Multifractal Components of Multiplicative Set Functions

$Page 1: Multifractal Components of Multiplicative Set Functions$
Math. Nachr. 229 (2001), 129 – 160

Multifractal Components of Multiplicative Set Functions

By Manuel Moran of Madrid

(Received March 27, 2000; accepted March 23, 2001)

Abstract. We analyze the multifractal spectrum of multiplicative set functions on a self–similarset with open set condition. We show that the multifractal components carry self–similar measureswhich maximize the dimension. This gives the dimension of a multifractal component as the solutionof a problem of maximization of a quasiconcave function satisfying a set of linear constraints. Ouranalysis covers the case of multifractal components of self–similar measures, the case of Besicovitchnormal sets of points, the multifractal spectrum of the relative logarithmic density of a pair of self–similar measures, the multifractal spectrum of the Liapunov exponent of the shift mapping and theintersections of all these sets. We show that the dimension of an arbitrary union of multifractalcomponents is the supremum of the dimensions of the multifractal components in the union. Themultidimensional Legendre transform is introduced to obtain the dimension of the intersection offinitely many multifractal components.

1. Preliminaries

1.1. Introduction

The research in this paper springs out from an attempt to clarify the relationshipbetween Besicovitch normal subsets of points of a self–similar set (see [Mo2] and Ex-ample 1.7), and multifractal components of a self–similar measure. We conjecturedthat multifractal components of self–similar measures can be understood in terms ofthe finer decomposition provided by Besicovitch normal sets of points, and in fact thisconjecture has happened to be confirmed by our research. We work with a generaliza-tion of the classical definition of multifractal component which contains some relevantsubsets of a self–similar set (see Definitions 1.3 and 1.4 and Examples 1.5 to 1.8). Themain result of the paper shows that the class of multifractal components and the classof self–similar measures are dual classes in a Frostman’s sense, i. e., the dimensionof any multifractal component is given by the maximum of the dimensions of the self–similar measures on the multifractal component (Theorem 2.6), and the dimension ofany self–similar measure µ is the minimum of the dimensions of the multifractal com-ponents with full µ–measure (this derives from the fact that there exists a Besicovitch

1991 Mathematics Subject Classification. 28A80.Keywords and phrases.

c©WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2001 0025-584X/01/22909-0129 $ 17.50+.50/0

130 Math. Nachr. 229 (2001)

normal set with full µ–measure). In Theorem 2.8 it is proved that the dimension ofthe union of any (possibly non–countable) family of multifractal components is givenby the supremum of the dimensions of the components in the family. It also holds thatthe dimension of any multifractal component is the maximum of the dimensions of theBesicovitch normal sets contained in the multifractal component (see Remark 2.9). Inview of these results, Besicovitch normal sets may be thought as the atoms, from adimensional point of view, in self–similar multifractality.

Multifractal analysis roughly consists in the analysis of the so called multifractalspectra, which gives the dimension of the sets in some family

Fα

of subsets of some

set E as a function of the parameter α. It started from ideas of B. Mandelbrot

[Ma] and T. Hasley and alt. [Ha]. We refer the interested reader to [Ol], [Fa], whereit is given account of recent progress in the growing field of multifractality. The firstmultifractal spectrum analyzed was that of a probability measure µ in a subset of IRn.In [Ha] the “strength” of the singularity of µ at x was thought as the limit

limr→0

logµ(B(x, r))log r

,

where B(x, r) denotes a ball centered at x and with a radius r. We may then askfor the set of points M(µ, a) where the singularity of µ has a given strength a, andthe multifractal spectrum of the measure µ is the Hausdorff dimension of M(µ, a) asa function of a. Among the first measures for which the multifractal spectrum wasrigorously computed are self–similar measures (in [Br] for self–similar measures in thereal line, and in [Caw] for self–similar measures in IRn). They have provided a canonicalmodel of multifractality. From our Theorem 2.6 it follows that the computation of themultifractal spectrum of a self–similar measure reduces to solve a family of problemsof maximization of a strictly quasiconcave function — the dimension of a self–similarmeasure — with a set of linear constraints, and some strong regularity properties of themultifractal spectrum of a self–similar measure derive from this fact. Furthermore, thecomputation of the dimension of the intersection of several multifractal componentsreduces to solve a problem of optimization with a larger set of constraints. Theseresults are true for arbitrary intersections, rather than in the generic sense, usuallyconsidered in the literature, of dimensions of intersections for almost every translatedcopy of the intersecting sets.

To obtain the results of this paper we use the ergodic properties of the multifractalcomponents derived from the asymptotic behaviour of the codes of their points. Thisapproach started with Besicovitch normal sets introduced in [Be], giving rise to a lineof research which considerably deepen in this field [Eg, Bi1, Vo, Coo, Ca, Mo2, Mo3].In particular CooleBrook and Cajar (see [Coo, Ca]) obtain a cornerstone in thistheory finding a class of subsets of the real line (the saturated sets) with the propertythat the Hausdorff dimension of an arbitrary family of such sets can be obtained asthe supremum of the dimensions of the sets in the family. These papers obtain theirresults in terms of the Hausdorff and Billingsley [Bi1] dimensions in an abstractcode space. Our result in Theorem 2.8 stating that the Hausdorff dimension is a setfunction stable for arbitrary unions of multifractal components can be considered anextension of the mentioned result to the setting of self–similar sets.

We also mention that there has recently been interest in finding dynamical systems

Moran, Multiplicative Multifractality 131

carrying invariant measures with a maximal dimension [Ga1, Ga2]. Theorem 2.6 showsthat the shift dynamics restricted to multifractal components provides an example ofnon compact sets in this situation.

In Section 3 we analyze pointwise relationships among cylindrical and sphericaldensities. This allows us to extend to the spherical multifractal components the resultsfor cylindrical multifractal components proved in Section 2. An antecedent of the ideasused here can be found in [Mo3].

Lastly, in Section 4 we adopt the classical approach to multifractality based onLegendre transforms, showing that the dimension of intersections of classical sphericalmultifractal components can also be computed by a straightforward extension of thesetechniques.

In the remainder of this section we introduce the framework of self–similar setswhere the analysis is carried out (Sections 1.2, 1.3 and 1.4), we describe self–similarmeasures (Section 1.5), proving some elementary results used in the text, and weintroduce multifractal components of multiplicative functions (Section 1.6), presentingsome examples which give an idea of the scope of the results obtained in the paper.

1.2. Self–similar subsets of IRn

Given a set Ψ = ψ1, ψ2, . . . , ψm of contracting similitudes of IRn, let SΨ be theset mapping defined by

SΨ(X) =⋃i∈M

ψi(X) , X ⊆ IRn ,

where M = 1, 2, . . . , m with m ≥ 2. There is a unique compact set E associatedwith Ψ, which is invariant for SΨ, i. e, such that SΨ(E) = E [Hu]. Such set is calledself–similar set generated by Ψ. We may assume, without loss of generality, that|E| = 1 (we write |X| to denote the diameter of X ⊆ E). We keep these notationsfrom now on.

In this paper we analyze the size of certain subsets of E, the multifractal componentsto be defined later, in terms of their Hausdorff dimension. We denote byHs(X), dimXand dimµ respectively the Hausdorff s–dimensional measure of a subset X of IRn, theHausdorff dimension of X and the Hausdorff dimension of a measure µ on IRn. Thislater is defined by dimµ = infdimX : µ(X) > 0. See refs [Fa], [Mat] for details onthe properties of these measures and dimensions. The packing dimension of a subsetX ⊆ IRn is denoted by DimX. The packing dimension Dimµ of a measure µ on IRn

is given by Dimµ = infDimX : µ(X) > 0. See [Tr], [Fa] or [Mat] for definitionsand properties of packing measures and dimensions.

1.3. The overlapping set

We denote by IM = M∞ the code space of infinite sequences with terms inM , and byM∗ the set of finite sequences with terms inM . Given a sequence i = i1i2 . . . ik ∈M∗

we write ψi for the similitude ψi1 ψi2 . . . ψik and ri for the contraction ratiori1ri2 . . . rik of ψi. We abbreviate ψi(E) to Ei for every i ∈ M∗. Given an i ∈ IM wewrite i(k) for the curtailed sequence i1i2 . . . ik ∈ Mk. We also use the notation i(k)for the curtailement of a sequence i ∈ M q with q ≥ k. We call geometric k–cylinders

132 Math. Nachr. 229 (2001)

the sets Ei, i ∈ Mk, k ∈ IN. We denote by Ck the class of all geometric k–cylinders,and we write C∗ for the whole class

⋃k∈IN Ck of geometric cylinders.

There is a natural coding map π : IM → E associated with Ψ given by

π(i) =∞⋂k=1

Ei(k) , i = i1i2i3 . . . ∈ IM .

When π(i) = x we say that i is an address of x. The primary overlapping set is definedby

O =⋃i =jψi(E) ∩ ψj(E) .

The primary overlapping set O is the set of points of E which have at least twoaddresses differing in their first entries. The overlapping set

O∗ =⋃

i∈M∗Ψi(O)

consist of all points of E with multiple addresses. From the invariance of the Hausdorffdimension of sets under similitudes, it follows that dimO = dimO∗ holds.

Let ∂X and AdhX denote respectively the boundary and the closure of X ⊆ Ein the topology of E as a metric subspace of IRn and let x ∈ ∂Ei, i ∈ M. Thenx ∈ Adh

⋃j =i Ej, so x ∈ Ej for some j = i, which shows that ∂Ei ⊆ O. An analogous

argument shows that ∂Ei ⊆ O∗ for any i ∈M∗.In order to keep a control on the size of the overlapping set we assume that the

system Ψ satisfies the strong open set condition, i. e. that there exists a non – emptyand bounded set O ⊆ IRn, which is open in the topology of IRn, with E ∩O = ∅, andsuch that SΨ(O) ⊆ O and ψi(O) ∩ ψj(O) = ∅ for i = j, i, j ∈ M. If s denotes thesimilarity dimension of E, i. e., the unique real number such that

∑i r

si = 1, then it is

known that dimE = s and 0 < Hs(E) <∞. This result is due to J.E. Hutchinson

[Hu], extending ideas of P.A. P. Moran [Mor].In [Mo1] it is shown that the strong open set condition holds if and only if there

exists a non–empty set W ⊆ E, which is open in E and satisfies the following twoconditions:

i) SΨ(W ) ⊆W,ii) ψ(W ) ∩O = ∅, for all ψ ∈ Ψ.Let

G = Adh

( ⋃i∈IN

T i(O)

),

where T (x), x ∈ E, denotes the image of x under the geometric shift (see Section 1.4for a definition). In [Mo1] it is proved that the largest non–empty open subset of Ewhich satisfies conditions i) and ii) is given by V = E −G, and that ∂V = G holds.We keep these notations from now onwards. We also use in Section 3 the followingproperty of V [Mo1].


iii) Let B(x, r) denote a ball of E as a metric subspace of IRn, centered at x ∈ Eand with radius r. If B(x, r) ⊆ V , then

ψi(B(x, r)) = B(ψi(x), rir

), i ∈M∗ ,

holds. The set ∂V also plays a relevant role for the results in Section 3.For i ∈ Mk, let l(i) denote the length k of i. By volume estimate [Hu], it can be

found a constant D such that, given a ball B(x, r), the cardinality of the set

Q(x, r) :=i ∈M∗ : Ei ∩B(x, r) = ∅ and |Ei| < r ≤ |Ei(l(i)−1)|

is less than D independently of x and r. In particular this shows that the cardinalityof the set π−1(x) of addresses of any x ∈ E is less than D, since otherwise if we takea suficiently small r the cardinality of Q(x, r) would not be less than D.

1.4. Invariant measures

We define the forward shift mapping τ in the code space by τ (i1i2i3 . . .) = i2i3i4 . . .The forward shift induces in the geometric space E the geometric forward shift T :=π τ π−1. The geometric shift is well defined as a point to point mapping on the setE−O∗. Every point x ∈ E−O∗ has a unique address ix ∈ IM such that π(ix) = x. Itis easy to check that, for x ∈ E−O∗, T (x) = ψ−1

ix(1)(x) holds. It is also easy to see that

T−1(A) = SΨ(A) for any A ⊆ IRn, and that if x ∈ T (A), with A ⊆ E, then ψ(x) ∈ Afor some ψ ∈ Ψ. Therefore, if for x ∈ E−O∗, y := T (x) ∈ O∗ holds, then there existsa ψ ∈ Ψ such that ψ(y) = x, which gives the contradiction x ∈ ψ(O∗) ⊆ O∗. HenceT (E −O∗) ⊆ E −O∗ holds.

Since we shall restrict our attention to the dynamics of T, we call invariant a prob-ability measure µ on E if µ

(T−1(A)

)= µ(A) for every µ–measurable A ⊆ E. Assume

that µ is an invariant measure with µ(O) = 0. Then µ(O∗) = 0 also holds. We haveseen above that the geometric shift defines a dynamical system in E −O∗. Moreover,for any µ–measurable A ⊆ E we have

µ(T−1(A) ∩ (E − O∗)

)= µ

(T−1(A)

)= µ(A) = µ(A ∩ (E − O∗)) ,

which shows that Birkhoff’s ergodic theorem holds in the probability space E − O∗

endowed with the restricted measure µ∣∣E − O∗ defined on the σ–algebra induced by

E − O∗ in the Borel subsets of E. Observe that since for a µ–measurable A ⊆ E,µ∣∣E − O∗(A ∩ (E − O∗)) = µ(A) holds, the statements of the Birkhoff’s ergodic

theorem for the restricted measure µ∣∣E − O∗ on the probability space E − O∗ also

hold µ–a. e. x ∈ E. We shall frequently use this fact.

1.5. Self–similar measures

Let P ⊆ IRm be the set of m–coordinate probability vectors, given by

P =

(xt)t∈M : xt ≥ 0, t ∈M,

∑t∈M

xt = 1

.

We may regard each p = (pi)i∈M ∈ P as a probability measure on M defined byp(t) = pt, for t ∈M . Given a p ∈ P we write νp for the infinite–fold product measure

134 Math. Nachr. 229 (2001)

νp = p×p×p . . . on IM. For every p ∈ P the projection π induces on E a self–similarmeasure µp = νpπ−1. We denote by S the set µp : p ∈ P of self–similar measures.We generalize this construction to the set of k–self–similar measures by consideringthe system of similitudes Ψ(k), k ∈ IN, made up of all possible compositions of ksimilitudes of the original system Ψ. The corresponding set mapping associated withΨ(k) is the k–th iterate SΨk of the set mapping SΨ. The unique invariant set for SΨk

is again E. The self–similar measures associated with Ψ(k) are obtained by regardingthe probability vectors p ∈ Pk ⊆ IRmk

as measures in the code space Mk, and thenconsidering the infinite fold product measures νp = p × p × p . . . defined on the codespace IM(k) = Mk × Mk ×Mk . . . The set S(k) of k–self–similar measures is thendefined by S(k) =

νp π−1 : p ∈ Pk

and the set of finitely generated self–similar

measures by S∗ =⋃

k∈IN S(k). It was shown independently by Patche [Pa] and [Mo2]that µp(O∗) = 0 holds for any self–similar measure µp ∈ S(k) with µp(Ei) > 0 for alli ∈Mk. We denote by SP the set of measures

SP =µ ∈ S with µ(Et) > 0 for t ∈M

.(1.1)

Self–similar measures are known to be invariant and ergodic measures. We give someproperties of self–similar measures in the following lemma.

Lemma 1.1. Let p ∈ Pk for some k ∈ IN. Theni) Let M ′ = i ∈ Mk : pi > 0 and Φ = ψi : i ∈ M ′. Then, the self–similar

measure η constructed from the system of similitudes Φ which gives probability η(Ei) =µ(Ei) for each i ∈M ′ coincides with µp.

ii) µ(∂Ei) = µ(O∗) = 0 for any µ ∈ S(q), q ∈ IN, and any i ∈M∗. For µ ∈ SP ,µ(∂V ) = 0 holds.

iii) If we make the convention 0 × log 0=0, then

dimµp = Dimµp =∑

i∈Mk pi logpi∑i∈Mk pi log ri

.

Proof . Part i) derives from the fact that η and µp agree over the field Fk generatedby the geometric cylinder sets Cq×k, q = 1, 2, 3, . . ., which is a generating field for theσ–field of Borel subsets of E. To prove part ii), let q = 1 and notice that the measure ηis supported on the unique compact set EΦ invariant with respect to the set mappingSΦ =

⋃i∈M ′ ψi, where M ′ is as in part i) for k = 1. Let O∗

Φ denote the overlappingset in EΦ. Then

O∗ −O∗Φ =

⋃k∈IN

⋃i=j,

(i,j)∈Mk×(Mk−M′k)

Ei ∩ Ej .

So O∗−O∗Φ ⊆ ⋃

k∈IN⋃

i∈(M−M ′)k Ei, which shows that η(O∗−O∗

Φ

)= 0, and therefore

η(O∗) = η(O∗Φ

)holds. Hence we have

µ(∂Ei) ≤ µ(O∗) = η(O∗) = η(O∗Φ) = 0 .

If q > 1 we use first the above result for the system of similitudes Ψ(q) with overlappingset O∗(q) to obtain µ

(O∗(q)) = 0, and then observe that O∗ ⊆ O∗(q). To prove the


last statement in part ii), notice that if µ ∈ SP then µ–a. e. x ∈ E has a T–orbit⋃q∈IN T

q(x) which is dense in E (see [Mo2] for a proof). In [Mo1] it is proved thatT (∂V ) ⊆ ∂V . Thus, if a point of ∂V had a T–orbit dense in E, ∂V would be densein E, and then V = ∅ in contradiction with the definition of V . The result of part iii)is well–know to hold for vectors p ∈ Pk with pi > 0, for all i ∈Mk (see [De], [Caw]).If pi = 0 for some i ∈ Mk we may apply this result to the measure η considered inpart i).

The bijection Π : S → P given by Π(µp) = p allow us to regard the set of self–similarmeasures as the set of probability m–dimensional vectors P. If we consider the usualtopology of P as a subspace of IRm, and S with the topology of weak convergence, wehave the following lemma.

Lemma 1.2. The mapping Π : S → P is a homeomorphism.

Proof . Let µ ∈ S, and µk→wµ. Since µ(∂Et) = 0 for t ∈ M , µk(Et) → µ(Et). This

gives the continuity of Π at every µ ∈ S. Since P is a Hausdorff topological space, Πis a homeomorphism.

1.6. Multifractal components of multiplicative functions

Definition 1.3. Given a class D of geometric cylinders and a real function f : D →IR+ we say that f is a multiplicative set function on D if f

(Ei∗j

)= f(Ei)f

(Ej

)for

every pair of cylinders Ei and Ej of D such that Ei∗j ∈ D, where the notation i ∗ jstands for the code i1i2 . . . ikj1j2 . . . jq if i ∈ Mk and j ∈ M q. Self–similar measuresand the diameter | · | are examples of set functions which are multiplicative on the setC∗ of geometric cylinder sets.

A multiplicative function on C∗ is completely determined by the values it takes onthe cylinder sets Et, t ∈ M . Therefore we may identify the set of such multiplicativefunctions with the set

R = (x1, x2, . . . , xm) : xt > 0, 1 ≤ t ≤ m .Notice that we disregard multiplicative functions that vanishes in some cylinder set.We consider the set R endowed with the topology derived from the Euclidean metric.Observe that the set SP defined in (1.1) is a subset of R. In Sections 2 and 3 weconsider relatively compact subsets Λ ⊆ R. They are bounded subsets of R, and suchthat, for some positive constant K, f(Et) > K for all t ∈ M and all f ∈ Λ.

The following definition is basic for this paper.

Definition 1.4. Given a multiplicative function f : C∗ → IR+, we define thegeneralized cylindrical multifractal component E(f) associated to f by

E(f) =x ∈ E : lim inf

k→∞k−1 log f

(Ei(k)

) ≥ 0 for all i ∈ π−1(x).

The sets E(f) and E(f) are defined replacing in the above expression “lim inf”respectively with “lim sup” and “lim”, and the symbol ≥ respectively with ≤ and

136 Math. Nachr. 229 (2001)

=. Given an arbitrary set of multiplicative functions Λ, we define the multifractalcomponent E(Λ) associated to Λ by E(Λ) =

⋂f∈Λ E(f). The multifractal components

E (Λ) and E(Λ) are defined in the obvious way. Given a set of multiplicative functionsΛ, let Λ−1 denote the set 1/f : f ∈ Λ. Then it is easy to check thatE(Λ) = E

(Λ−1

)and E(Λ) = E

(Λ ∪ Λ−1

)hold, so that we may restric our attention to multifractal

components of the form E(Λ).We now give some examples of cylindrical multifractal components.

Example 1.5. Cylindrical multifractal component of a self–similar mea-sure. If µ ∈ SP, and a is a positive number, the multifractal component of themultiplicative function f(C) = µ(C) |C|−a, C ∈ C∗, are useful. For instance, E(f) isgiven by


k→∞k−1

(logµ

(Ei(k)

)− a log∣∣Ei(k)

∣∣) ≥ 0 for all i ∈ π−1(x).

In Lemma 3.3 it is shown that these multifractal components coincide with the setsx ∈ E : lim sup

k→∞

logµ(Ei(k)

)log |Ei(k)| ≤ a for all i ∈ π−1(x)

.

The multifractal components of the form E(Λ) with Λ = f, 1/f, are then the setsx ∈ E : lim

k→∞logµ

(Ei(k)

)log |Ei(k)| = a for all i ∈ π−1(x)

,

which are the cylindrical multifractal components usually considered in the literature.It is also shown in Theorem 3.4 and Corollary 4.3 that the multifractal cylindricalcomponents

x ∈ E : limk→∞

logµ(Ei(k)

)log |Ei(k)| = a for all i ∈ π−1(x)

,

have the dimension of the spherical multifractal components defined byx ∈ E : lim

r→0

logµ(B(π(i), r))log r

= a for all i ∈ π−1(x).

Theorem 2.6 gives the dimension of these multifractal components, and also of inter-sections of these components, and Theorem 3.4 shows that the dimension of a unionof an arbitrary family of multifractal components is the supremum of the dimensionof the components in the family.

Example 1.6. Cylindrical multifractal spectrum of pairs of measures. Wemay also consider multiplicative functions of the form µ(C)λa(C), where µ, λ ∈ SPand a < 0. Theorem 2.6 allows us to obtain the cylindrical multifractal spectrum ofpairs of self–similar measures. A approach to relative multifractality general measurescan be seen in [Co].


Example 1.7. Besicovitch normal sets. Given a real number λ ∈ [0, 1], considerthe multiplicative function defined by

f(Et) = exp(−λ) if t = j , and f(Ej) = exp(1 − λ) .Then k−1 log f

(Ei(k)

)= δj(i, k) − λ, where

δj(i, k) = k−1#q : 1 ≤ q ≤ k and iq = j .Therefore


k→∞δj(i, k) ≥ λ for all i ∈ π−1(x)

,

and

E(f, 1/f) =x ∈ E : lim

k→∞δj(i, k) = λ for all i ∈ π−1(x)

.

Given a positive probability vector p ∈ IRm, let fi(Et) = exp(1 − pi) if t = i, andfi(Et) = exp(−pi) if t = i. Then E(f1, f2, . . . , fm, 1/f1, 1/f2, . . . , 1/fm) is the Besico-vitch normal set associated with p, defined in Moran and Rey as

Bp =x ∈ E : lim

k→∞δj(i, k) = pj , 1 ≤ j ≤ m for all i ∈ π−1(x)

.

It is easy to see that, among the measures in S, only µp can give full measure to this set(this is an easy consequence of the strong law of large numbers), so, as a consequenceof Theorem 2.6, the dimension of Bp is given by dimµp (this result is well–known, see[De]).

Example 1.8. Multifractal spectrum of the Liapunov exponent of thegeometric shift. Take as a multiplicative function f(Et) = a−1rt, a > 0, t ∈ M .Then the multifractal component E(f, 1/f) can be interpreted as the set of pointsx ∈ E where the Liapunov exponent of the shift mapping T takes the value log ain any direction of the tangent space at x (see [Ru] for the definition of Liapunovexponents, and [Mo2] for a computation of the Liapunov exponents of a self–similarmeasure). So the results in this paper allows us to compute the multifractal spectrumof Liapunov exponents.

2. Hausdorff dimension of cylindrical multifractal components

In this section we analyze cylindrical multifractal components. The main resultsare in Theorems 2.6 and 2.8. We start by proving some lemmas which allow us todischarge the proofs of the theorems of some technical details.

Lemma 2.1. Assume that µ is a probability measure with µ(O∗) = 0. Assume alsothat µ is invariant on the finite algebra Ak generated by the set of geometric cylinders

138 Math. Nachr. 229 (2001)

Dk = Ei : i ∈ M q, 1 ≤ q ≤ k (i. e. µ

(T−1(Ei)

)= µ(Ei) for any i ∈ Mk−1

). Let f

be a multiplicative set function defined on Dk. Then∑i∈Mk

µ(Ei) log f(Ei) = k∑i∈M

µ(Ei) log f(Ei) .

Proof . For every q < k we may write∑i∈Mq+1

µ(Ei) log f(Ei) =∑

i∈Mq+1

µ(Ei) log f(Ei(q)∗iq+1

)=

∑i∈Mq+1

µ(Ei) log f(Ei(q)

)+

∑i∈Mq+1

µ(Ei) log f(Eiq+1

)

=∑i∈Mq

[ ∑iq∈M

µ(Ei∗iq

)]log f(Ei)

+∑iq∈M

[ ∑i∈Mq

µ(Ei∗iq

)]log f

(Eiq

).

Since∑

iq∈M µ(Ei∗iq

)= µ(Ei) and, by virtue of the invariance of µ,

∑i∈Mq µ

(Ei∗iq

)=

µ(Eiq

), we get∑i∈Mq+1

µ(Ei) log f(Ei) =∑i∈Mq

µ(Ei) log f(Ei) +∑i∈M

µ(Ei) log f(Ei) ,

which, recursively applied, gives the desired result.

Lemma 2.2. If η is an arbitrary probability measure on E, and f is a function onC∗, then the identity

∫ ∑

i∈π−1(x)

log f(Ei(k)

) dη(x) =∑

i∈Mk

log f(Ei)η(Ei)

holds for all k ∈ IN.

Proof . Every k–cylinder set Ei may be decomposed into a disjoint union of a setAi of atoms of the partition A determined on E by the class Ck of k–cylinders. Thisgives

∫ ∑

i∈π−1(x)

log f(Ei(k)

)dη(x) =∑A∈A

η(A)

∑i∈π−1(x) :x∈A

log f(Ei(k)

)

=∑

i∈Mk

[log f(Ei)

∑A∈Ai

η(A)

]

=∑

i∈Mk

log f(Ei)η(Ei) .


Lemma 2.3. Let η be an invariant probability measure with η(O∗) = 0 and f amultiplicative function on C∗. Then∫

lim infk→∞

k−1 log f(Eix(k)

)dη(x) =

∫limk→∞

k−1 log f(Eix(k)

)dη(x)

=∑t

η(Et) log(f(Et)) .

Proof . Let F : E − O∗ → IR be defined by F (x) = f(Eix(1)

). Observe that, for

x ∈ E − O∗, we may write

k−1 log f(Eix(k)

)= k−1

k−1∑i=0

logF(T i(x)

).(2.1)

Ergodic Theorem of Birkhoff guarantees the existence of a set B ⊆ E − O∗ of fullη–measure and an integrable function F ∗ : B → IR and such that

k−1k−1∑i=0

logF(T i(x)

) −→ F ∗(x)

for every x ∈ B and also that∫F ∗(x) dη(x) =

∫logF (x) dη(x) =

∑t∈M

η(Et) log(f(Et)) .

Notice, that by Lemma 1.1 ii), Lemma 2.3 holds for any η ∈ S even if η ∈ SP.

Lemma 2.4. Let E(f) be as in Definition 1.4 for some multiplicative functionf ∈ R. The following properties hold.

i) If η is a probability measure on E(f) then

lim infk→∞

∑i∈Mk

η(Ei)k−1 log f(Ei) ≥ 0 .

ii) If η is an invariant probability measure on E(f) and η(O∗) = 0, then∑t∈M

η(Et) log f(Et) ≥ 0 .

iii) Assume that for an invariant and ergodic probability measure η on E we haveη(O∗) = 0 and ∑

t∈Mη(Et) log f(Et) ≥ 0 .

Then η(E(f)) = 1.

140 Math. Nachr. 229 (2001)

Proof . Assume that η is a probability measure on E(f). Since f is a multiplicativefunction we have k−1 log f

(Ei(k)

) ≥ min

log f(Et) : t ∈Mfor all i ∈ IM. By Fatou’s

lemma and Lemma 2.2 we may write

0 ≤∫

∑i∈π−1(x)

lim infk→∞

k−1 log f(Ei(k)

)dη(x)

≤ lim infk→∞

k−1

∫ ∑

i∈π−1(x)

log f(Ei(k)

)dη(x)= lim inf

k→∞

∑i∈Mk

η(Ei)k−1 log f(Ei) .

ii) Since η(E(f)) = 1, Lemma 2.3 gives

0 ≤∫

lim infk→∞

k−1 log f(Eix(k)

)dη(x) =

∑t∈M

η(Et) log f(Et) .

iii) By virtue of Lemma 2.3 we have

0 ≤∑t∈M

η(Et) log f(Et) =∫

lim infk→∞

k−1 log f(Eix(k)

)dη(x) .

Since η is ergodic, if we apply to the function f the decomposition (2.1) used inLemma 2.3, we see, by Birkhoff’s ergodic theorem, that the integrand in the lastexpression has to be constant η–a. e., and therefore non–negative, as required.

Lemma 2.5. i) Let Λ ⊆ R, and define the set S(E(Λ)) by

S(E(Λ)) = η ∈ S : η(E(f)) = 1 for all f ∈ Λ .Then S(E(Λ)) is a compact subset of S, Π(S(E(Λ))) is a convex subset of P (seedefinition of Π in Lemma 1.2) and

S(E(Λ)) =

η ∈ S :

∑i∈M

η(Et) log f(Et) ≥ 0 for all f ∈ Λ

.

ii) Let η be a probability measure on E. Then for every k ∈ IN and every α ∈ IRthe set

A(η, k) :=

f ∈ R :

∑i∈Mk

η(Ei)k−1 log f(Ei) > α

is an open set in R.iii) Let M denote the set of real functions on IM endowed with the supremum norm.

Then the mapping B : R → MB(f)(i) = lim inf

k→∞k−1 log f

(Ei(k)

), i ∈ IM


is a continuous mapping.iv) Let Λ be a subset of R and let Adh (Λ) be the closure of Λ in R. Then E(Λ) =

E(Adh (Λ)).v) Let Λ be a subset of R. Then S(E(Λ)) = η ∈ S : η(E(Λ)) = 1.Proof . i) The set S(E(Λ))) is the intersection of the sets S(E(f)), f ∈ Λ. Using

parts ii) and iii) of Lemma 2.4 we see that η ∈ S(E(f)) if and only if∑t∈M η(Et) log f(Et) ≥ 0 holds. The linearity of

∑t∈M η(Et) log f(Et) as a func-

tion of the coordinates η(Et) of η ∈ S shows that S(E(f)) is closed and Π(S(E(f)))is convex, and part i) follows.

ii) This is a consequence of the continuity of the expression∑i∈Mk

η(Ei)k−1 log f(Ei)

with respect to the coordinates of f ∈ R.iii) Let fq be a convergent sequence of functions in R with a limit f ∈ R. Given

ε > 0, there exists a q0 such that

exp(−ε) < f(Et)fq(Et)

< exp(ε)

holds for every t ∈M for all q > q0. Consider the multiplicative functions f∗ and f∗∗

defined by f∗(Et) = f(Et) exp(−ε), f∗∗(Et) = f(Et) exp(ε), t ∈M . They satisfy

k−1 log f(Ei(k)

)− ε = k−1 log f∗(Ei(k)

)< k−1 log fq

(Ei(k)

)< k−1 log f∗∗

(Ei(k)

)= k−1 log f

(Ei(k)

)+ ε

for any i ∈ IM for all k ∈ IN, which shows that d(B (f),B (

fq))< ε for q > q0. This

proves part iii).iv) Let x ∈ E(Λ), and f ∈ Adh (Λ). Then there exists a sequence

fq

of functions

in Λ tending to f , and using part iii), from B(fq

) → B(f), it follows that B(f)(i) =limq→∞B

(fq

)(i) ≥ 0 for i ∈ π−1(x), so x ∈ E(Adh (Λ)).

v) Let Λ ⊆ R, and let Λ′ ⊆ Λ be a countable subset of Λ which is dense in Λ. Bypart iv) we know that E(Λ) = E(Λ′). Let η ∈ S(E(Λ′)). Then η(E(f)) = 1 for everyf ∈ Λ′ and, since Λ′ is countable, η(E(Λ′)) = 1 holds. Also, if η(E(Λ)) = 1 and f ∈ Λ,since E(Λ) ⊆ E(f), η(E(f)) = 1 holds, so η ∈ S(E(Λ)).

Theorem 2.6. Let Λ be a relatively compact subset of multiplicative functions suchthat E(Λ) = ∅. Then

dimE(Λ) = maxdim η : η ∈ S(E(Λ)) .Proof . By part iv) of Lemma 2.5 we may assume that Λ is a compact subset ofR. We

also know, by part i) of the quoted lemma, that S(E(Λ)) is a compact set, and using thecontinuity of the mapping dim : S → IR it is sufficient to prove dimE(Λ) = supdimη :η ∈ S(E(Λ)). We shall prove that S(E(Λ)) = ∅ and dimE(Λ) ≤ supdim η :η ∈ S(E(Λ)) hold since the opposite inequality easily follows from the definitionsand part v) of Lemma 2.5.

142 Math. Nachr. 229 (2001)

Let s = 0 if dimE(Λ) = 0 and let 0 < s < dimE(Λ) if dimE(Λ) > 0. Then asE(Λ) = ∅ , Hs(E(Λ)) > 0 holds in both cases, and by Frostman’s Lemma (see[Mat]) there is a probability measure η on E(Λ) such that dim η ≥ s. We only haveto prove that, given an arbitrary ε > 0, there exists a measure λ(ε)s ∈ S(E(Λ)) withdimλ(ε)s ≥ s−ε (obviously, in the case s = 0, the proof reduces to check that S(E(Λ))is non – empty).

Given k ∈ IN we denote by Pk the group of permutations of the set 1, 2, . . . , k. Fori1i2 . . . ik ∈ Mk and σ ∈ Pk we denote by σ(i) the sequence iσ(1)iσ(2) . . . iσ(k). Thisdefines an action of the group Pk on the set Mk. The orbit of a sequence i ∈ Mk

under this action is the set σ(i) : σ ∈ Pk, and we denote by Ωk the set of all suchorbits. With these notations we define the measure µk as that self–similar measure inS(k) giving to every cylinder set Ei, i ∈ i, i ∈ Ωk the common weight

µk(Ei) = (# i)−1∑j∈i

η(Ej

).

Notice that since ri = rσ(i) for every σ ∈ Pk, we have∑

i∈Mk

η(Ei) log ri =∑

i∈Mk

µk(Ei) log ri .(2.2)

We now prove the following inequality:∑i∈Mk

η(Ei) log η(Ei) ≥∑

i∈Mk

µk(Ei) logµk(Ei) .(2.3)

Since x logx ≤ 0 holds for 0 ≤ x ≤ 1, the problem

Pk ≡ min

∑i∈Mk

xi log xi : 0 ≤ xi ≤ 1 for all i ∈Mk

and∑j∈i

xj =∑j∈i

η(Ej

)for every i ∈ Ωk

can be solved by gathering the solutions of the problems

P (i) ≡ min

∑j∈i

xj log xj : 0 ≤ xj ≤ 1 for all j ∈ i and∑j∈i

xj =∑j∈i

η(Ej

), i ∈ Ωk .

Since the function to be minimized in Pk is the sum of the non–positive functions tobe minimized in the problems P (i), for j ∈Mk, the optimal value of xj for the problemPk is given by the optimal value of xj for the unique problem P (i) such that j ∈ i.The optimal solution for each such problem is well–known to be attained if xj = xi,for all i, j ∈ i, which shows that (2.3) holds.

Our next step is to prove that, given an arbitrarily small ε > 0

s− ε ≤ dimµk(2.4)


holds for k large enough. If we denote by B(x, r) a closed ball centered at x and withradius r, then it is well–known that

lim infr→0

log η(B(x, r))log r

≥ s

holds for η–almost every x ∈ E. For any i ∈ IM and any q, Ei(q) ⊆ B(π(i), ri(q)

)holds.

From this it follows that

s ≤ lim infk→∞

log η(Ei(k)

)log ri(k)

holds for all i ∈ π−1(x) for η–a. e. x. Hence

0 ≤∑

i∈π−1(x)

[(s− 2−1ε

)log ri(k)− log η

(Ei(k)

)]

holds η–a. e. x for k large enough. Therefore we also have

0 ≤ lim infk→∞

∑i∈π−1(x)

[(s− 2−1ε

)log ri(k)− log η

(Ei(k)

)]

η–a. e. x. Using Fatou’s lemma, Lemma 2.2, (2.2) and (2.3), we have

0 ≤ lim infk→∞

∫ ∑i∈π−1(x)

[(s− 2−1ε

)log ri(k) − log η

(Ei(k)

)]dη(x)

= lim infk→∞

∑i∈Mk

[(s− 2−1ε

)η(Ei(k)

)log ri(k)− η

(Ei(k)

)log η

(Ei(k)

)]≤ lim inf

k→∞

∑i∈Mk

[(s− 2−1ε

)µk(Ei) log ri − µk(Ei) logµk(Ei)

].

Therefore, using Lemma 1.1 iii), for any δ > 0,

dimµk =∑

i∈Mk µk(Ei) logµk(Ei)∑i∈Mk µk(Ei) log ri

≥ δ∑i∈Mk µk(Ei) log ri

+ s− 2−1ε

holds for k large enough. Since∑

i∈Mk µk(Ei) log ri ≤ k logR where R = max ri :i ∈M, we get dimµk ≥ δ/(k logR) + s− 2−1ε for large k, and dimµk ≥ s− ε holdsfor large k.

Our next step is to construct, using the measures µk, a measure λ(ε)s ∈ S(E(Λ)) withdimλ(ε)s ≥ s − ε. Observe that, for any k, the measure µk is invariant on the finitefield Ak generated by the cylinders in the class Dk (see definitions in Lemma 2.1). Weprove this assertion. Given Ei with i ∈ M q and q < k, setting M0 = ∅ if q = k − 1,we have

µk(T−1(Ei)

)= µk

m⋃

j=1

Ej∗i

= µk

m⋃

j=1

⋃k∈Mk−q−1

Ej∗i∗k

=

144 Math. Nachr. 229 (2001)

=m∑j=1

∑k∈Mk−q−1

µk(Ej∗i∗k

)=

m∑j=1

∑k∈Mk−q−1

µk(Ei∗j∗k

)= µk(Ei) .

The field Ak is also generated by the class of sets Eq,t : 1 ≤ q ≤ k, 1 ≤ t ≤ m,where Eq,t = i ∈ IM : iq = t. Think of the classes of sets Eq = Eq,t : 1 ≤ t ≤ m aspartitions of E. Then the generating partition P (Ak) consisting of all intersections,

k⋂i=1

Ai , Ai ∈ Ei , 1 ≤ i ≤ k ,

which coincides with Ck, is known to have an entropy

H(α, P (Ak)) = −∑

i∈Mk

α(Ei) logα(Ei)

with respect to a given probability measure α which satisfies

H(α, P (Ak)) ≤k∑

i=1

H(α, Ei)

(see Walters, Theorem 4.3). Furthermore, if α is a measure invariant on Ak, sinceT−q+1(E1) = Eq , H(α, Eq) = H(α, E1) = H(α, C1), q = 2, . . . , k hold (see quoted ref.).Thus H(α, P (Ak)) ≤ kH(α, C1), which for α = µk gives∑

i∈Mk

µk(Ei) logµk(Ei) = −H(µk, P (Ak))

≥ −kH(µk, C1)= k

∑i∈M

µk(Ei) logµk(Ei) .

(2.5)

By Lemma 2.1 we have

k∑i∈M

µk(Ei) log ri =∑

i∈Mk

µk(Ei) log ri .(2.6)

Using (2.5) and (2.6) we get

dimµk =∑

i∈Mk µk(Ei) logµk(Ei)∑i∈Mk µk(Ei) log ri

≤∑

i∈M µk(Ei) logµk(Ei)∑i∈M µk(Ei) log(ri)

.(2.7)

Let λk ∈ S be the probability measure defined by λk(Et) = µk(Et), t ∈ M . By (2.7)we know that dimλk ≥ dimµk, so for large k, from (2.4) it follows

dimλk ≥ s− ε .(2.8)

Using part i) of Lemma 2.4 we see that given a j ∈ IN and an f ∈ Λ,

−1j<

∑i∈Mk

η(Ei)k−1 log f(Ei) =∑

i∈Mk

µk(Ei)k−1 log f(Ei)(2.9)


holds for large k. Part ii) of Lemma 2.5 shows that the set of functions f ∈ R forwhich (2.9) holds for a given k is open. Therefore the family of such sets, obtainedletting f and k range respectively over Λ and IN, is an open covering of the compactset Λ, which shows that there exists a K(j) for which (2.9) holds for every f ∈ Λ ifk ≥ K(j). Now, Lemma 2.1 gives∑

t∈MλK(j)(Et) log f(Et) = K(j)−1

∑i∈MK(j)

µK(j)(Ei) log f(Ei) > −1j

(2.10)

for all f ∈ Λ. We may assume, choosing a suitable subsequence of the sequence of self–similar measures

λK(j)

j∈IN which we also denote by

λK(j)

j∈IN, that limj λK(j)(Et)

exists for every t ∈ M . Let λ(ε)s ∈ S be defined by λ(ε)s (Et) = limj λK(j)(Et), t ∈ M .Then by (2.8) the measure λ(ε)s satisfies dimλ(ε)s ≥ s−ε. Furthermore if we take limitswhen j tends to ∞ in the first and third terms in (2.10) we have∑

t∈Mλ(ε)s (Et) log f(Et) ≥ 0 , f ∈ Λ ,

which, by part iii) of Lemma 2.4, shows that λ(ε)s ∈ S(E(Λ)). Since ε can be takenarbitrarily small, this shows that s ≤ supdim η : η ∈ S(E(Λ)), and the proof iscompleted.

Corollary 2.7. Given a relatively compact set Λ ∈ R such that E(Λ) = ∅ we have

i) dimE(Λ) = max∑

t∈M xt log xt∑t∈M xt log rt

,

where the maximum is taken over the xt, t = 1, 2, 3, . . . , m, which satisfya)

∑t∈M xt = 1, xt ≥ 0, for t = 1, 2, . . . , m.

b)∑

t∈M xt log f(Et) ≥ 0, for all f ∈ Λ.

ii) We may replace E(Λ) withE(Λ) in Theorem 2.6 and part i) above, replacing inthis last case “≥” with “≤” in b).

iii) If E(Λ) = ∅, then we may replace E(Λ) with E(Λ) in Theorem 2.6 and part i)above, replacing in this last case “≥” by “=” in b).

Proof . Part i) is only a restatement of the theorem in terms of the characterizationof S(E) given in Lemma 2.5 i). Part ii) follows from the theorem and part i) of thecorollary applied to the multiplicative functions 1/f : f ∈ Λ. Part iii) follows fromi) and ii).

Theorem 2.8. LetΛα

α∈I be an arbitrary family of relatively compact sets of

multiplicative functions with⋃α∈I E

(Λα

) = ∅. Then

dim⋃α∈I

E(Λα

)= sup

dimE

(Λα

): α ∈ I .

Proof . We may assume all sets E(Λα) to be non–empty. We know by Corollary 2.7that the dimension of each E(Λα) is given by the dimension of some self–similar

146 Math. Nachr. 229 (2001)

measure in S(E(Λα)). Let us regard the measures µp ∈ S as points in P by identifying

each µp to the probility vector p. Then we may regard S(E(Λα)) as the polyhedron⋂f∈Λα

H(f), where H(f) denotes the intersection of P and the half–spacex ∈ IRm :∑

t∈M xt log f(Et) ≥ 0. We now prove that we can approximate in the Hausdorff

metric any set S(E(Λα)), α ∈ I, by a suitable polyhedron taken from a countable andindependent from α family of polyhedrae. Let ε > 0 and let Λ denote a given set Λα,α ∈ I. Notice that, since IRn is a separable metric space, we can find a countableand dense subset Ξ ⊆ Λ. Furthermore, we can find an increasing sequence Ξnof finite subsets of Ξ such that the sequence of compact sets S(Ξn) decreases toS(Λ). The sequence dH(S(Ξn),S(Λ)), where dH denotes the Hausdorff metric, tendsto zero. Therefore, there exists a set S(Ξn) ⊇ S(Λ), with dH(S(Ξn),S(Λ)) < ε/2.The set S(Ξn) is a convex polyhedron with faces of the form H(fi), with fi ∈ Λfor 1 ≤ i ≤ k, and it is easy to see that there exists a set ∆ = gi : 1 ≤ i ≤ kof multiplicative functions with rational coordinates gi(Et), t ∈ M , and a convexpolyhedron S(∆) =

⋂ki=1H(gi) satisfying dH(S(∆),S(Ξn)) < ε/2 and S(Ξn) ⊆ S(∆).

Therefore, for each Λα we may find a polyhedron S(∆α) taken from a countableand independent from α family of polyhedrae such that dH(S(∆α),S(Λα)) < ε andS(Λα) ⊆ S(∆α). Notice that by the uniform continuity of the mapping dim : S → IR,we may also require

dimE(∆α) < dimE(Λα) + ε .

Hence we have

dim⋃α∈I

E(Λα) ≤ dim⋃α∈I

E(∆α) = supdimE(∆α) : α ∈ I

≤ supdimE(Λα) : α ∈ I + ε

where the equality in the string holds because the set ∆α : α ∈ I is countable even ifI is non–countable. Letting ε tend to zero we get dim

⋃α∈I E(Λα) ≤ supdimE(Λα) :

α ∈ I. The opposite inequality is trivial.

Remark 2.9. Let η = µp ∈ S(E(Λ)) for some relatively compact subset Λ ofmultiplicative functions, and let f ∈ Λ. We may write

k−1 log f(Ei(k)

)=

m∑t=1

δt(i, k) log f(Et) , for i ∈ IM.

For a point π(i) ∈ Bp, δt(i, k) → pt = η(Et), so

k−1 log f(Ei(k)

) −→m∑t=1

η(Et) log f(Et)

which, by Lemma 2.4, shows that π(i) ∈ E(f). Therefore, Bp ⊆ E(Λ), and thenthe dimension of the multifractal components is the maximum of the dimensions ofthe Besicovitch normal sets they contain. As a particular case of Theorem 2.8, thedimension of an arbitrary union of Besicovitch sets is the supremum of the Besicovitchsets in the union.


Remark 2.10. Notice that the function to be maximized in Corollary 2.7 is thequotient of a strictly convex function by a linear negative function, so it is itself astrongly quasiconcave function. To see this, let D : P → IR be defined by D(x) =H(x)/G(x), with H(x) =

∑i∈M xi log xi convex and G(x) =

∑i∈M xi log ri linear.

Then, for non–negative α and β with α+ β = 1, and for x, y ∈ P, x = y, we have

D(αx + βy) >αH(x) + βH(y)αG(x) + βG(y)

≥ minαH(x)αG(x)

,βH(y)βG(y)

= minD(x), D(y) ,

as required. Given a multifractal component E(Λ) corresponding to a finite set Λ ofmultiplicative functions, since the maximum of a strongly quasiconcave function ona convex set is attained at a unique point (see [Ba]), if we regard the coordinates ofthe functions in Λ as parameters in the set of constraints, the dimension dimE(Λ) isa smooth function of these parameters. This is a standard technique in optimizationtheory. It is well–known that the solution of such problems are smooth functions ofthe parameters, and their derivatives can be obtained as a function of the parameters(see quoted reference).

Corollary 2.11. Let Λ be a relatively compact subset of multiplicative functions,and define E∗(Λ) by

E∗(Λ)=x ∈ E : lim inf

k→∞k−1 log f

(Ei(k)

)= 0 for all f ∈ Λ and for all i ∈ π−1(x)

.

If E∗(Λ) = ∅ then dimE∗(Λ) = dimE(Λ).

Proof . Since E(Λ) ⊆ E∗(Λ), we only have to prove dimE∗(Λ) ≤ dimE(Λ). Theset Λ is bounded, so we may find an α such that lim supk→∞ k−1 log f

(Ei(k)

) ≤ α

for all f ∈ Λ and all i ∈ IM. Hence E∗(Λ) ⊆ E(Λ′) if Λ′ =⋃

f∈Λf, exp(α)f−1,

where exp(α)f−1 is the multiplicative function which gives weight exp(α)/f(Et) tothe cylinder Et, t ∈M . Let Λ∗ be a countable and dense subset of Λ′. The dimensionof E(Λ′) is given by the dimension of a self–similar measure η ∈ S(Λ∗). By Birkhoff’sergodic theorem, for η–a. e. π(i) ∈ E,

0 = lim infk→∞

k−1 log f(Ei(k)

)= lim

k→∞k−1 log f

(Ei(k)

)holds for all f ∈ Λ∗. This shows that η ∈ S(E(Λ∗)) so, using that Λ∗ ∩ Λ is dense inΛ we have

dimE∗(Λ) ≤ dimE(Λ′) = dim η ≤ dimE(Λ∗) ≤ dimE(Λ∗ ∩ Λ) = dimE(Λ) .

3. Spherical and cylindrical densities

We show in this section that cylindrical densities do not coincide with sphericaldensities in negligible subsets, and we obtain estimates of the size of the exceptionalsubsets. In Theorem 3.4 we show how these results may be applied to extend to

148 Math. Nachr. 229 (2001)

spherical multifractal components the results of Section 2. We first start with twobasic lemmas used later on.

Lemma 3.1. i) Given a subset A ⊆ E, a natural number q and an x ∈ E −O∗, let

κ(x, A, q) = mink : T q+k(x) ∈ A

.(In the case when T q(x) ∈ A, we set κ(x, A, q) equal to zero, and we set κ(x, A, q)equal to infinity in the case when T q+k(x) /∈ A for any k

). Let η be an invariant

measure such that η(V ) = 1 (see 1.2 for a definition of V ). Let

Γ =x ∈ E −O∗ : for some p, lim

q→∞ q−1κ(x, Ap, q) = 0

,

where Ap = x ∈ V : d(x, ∂V ) > 1/p and Ap = E if ∂V = ∅. Then η(Γ) = 1.ii) Let Ω be defined by

Ω = x ∈ E − O∗ : for all p there exists a q such that T i(x) ∈ Ap for i > q .Then dimΩ = dim∂V .

Proof . i) Notice that O = O∩SΨ(E) = (O∩SΨ(V ))∪ (O∩SΨ(∂V )), and since, asremarked in Section 1.2, ψi(V ) ∩O = ∅, i ∈ M , we see that O ⊆ SΨ(∂V ). Therefore,from η(∂V ) = 0 it follows η(O) = 0, and by the invariance of η, η(O∗) = 0. We assumethat the set H = E − (Γ ∪O∗) has a measure η(H) = α > 0, and we show that thisgives a contradiction. Since V =

⋃p∈IN Ap and η(V ) = 1, there exists a p such that

η(Ap) > 1 − α. By Birkhoff’s ergodic theorem, if we write ξq(x) for the expressionq−1

∑qi=1 IAp(T i(x)) then limq→∞ ξq(x) exists for η–a. e. x. If ξ∗(x) denotes this limit,

then ∫ξ∗(x) dη(x) = η(Ap) .(3.1)

Let x ∈ H with ξq(x) → ξ∗(x). Since x ∈ H , lim supq→∞ q−1κ(x, Ap, q) > 0. There-fore we may find an ε > 0 such that q−1κ(x, Ap, q) > ε for infinitely many val-ues of q. Let q > 2/ε be one of such values. From the definitions it follows thatξq+k(x) = q(q + k)−1ξq(x) for some k > εq − 1, so

ξq+k(x) < q(q + εq − 1)−1ξq(x) <(

1 +ε

2

)−1

ξq(x) .

This shows thatξq(x)

q∈IN can be a Cauchy sequence only in the case that it con-

verges to zero. Consider the sets

Hp :=y ∈ H : ξq(y) → 0

,

Lp :=x ∈ H : lim inf

qξq(x) < lim sup

qξq(x)

.

By the argument given above we see that H = Hp∪Lp. By Birkhoff’s ergodic theorem,η(Lp) = 0. By the definitions we know that ξ∗(x) = 0 for x ∈ Hp, and ξ∗(x) ≤ 1


for η–a. e. x ∈ E, so∫Hξ∗(x) dη(x) = 0 and

∫Γξ∗(x) dη(x) ≤ η(Γ). Thus, using the

inequality η(Ap) > 1 − α and (3.1), we get the contradiction

1 − α <

∫ξ∗(x) dη(x) =

∫Γ

ξ∗(x) dη(x) ≤ η(Γ) = 1 − α .

ii) In [Mo1] it is proved that ∂V =⋂∞

k=1〈Gk〉, where 〈Gk〉 denotes the self–similarset generated by the system of similitudes Gk = ψi : i ∈ Mk, Ei ∩ ∂V = ∅. If,for x ∈ E − O∗, T i(x) ∈ Ap for i > q then T q(x) belongs to the self–similar set Fp,kgenerated by the system of similitudesFp,k = ψi : i ∈Mk and Ei∩(E−Ap) = ∅ and,for all k ∈ IN, x ∈ SΨq(Fp,k). Therefore, if SΨ∗ denotes the set mapping

⋃q∈IN SΨq ,

we have

Ω ⊆∞⋂p=1

SΨ∗(Fp,k)

for all k ∈ IN. Given k ∈ IN, consider the class of indices Ak = i ∈Mk, Ei∩∂V = ∅.Since Ap increases to V when p tends to infinity,

⋃i∈Ak

Ei ⊆ Ap for a sufficiently largep. If Ei ∩ (E − Ap) = ∅, then i ∈ Ak, so, Fp,k ⊆ Gk for a large enough p. Thus wemay write

Ω ⊆∞⋂k=1

∞⋂p=1

SΨ∗(Fp,k) ⊆∞⋂k=1

SΨ∗(〈Gk〉) ,which shows that dim Ω ≤ dimSΨ∗(〈Gk〉) = dim〈Gk〉 for all k. In [Mo1] it is shownthat limk→∞ dim〈Gk〉 = dim ∂V holds. Therefore we have

dimΩ ≤ limk→∞

dim〈Gk〉 = dim∂V .

Since ∂V −O∗ ⊆ Ω and dimO∗ ≤ dim∂V , the opposite inequality follows.

Recall that B(x, r) denotes the ball of E as a metric subspace of IRn centered at xand with radius r.

Lemma 3.2. Let µ and λ be monotonic set functions defined on the σ–algebra ofBorel subsets of E, and assume that λ is multiplicative on the subclass C∗ of geometriccylinder sets and 0 < λ(Et) < 1, t ∈ M .

i) If x = π(i) ∈ Γ (see definition of Γ in Lemma 3.1) we have

lim supq→∞

logµ(Ei(q)

)logλ

(Ei(q)

) = lim supr→0

logµ(B(x, r))logλ(B(x, r))

.

ii) If we interchange the hypotheses on λ and µ then

lim infq→∞

logµ(Ei(q)

)logλ

(Ei(q)

) = lim infr→0


holds for every x = π(i) ∈ Γ.

150 Math. Nachr. 229 (2001)

iii) If µ is a monotonic set function and x = π(i) ∈ E − (Ω ∪O∗), then

limr→0

logµ(B(x, r))log r

= lim infq→∞

logµ(Ei(q)

)log |Ei(q)|

if the limit in the left–hand term exists, and

lim supr→0

logµ(B(x, r))log r

= limq→∞

logµ(Ei(q)

)log |Ei(q)|

if the limit in the right–hand term exists.

The main applications of this lemma are to the case when both multiplicative setfunctions λ and µ are in SP or µ is a measure and λ is chosen to be the function thatgives the diameter of the subsets of E, as in part iii).

Proof . i) Let i ∈ IM be such that π(i) = x ∈ Γ. We know that there exists anatural number p such that limq→∞ q−1κ(x, Ap, q) = 0, where Ap is as in Lemma 3.1.Let q ∈ IN. If we abbreviate κ(x, Ap, q) to κq, then y := T q+κq (x) ∈ Ap. From thedefinition of the set Ap we know that B

(y, p−1

) ⊆ V . From property iii) of the set Vgiven in Section 1.2 it follows that

B(x, p−1ri(q+κq)

)= ψi(q+κq )

(B(y, p−1

)) ⊆ Ei(q+κq) ⊆ Ei(q) .(3.2)

Therefore

logµ(Ei(q)

)logλ

(Ei(q)

) ≤ logµ(B(x, p−1ri(q+κq)

))logλ

(Ei(q)

) .(3.3)

Observe also that if U = maxri : i ∈M and K = mink : k logU < − log p then

ri(q+κq+K) < p−1ri(q+κq) .

So, taking into account that |E| = 1, we have∣∣Ei(q+κq+K)

∣∣ < p−1ri(q+κq). Hence,from the monotonicity of λ it follows

λ(Ei(q+κq+K)

) ≤ λ(B(x, p−1ri(q+κq)

)).

Using that

logλ(Ei(q)

)+ (κq +K)uλ ≤ logλ

(Ei(q+κq+K)

),

where uλ = minlogλi : i ∈M, and (3.3) we get

logµ(Ei(q)

)logλ

(Ei(q)

) ≤ logµ(B(x, p−1ri(q+κq)

)logλ

(B(x, p−1ri(q+κq)

) − (κq +K)uλ.(3.4)

Since x ∈ Γ and, by (3.2)

logλ(B(x, p−1ri(q+κq)

) ≤ logλ(Ei(q)

) ≤ qUλ ,


where Uλ = maxlogλ(Ei) : i ∈M, holds, we have

lim supq→∞

logµ(Ei(q)

)logλ

(Ei(q)

) ≤ lim supr→0


.

We now obtain an inequality in the opposite direction. Given any r > 0 let

qr = maxq ∈ IN : r < p−1ri(q+κq)

.

The value qr is well defined for any sufficiently small r. If K is defined as above andwe write q∗r for qr + 1 + κqr+1 we have

ri(q∗r+K) < p−1ri(q∗r ) < r < p−1ri(qr+κqr ).

Therefore, for a sufficiently small r, Ei(q∗r+K) ⊆ B(x, r) ⊆ Ei(qr) holds. Then, usingthat

λ(Ei(qr)

)+

(1 + κqr+1 +K

)uλ ≤ logλ

(Ei(q∗r+K)

)we get


≤ logµ(Ei(q∗r+K)

)logλ

(Ei(q∗r+K)

)− (1 + κqr+1 +K

)uλ.

By a similar argument to that given above this gives

lim supr→0


≤ lim supq→∞

logµ(Ei(q)

)logλ

(Ei(q)

) .ii) If the hypotheses on µ and λ are interchanged, we may write for x = π(i) ∈ Γ,

a = lim infr→0


=[lim supr→0

logλ(B(x, r))logµ(B(x, r))

]−1

=

[lim supq→∞

logλ(Ei(q)

)logµ

(Ei(q)

)]−1

= lim infq→∞

logµ(Ei(q)

)logλ

(Ei(q)

) .iii) If π(x) ∈ E − (Ω ∪ O∗), then there exists a p such that T q(x) ∈ Ap infinitely

often. If we apply the argument used in i) to obtain (3.4) for κq = 0 we get for thesevalues of q

logµ(Ei(q)

)log

∣∣Ei(q)

∣∣ ≤ logµ(B(x, p−1ri(q)

)log

∣∣B(x, p−1ri(q)

)∣∣−K log ρ

where ρ = minri : i ∈M. Hence

lim infq→∞

logµ(Ei(q)

)log |Ei(q)| ≤ lim

r→0

logµ(B(x, r))log r

(3.5)

if the limit exists, and

limq→∞

logµ(Ei(q)

)log |Ei(q)| ≤ lim sup

r→0

logµ(B(x, r))log r

(3.6)

152 Math. Nachr. 229 (2001)

if the limit exists. On the other hand, the monotonicity of µ gives

logB(π(i), ri(q)

)log ri(q)

≤ logµ(Ei(q)

)log

∣∣Ei(q)

∣∣for all q. Hence

lim infq→∞

logµ(Ei(q)

)log

∣∣Ei(q)

∣∣ ≥ limr→0

logµ(B(x, r))log r

if the limit exists, which together with (3.5) gives the first statement, and

limq→∞

logµ(Ei(q)

)log

∣∣Ei(q)

∣∣ ≥ lim supr→0

logµ(B(x, r))log r

if the limit exists, which together with (3.6) gives the second statement.

The following lemma links up the sets considered in Lemma 3.2 with the cylindricalmultifractal components used in Section 2.

Let R∗ be the set of multiplicative functions defined by

R∗ =µ | · |−α : µ ∈ SP , α > 0

.

Let Λ ⊆ R∗. We define the sets:

C (Λ) =

x ∈ E : lim sup

k→∞

logµ(Ei(k)

)log

∣∣Ei(k)

∣∣ ≤ α for all µ | · |−α ∈ Λ

and for all i ∈ π−1(x)

,

C(Λ) =

x ∈ E : lim inf

k→∞logµ

(Ei(k)

)log

∣∣Ei(k)

∣∣ ≥ α for all µ | · |−α ∈ Λ


,

C(Λ) =

x ∈ E : lim

k→∞logµ

(Ei(k)

)log

∣∣Ei(k)

∣∣ = α for all µ | · |−α ∈ Λ


.

Lemma 3.3. Let Λ ⊆ R∗. Then

C(Λ) = E(Λ) , C(Λ) = E(Λ) and C(Λ) = E(Λ) .

Proof . We only prove thatC(Λ) = E(Λ) holds. Let i ∈ π−1(x) for an x ∈ C(Λ),and let µ | · |−α ∈ Λ . Then, for any ε > 0,

k−1 log(µ(Ei(k)

)∣∣Ei(k)

∣∣−α−ε)> 0


holds for large k, which shows that

lim infk→∞

k−1 log(µ(Ei(k)

)∣∣Ei(k)

∣∣−α−ε)

≥ 0 .

From part iii) of Lemma 2.5 it follows that

limε→0

lim infk→∞

k−1 log(µ(Ei(k)

)∣∣Ei(k)

∣∣−α−ε)

= lim infk→∞

k−1 log(µ(Ei(k)

)∣∣Ei(k)

∣∣−α)

≥ 0 ,

which proves that x ∈ E(Λ). Let i ∈ π−1(x) for an x ∈ E(Λ), and let µ | · |−α ∈ Λ.Then, given ε > 0,

k−1 logµ(Ei(k)

)> αk−1 log

∣∣Ei(k)

∣∣− εholds for large k. Since k−1 log

∣∣Ei(k)

∣∣ < C := max log |Et| this gives

logµ(Ei(k)

)log

∣∣Ei(k)

∣∣ < α− ε

C

for large k, as required. This shows that x ∈ C(Λ).

Theorem 3.4. Let Λ ⊆ R∗ be a relatively compact set of multiplicative functions.Let

M =

x ∈ E : lim inf

r→0

logµ(B(π(i), r))log r

≥ α for all µ | · |−α ∈ Λ


.

Then the following properties holdi) If dimM(Λ) > dim∂V then dimM(Λ) = dimE(Λ).ii) Let M(Λ) be defined replacing “ lim inf” with “ lim” in the above definition of

M(Λ) and assume that dim(M(Λ)) > dim∂V . Then

dimM(Λ) = dimE(Λ) .

iii) If Λαα∈I is an arbitrary family of relatively compact sets of multiplicativefunctions and dimM(Λα) > dim∂V holds for all α ∈ I, then

dim⋃α∈I

M(Λα) = supdimM(Λα) : α ∈ I .

If dimM(Λα) > dim∂V holds for all α ∈ I then

dim⋃α∈I

M(Λα) = supdimM(Λα) : α ∈ I .

Proof . i) From the monotonicity of the measures in S it easily followsM(Λ) ⊆ C(Λ),and by Lemma 3.3 we know that C(Λ) =E(Λ). Therefore, dim M(Λ) ≤ dimE(Λ)

154 Math. Nachr. 229 (2001)

holds. We now prove the opposite inequality. By Theorem 2.6 and Lemma 2.5 v)we know that there exists a η ∈ S(E(Λ)) with dim η = dimE(Λ) and η(E(Λ)) = 1.Since dimE(Λ) ≥ dimM(Λ) > dim∂V holds, we get η(∂V ) = 0, so by Lemma 3.1 i),η(Γ) = 1 holds. Hence, by Lemma 3.2, we have

dimE(Λ) = dim η ≤ dim(E(Λ) ∩ Γ

)= dim(M(Λ) ∩ Γ) ≤ dim M(Λ) ,

and part i) follows.ii) By Lemma 3.2 iii), M(Λ) − (Ω ∪ O∗) ⊆ E∗(Λ) holds (see Corollary 2.11 for a

definition of E∗(Λ)). Since dimM(Λ) > dim∂V , Lemma 3.1 ii) gives dim(M(Λ) −(Ω ∪O∗)) = dimM(Λ). Hence, by Corollary 2.11 we get

dimM(Λ) = dim(M(Λ)− (Ω ∪O∗)) ≤ dimE∗(Λ) = dimE(Λ) .

By Lemma 3.2 i) and ii) E(Λ) ∩ Γ ⊆ M(Λ). An analogous argument to that givenin proving part i) above shows that dim(E(Λ) ∩ Γ) = dimE(Λ). Hence we also havedimE(Λ) ≤ dimM(Λ).

iii) We know that for every α ∈ I, M(Λα) ⊆E(Λα) holds. Then, by Theorem 2.8and part i) above we have

dim⋃α∈I

M(Λα) ≤ dim⋃α∈IE (Λα) = supdimE(Λα) : α ∈ I

= supdimM(Λα) : α ∈ Iand this together with the trivial inequality

dim⋃α∈I

M(Λα) ≥ supdimM(Λα) : α ∈ I ,

proves the first statement in part iii). The second statement is obtained in an analogousway.

Remark 3.5. Part ii) in the above theorem is known to hold when Λ is reduced toa singleton (see [Pa]) even if we drop the assumption dimM(Λ) > dim∂V . It seemsthat this result requires global properties of the packings of the sets M(Λ), and itcannot be derived from the pointwise approach to cylindrical and spherical densitiesadopted in this section. We use Patche’s techniques to extend part ii) to the generalcase in Corollary 4.3. Under the assumptions of the theorem, the monotonicity ofmeasures gives E(Λ) ⊆ M(Λ), so dimE(Λ) ≤ dimM(Λ) holds. It would be nice tobe able to prove that the equality also holds, at least if dimE > dim∂V . This couldbe done by proving that dim

(M(Λ) ∩ Γ

)= dimM(Λ) holds, which could be derived

from the statement dimΓ ≤ dim∂V , a strengthening of Lemma 3.1. This is likely tobe true, although not easy to prove.

4. Legendre transforms

In this section we show that the classical approach to multifractality, based on thethermodynamical formalism and Legendre transforms, can be applied to compute the


multifractal spectrum of the sets E(Λ). This gives an additional information on themeasure of S(E(Λ)) for which the dimension of E(Λ) is attained, since a more explicitexpression of this measure in terms of the measures in Λ can be obtained (see expression(4.4)).

We start observing that by Corollary 2.7 iii) we may assume, without loss of gener-ality, that Λ is a finite set of multiplicative functions. In fact, either the cardinality ofΛ may be reduced to n − 1 by suppression of redundant multiplicative functions, orS(E(Λ)) is empty, and in consequence E(Λ) must be empty.

It is well–known that if µ ∈ SP and µ(Et) = pt, 1 ≤ t ≤ m, and φ : IR2 → IR isgiven by

φ(q, β) =m∑t=1

pqtrβt ,

the equality φ(q, β) = 1 implicitly defines β as a convex function β(q) on IR, withβ(−∞) = ∞ and β(∞) = −∞, and then

ϕ(α) := dim(E(µ | · |−α

))= Dim

(E(µ | · |−α

))is given by the Legendre transform of β, defined on the interval

[−β′(∞),−β′(−∞)].Let λi ∈ SP, for 1 ≤ i ≤ k, and write pti for λi(Et), 1 ≤ t ≤ m, 1 ≤ i ≤ k. Let

q := (q1, q2, . . . , qk). We define the real function Φ : IRk+1 → IR by

Φ(q, β) =m∑t=1

k∏i=1

pqi

ti rβt .

Lemma 4.1. For every q ∈ IRk there exists a unique β(q) such that Φ(q, β(q)) = 1holds. The function β(q) is convex on q, and there exists a subset D ⊆ IRk such thatfor every a := (a1, a2, . . . , ak) ∈ D the minimimum

min

k∑

i=1

aiqi + β(q) : q ∈ IRk

is attained.

Proof . The function Φ(q, β) defined on IRk+1 is easily seen to be strictly decreasingon β, with Φ(q,−∞) = ∞, Φ(q,∞) = 0, and it is a convex function convex on(q, β), since the sumands are convex functions on (q, β) (take logarithms to checkthis). Therefore Φ(q, β) = 1 implicitly defines β as a function β(q). Let Γβ be theepigraph of β(q), defined by

Γβ = (q, y) : y ≥ β(q) .From the convexity of Φ it follows that the set (q, y) : Φ(q, y) ≤ 1 is convex, andsince Φ is decreasing in β, Φ(q, y) ≤ 1 if and only if y ≥ β(q). This shows that Γβ

convex, so that β(q) is convex on q. By convexity, min∑k

i=1 aiqi + β(q) : q ∈ IRk

is attained if and only if

grad q

(k∑

i=1

aiqi + β(q)

)= 0 ,

156 Math. Nachr. 229 (2001)

which happens to occur when the equalities

∂β(q)∂qi

= −ai , 1 ≤ i ≤ k ,(4.1)

are satisfied. So, the minimum is attained if a ∈ D :=− grad (β(q)) : q ∈ IRk

.

Notice that, by differentiation of the equality Φ(q, β(q)) = 1, we get

∂β

∂qi= −

∑mt=1 log pti

∏kj=1 p

qj

tjrβ(q)t∑m

t=1 log rt∏k

j=1 pqj

tjrβ(q)t

.(4.2)

Hence, if the Jacobian J(−grad (β(q))) := det((

−∂2β(q)∂qi∂qj

))does not vanish for all

q ∈ IRk then D contains the open subset

−grad (β(q)) : J(−grad (β(q))) = 0 .On the other hand it is easy to check that D ⊆ R where R is the rectangle

R =a ∈ IRk : min

t∈M

log ptilog rt

≤ ai ≤ max

t∈M

logptilog rt

.

We are now ready to prove the main result of this section.

Theorem 4.2. If a ∈ D and

Λa =λi | · |−ai : 1 ≤ i ≤ k ,(4.3)

then the Legendre transform of β(q) defined by

ϕ(a) := min

k∑i=1

aiqi + β(q) : q ∈ IRk

gives the Hausdorff and packing dimensions of E(Λa).

Proof . Let us abbreviate Λa to Λ and let η ∈ S be defined by

η(Et) =k∏

i=1

pqi

tirβ(q)t for t ∈M ,(4.4)

where q satisfies (4.1). From (4.1) and (4.2) it easily follows that dim η = ϕ(a), whichshows that ϕ(a) ≥ 0. We now prove that dim(E(Λ)) ≥ dim η holds. Let Bη be theBesicovitch normal set associated to η (i. e. Bη := Bp for pt = η(Et), 1 ≤ t ≤ m).Then, if π(i) ∈ Bη for i ∈ IM, by Remark 2.9, (4.1) and (4.2) we have

limk→∞

k−1 logλi(Ei(k)

)∣∣Ei(k)

∣∣−ai =m∑t=1

η(Et) log pti − aim∑t=1

η(Et) log rt

=m∑t=1

k∏i=1

pqi

ti rβ(q)t log pti − ai

m∑t=1

k∏i=1

pqi

ti rβ(q)t log rt

= 0


for all 1 ≤ i ≤ k. Therefore Bη ⊆ E(Λ) and then η ∈ S(E(Λ)), which gives dimE(Λ) ≥dim η. Since dimE(Λ) ≤ DimE(Λ) we only have to prove that DimE(Λ) ≤ ϕ(a)holds. Let x = π(i) ∈ E(Λ) for i ∈ IM. Then, by Lemma 3.3 and the monotonicity ofη we have

ϕ(a) =k∑i=1

aiqi + β(q) = β(q) +k∑i=1

qi limj→∞

logλi(Ei(j)

)log

∣∣Ei(j)

∣∣ = limj→∞

log η(Ei(j)

)log

∣∣Ei(j)

∣∣≥ lim sup

r→0

log η(B(x, r))log r

.

This shows (see [Fa]) that DimE(Λ) ≤ ϕ(a).

Following ideas in [Pa] we extend this result to spherical multifractal components.We introduce some previous notations. For i ∈ Mk we denote by l(i) the length k ofthe sequence i. Given ρ < 1 let Q(ρ, n) be the subset of M∗ defined by

Q(ρ, n) =i ∈M∗ : ri < ρn ≤ ri(l(i)−1))

.

It is easy to see, using Lemma 1.1 ii), that for any measure µ ∈ S,∑

i∈Q(ρ,n) µ(Ei) = 1holds.

Corollary 4.3. Let M(Λa) be as in Theorem 3.4 for the set Λa defined in (4.3) foran a ∈ D. Then DimM(Λa) = dimM(Λa) = ϕ(a).

Proof . We give a proof for the case k = 2, and assume that Equations (4.1) holdfor a q = (q1, q2) with q1 ≥ 0 and q2 < 0. The extension to the general case is thentrivial. Let u = max

r−1t : t ∈ M

, ρ < 1/2, and ε > 0. Let us abbreviate Λa to Λand consider the sets

Mq(Λ) =x ∈ M(Λ) : r(ai+ε) ≤ λi(B(x, r)) ≤ r(ai−ε) for all r ≤ 2ρq and 1 ≤ i ≤ 2

for q ∈ IN. In particular, for x ∈Mq(Λ),

ρn(ai+ε) ≤ λi(B(x, ρn)) ≤ ρn(ai−ε) ,

λi(B(x, ρn)) ≥ c2ρ2nελi(B(x, 2ρn))

(4.5)

hold for all n ≥ q and 1 ≤ i ≤ 2, where c2 is a constant independent of i andn. Let

B(xj, rj

)j∈IN be a ρq–packing of Mq(Λ). For all j let nj > q be such

that ρnj < rj ≤ ρnj−1. ThenB(xj, ρ

nj)j∈IN is a ρq–packing of Mq(Λ). For γ =

a1q1 + a2q2 + β + ε(q1 − 3q2) with β = β(q), by (4.5), we have∞∑j=1

rγj ≤ ρ−γ∞∑j=1

ρnjγ

= ρ−γ∞∑j=1

ρnj(β−ε2q2)ρnjq1(a1+ε)ρnjq2(a2−ε)

≤ ρ−γ∞∑j=1

ρnj(β−ε2q2)λq11(B(xj, ρ

nj))λq22

(B(xj, ρ

nj)).

158 Math. Nachr. 229 (2001)

We now get upper bounds for the two last factors in the terms of the sum. Let usabbreviate B

(xj , ρ

nj)

to Bj and let Q(Bj

):=

i ∈ Q(

ρ, nj)

: Bj ∩ Ei = ∅. Byvolume estimating the cardinality of Q(

Bj

)is bounded by a constant c1 independent

of Bj . Then

λ1(Bj

) ≤ λ1

⋃

i∈Q(Bj)

Ei

≤ c1 max

λ1(Ei) : i ∈ Q(

Bj

).

Notice that, for any i ∈ Q(Bj

), Ei ⊆ B

(xj, 2ρnj

)and then, by (4.5), λ2

(Bj

) ≥c2ρ

nj2ελ2(Ei). Notice also that ρnjβ ≤ u|β|rβi for i ∈ Q(Bj

). Using these inequalities

we get

∞∑j=1

rγj ≤ ρ−γcq11 cq22

∞∑j=1

ρnjε maxρnjβλq11 (Ei)λ

q22 (Ei) : i ∈ Q(

Bj

)

≤ ρ−γcq11 cq22 u

|β|∞∑j=1

ρnjε maxη(Ei) : i ∈ Q(

Bj

),

where η is the measure given by (4.4). Each index i for which the maximum in theabove sum is attained belongs to some set Q(ρ, n). By volume estimating, there existsa constant C independent from i and j such that the cardinality of the set of indices

i ∈ IN : i = nj , and Bj ∩ Ei = ∅

is less than C. We then have

∞∑j=1

rγj ≤ ρ−γcq11 cq22 u

|β|C∞∑n=q

ρnε∑

i∈Q(ρ,n)

η(Ei)


|β|C∞∑n=q

ρnε


|β|C(1 − 1/2ε

)−1

< ∞ .

Hence Dim(Mq(Λ)

) ≤ a1q1 + a2q2 + β + ε(q1 − 3q2). Since M(Λ) ⊆ ⋃∞q=1Mq(Λ) and

ε is arbitrarily small this gives

dimM(Λ) ≤ DimM(Λ) ≤ ϕ(a) .

By Lemma 3.2 we know that M(Λ) ∩ Γ = E(Λ) ∩ Γ. Since η ∈ SP, by part ii) ofLemma 1.1 we know that η(∂V ) = 0. Then, using arguments given in Theorem 3.4,dim(E(Λ) ∩ Γ) = dimE(Λ) follows. Theorem 4.2 then gives

ϕ(a) = dimE(Λ) = dim(M(Λ) ∩ Γ) ≤ dimM(Λ) .


5. Conclusions

The results we present for the canonical self–similar case allows us to make clearthe underlying basic ideas. We do hope that our approach can be fruitfully used inother settings where the multifractal analysis of self–similar measures has played therole of a first step. In particular, it seems likely that our results can be extended tothe setting of self–conformal constructions (see [Pa]).

A natural problem posed by Theorem 2.6 is to find classes of sets which are dualin a Frostman’s sense for the class of ergodic measures, and for the wider classof invariant measures, since the geometry of dimensionally relevant subsets of a setcarrying an ergodic measure or an invariant measure of maximal dimension, can stillbe analyzed via Birkhoff’s Ergodic Theorem.

Acknowledgements

I am grateful to Pertti Mattila for the support given to this research and to the Depart-ment of Mathematics of the University of Jyvaskyla for their kind hospitality during the staywhen this paper was written.

This research has been partially supported by Direccion General de Ensenanza Superior

Cientıfica y Tecnica, PB97–0301.

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Dpto Analisis Economico IUniversidad ComplutenseCampus de Somosaguas28223 MadridSpainE–mail:[email protected]

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