant set Λ . In addition, there is an
uncountable set of non-periodic orbits in Λ , and there are orbits that are dense in Λ hence
Λ is transitive.
The geometric horseshoe arises in many dynamical systems, one of them being the Hénon
map but only for certain parameter values26. It also arises from a transverse intersection of
the stable and unstable manifolds of a periodic point – see illustration of the process below.
Such an intersection is called a homoclinic point.
Let us consider a C1 diffeomorphism
: n nP R R→ with a hyperbolic fixed
point say x=0 and its stable and
unstable manifolds intersect transversely
at x0. Since (0)sW and (0)uW are
invariant under P iterates of x0 , 2
0 0( ), ( )P x P x … as well as 1 2
0 0( ) , ( )P x P x− − , … also lie in
(0) (0)s uW W∩ . The only way this can
happen is if the manifold loops back and
crosses itself at a new homoclinic point.
Another loop must be formed, with
0( )P x and then 20( )P x another
homoclinic. Since 20( )P x is closer to
Page 55 of 61
the hyperbolic point than 0( )P x , the distance between 2 0( )P x and 0( )P x is less than that
between x0 and 0( )P x .
Area preservation requires the area
to remain the same, so each new
curve (which is closer than the
previous one) must extend further. In
effect, the loops become longer and
thinner. The network of curves
leading to a dense area of homoclinic
points is known as a homoclinic
tangle part of which is shown in
Figure 5.4 where (0)sW and (0)uW
accumulate on themselves.
So we see that the existence of one
transverse homoclinic point for P
implies the existence of an infinite number of homoclinic points which accumulate at x=0
i.e. we have a transverse homoclinic orbit of P. In a homoclinic tangle, a high enough
iterate of P will lead to a horseshoe map. Furthermore, the existence of a horseshoe map
results in chaotic dynamics.
Figure 5.5 If we start with a small ball of initial points
centred around the fixed point and iterate the map, the
ball will be stretched and squashed along the line Wu.
Similarly the small ball of initial points iterated
backward in time will trace the stable manifold.
Figure 5.6 – An iterate of P exhibiting a horseshoe map.
Page 56 of 61
Figure 5.1 - Illustration of the stable manifold
Two points on the same stable manifold for example approach each other exponentially in time.
6 Some key concepts and theorems
6.1 Stable and unstable manifolds for points of a h yperbolic invariant set
Definition 6.1. Given f ∈Diffr(M) and x∈M the stable and respectively unstable manifolds of x are as defined as follows:
( ) lim ( ( ), ( )) 0 s n nf
nW x y M d f x f y
→+∞= ∈ → (4.55)
+( ) lim ( ( ), ( )) 0 u n n
fn
W x y M d f x f y− −
→ ∞= ∈ → (4.56)
When dealing with only one diffeomorphism we shall denote these sets ( )sW x and
uW (x). Conform the Stable Manifold Theorem for a hyperbolic invariant set Λ , there are stable and unstable manifolds through each point p∈ Λ and they are immersed copies of the stable and
respectively unstable eigenspaces, spE ,
upE for
that point. Theorem 6.1 (Stable Manifold Theorem for a Hyperbolic set). Let f: M
→ M be a Ck
diffeomorphism. Let Λ be a hyperbolic invariant set for f with hyperbolic constants 0 < λ < 1 and C≥ 1. Then there is a ε > 0 such that for each p∈ Λ , there are two Ck embedded
disks ( )sW pε and ( )uW pε which are tangent to spE
,upE respectively. In order to consider these disks as graphs of functions we identify a
neighbourhood of each point p with spE (ε )× u
pE (ε ). Using this identification, ( )sW pε is
the graph of a Ck function : ( ) s s up p pE Eσ ε → with 0(0 ) 0s
p p pσ = and ( ) 0spD σ = .
( ) ( (y) , y) : y E ( ) s s sp pW xε σ ε= ∈
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Also the function and its first k derivatives vary continuously as p varies. Similarly there is
a Ck function : ( ) u u sp p pE Eσ ε → with (0 ) 0u
p p pσ = and 0( ) 0upD σ = and with the function
upσ and its first k derivatives varying continuously as p varies such that:
( ) ( (y) , y) : y E ( ) s s sp pW xε σ ε= ∈
After the local stable and unstable manifolds are obtained by the above theorem, the global stable and unstable manifolds ( )sW p and ( )uW p respectively, are determined using the Definition 2.3 above as follows:
( ) : ( ( ), ( )) 0 as js j jW p q M d f q f p≡ ∈ → → ∞ = Un≥ 0 ( ( ( )))n s nf W f pε−
and
( ) : ( ( ), ( )) 0 as ju j jW p q M d f q f p− −≡ ∈ → → ∞ = Un≥ 0 ( ( ( ) ))n u nf W f pε−
Definition 6.2. A Cr map : M Nϕ → from one manifold into another is called an
immersion provided the derivative of ϕ at each point is an isomorphism. The image of a
one to one immersion is called an immersed submanifold. If the immersion is a
homeomorphism then it is called an embedding and its image is called an embedded
submanifold. Proposition 6.2. Let Λ be a hyperbolic invariant set for a diffeomorphism f. Then for each
p∈ Λ , ( )sW p is an immersed copy of spE and ( )uW p is an immersed copy of upE .
Proof. By the Stable Manifold theorem each local stable and unstable manifold ( )sW pε and
( )uW pε are embedded images of the closed disks spE (ε ) and
upE (ε ) respectively. Because
f is a diffeomorphism ( ( ))j sf W pε is thus the embedded image of a closed disk spE (ε ) by
the map , =fs jp jσ
spσ . Thus ( )sW p is the union of these sets, each of which is an
embedded copy of a disk, and so ( )sW p is the immersed copy of the linear subspace spE .
The proof for the unstable manifold is similar.
This means that the map : ( ) s sp pE Mσ ε → is one-to-one but need not have a continuous
inverse. The fact that the stable and unstable manifolds of points are immersed copy of the linear spaces implies that they can not be circles or cylinders.
Page 58 of 61
Definition 6.3 A map f : M
→ M is (topologically) transitive on an invariant set N provided the forward orbit of some point p is dense in Y. The Birkhoff Transitivity Theorem proves that a map f is transitive on N if and only if, given two open sets U and V in N there is a positive integer n such that ( )nf U V∩ ≠ ∅ . This indicates that f mixes up the points of U and the map is one piece dynamically. The sets iΩ are called basic sets. These sets have the following properties:
i) They are closed invariant isolated hyperbolic sets and ii) f | iΩ is transitive.
Corollary 6.3 If f : M
→ M is as above one can write M canonically as a finite disjoint
union of invariant subsets 1
( )k
si
iM W
== ΩU =
1( )
ku
jj
W=
ΩU
( ) ( ) , s mi iW x M f x mΩ = ∈ → Ω → ∞ (4.57)
( ) ( ) , u mj jW x M f x m−Ω = ∈ → Ω → ∞ (4.58)
For the proof of this corollary we have to define the ω - limit set of x, the α - limit set of x
and the limit set of f as follows:
Definition 6.4 A point y is an ω -limit point of x if there exists a sequence of nk going to infinity as k goes to infinity such that
lim ( ( ), ) 0kn
kd f x y
→∞=
The set of all ω -limit point of x for f is called the ω -limit set of x and is denoted by ( )xω or ( , )x fω . If the map f is invertible then the α - limit set of x for f is defined in the
same way but with nk going to minus infinity. The closure of all ω - limit sets is called the limit set of f. ( ) ( ( ) :L f cl x x Mω= ∈U (4.59)
If f is invertible the limit set of f is then: ( ) ( ( ) ( ) :L f cl x x x Mω α= ∈U U (4.60)
Page 59 of 61
If we denote Per(f) as the set of periodic points of f, i.e.
( ) : ( ) , k 1kPer f x f x x some= = ≥ Then we can express the expressions (4.57) and (4.58) above as:
( ) : ( )si iW x xωΩ = ∈Ω (4.61)
and ( ) : ( )u
i iW x xαΩ = ∈Ω (4.62)
respectively. Using the definitions it is easy to see that ( ) ( ) ( )Per f L f f⊂ ⊂ Ω
Proof to Corollary 6.3. Let p∈M, then ω (x) ⊂ L(f). If f satisfies Axiom A then ( )Per f
= Ω (f). Hence by Spectral Decomposition Theorem L(f)= 1 2 kΩ Ω Ω U . . . . .U . The sets
iΩ disjoint and a bounded distance apart, say a maximum d. We want to show that ω (x) is
only contained in one iΩ . We assume the opposite and get a contradiction, i.e. we assume
that ω (x) ∩ jΩ ≠ ∅ and ω (x)∩ kΩ ≠ ∅ for j k≠ . Let D( iΩ , d/3) be the d/3
neighbourhood of iΩ . Because the sets iΩ are invariant, there are neighbourhoods Ui of
each iΩ such that cl(Ui) ⊂ D( iΩ , d/3) and f(cl(Ui)) ∩ D( tΩ , d/3) =∅ for t≠ i. With the
above assumptions on p, there is an increasing sequence of interates ni with 2 ( )injf p U∈
and 2 1 ( )inkf p U+ ∈ . Let mi be the largest integer m such that n2i < m <n 2i+1 .with
1( )mjf p U− ∈ . Then ( )im
jf p U∉ by the choice of mi. On the other hand, 1( )imjf p U− ∉ so
( )imf p f= 1( ) ( , / 3)im
t tf p D d U− ∉ Ω ⊃ for t≠ j. Therefore ( ) \imt tf p M UU . By taking a
subsequence of the mi, we get that ( )imf p accumulates on a point q that is not in any of the
tΩ . We have a contradiction because q is in ( )pω . Thus we have shown that ( )pω is
contained in a single iΩ . Therefore ( )sjp W∈ Ω . The proof that α (x) ⊂ kΩ some k is
similar so ( )ukp W∈ Ω .
Page 60 of 61
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