VARIATIONAL PROPERTIES OF MULTIFRACTAL ... - Semantic Scholar

Nonlinearity 14 (2001), 259–274.

VARIATIONAL PROPERTIES OF MULTIFRACTAL SPECTRA LUIS BARREIRA Abstract. For hyperbolic diffeomorphisms, we describe the variational properties of the dimension spectrum of equilibrium measures on locally maximal hyperbolic sets, when the measure or the dynamical system are perturbed. We also obtain explicit expressions for the first derivative of the dimension spectra and the associated Legendre transforms. This allows us to establish a local version of multifractal rigidity, i.e., of a “multifractal” classification of dynamical systems based on their multifractal spectra.

1. Introduction Given a finite measure µ on a compact manifold M , the dimension spectrum or multifractal spectrum for pointwise dimensions is defined by log µ(B(x, r)) Dµ (α) = dimH x ∈ M : lim =α , (1) r→0 log r

where B(x, r) ⊂ M is the ball of radius r centered at x, and dim H Z denotes the Hausdorff dimension of the set Z (see Section 4 for the definition). Dimension spectra are one of the primary components of multifractal analysis, a theory initially developed by physicists and applied mathematicians as a powerful tool for the numerical study of dynamical systems. Dimension spectra are examples of more general multifractal spectra, which include multifractal spectra for local entropies and multifractal spectra for Lyapunov exponents. These spectra encode information about various characteristics of the dynamics, as well as the associated invariant sets and invariant measures. Among them are the well-known Hausdorff dimension, correlation dimension, information dimension, and entropy of invariant measures. See [1, 10] for more details and further references. In this paper we study the effect of perturbations on dimension spectra. In particular, we address the question of how the dimension spectrum and its Legendre transform vary when the dynamical system f or the measure µ are perturbed. This question is of primary importance in the numerical study of dynamical systems. In fact, since numerical data, and in particular “numerical” multifractal spectra, may always be affected by small perturbations it is crucial to understand the influence of these perturbations. In particular we shall give a characterization of the variational properties of dimension spectra when: 1. µ is perturbed for a fixed dynamical system f ; 2000 Mathematics Subject Classification. Primary: 37C45, 37D35. Key words and phrases. dimension theory, multifractal analysis, multifractal rigidity. Partially supported by FCT’s Funding Program and NATO grant CRG970161. 1

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2. f is perturbed in such a way that µ remains invariant. Multifractal spectra contain an enormous amount of “physical” information in a unique function. A priori, this codification could cause a drastic loss of information about the dynamics. We believe that it is both a useful and challenging problem to recover this information, i.e., to obtain information about a dynamical system from its multifractal spectra. This study is called multifractal rigidity. Let us describe one of the main problems of multifractal rigidity. Consider probability measures µ1 and µ2 invariant under a dynamical system f such that Dµ1 = Dµ2 . The problem is to decide whether µ1 = µ2 . An affirmative solution to this problem would provide a multifractal classification of invariant measures. Some partial results in this direction were obtained by Barreira, Pesin, and Schmeling [1, 2] for certain restricted classes of hyperbolic dynamical systems and Gibbs measures. In this paper we establish a “local” version of multifractal rigidity, valid for arbitrary equilibrium measures of axiom A surface diffeomorphisms. We now illustrate this phenomenon with a rigorous statement in the case of the Smale horseshoe (for definiteness we shall assume that the Smale horseshoe map is piecewise affine, and that its derivative matrix is diagonal with the absolute values of the eigenvalues being 3 and 3 −1 at every point). We code the Smale horseshoe by a two-sided full shift on two symbols. Theorem 1. For the Smale horseshoe, let η 7→ µ η be a C 1 -family of Bernoulli measures on two symbols. If η 7→ Dµη is constant, then η 7→ µη is constant. Theorem 1 is a particular case of more general statements established in Section 5. Unlike with the work in [1, 2] the statement in Theorem 1 readily extends to horseshoes modeled by Bernoulli shifts on an arbitrary number of symbols. The proof is based on the study of the variational properties of multifractal spectra. See Section 5 for details. The structure of the paper is as follows. Section 2 recalls some notions and results from dimension theory. In Section 3 we describe the variational properties of multifractal spectra, under perturbations of the potential and under perturbations of the dynamical system. For the Hausdorff and box dimensions, a related study is described in Section 4. These results are applied in Section 5 to obtain local versions of multifractal rigidity. Acknowledgment. I would like to thank Jörg Schmeling for several interesting discussions on the subject. 2. Dimension spectra of hyperbolic diffeomorphisms Let f be a C 1+ε axiom A diffeomorphism of the compact manifold M , for some ε > 0, and Λ ⊂ M a basic set for f . This means that Λ is f -invariant, locally maximal, and hyperbolic for f , with f |Λ topologically mixing and such that the periodic points of f |Λ are dense in Λ. In particular, there exist a df -invariant decomposition Tx M = E s (x)⊕E u (x) into stable and unstable subspaces for each x ∈ Λ, and constants c > 0 and λ ∈ (0, 1) such that kdx f n |E s (x)k ≤ cλn

and kdx f −n |E u (x)k ≤ cλn ,

VARIATIONAL PROPERTIES OF MULTIFRACTAL SPECTRA

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for every x ∈ Λ and n ∈ N. It is well known that the distributions x 7→ E s (x) and x 7→ E u (x) are Hölder continuous (see, for example, [5]). This implies that the functions ψs : Λ → R and ψu : Λ → R defined by ψs (x) = logkdx f |E s (x)k

and ψu (x) = − logkdx f |E u (x)k

(2)

are also Hölder continuous. Consider now a Hölder continuous function ϕ : Λ → R on the basic set Λ. We denote by µf,ϕ the unique equilibrium measure of ϕ with respect to f |Λ, i.e., the unique f -invariant probability measure on Λ such that Z Pf |Λ (ϕ) = hµf,ϕ (f |Λ) + ϕ dµf,ϕ , Λ

where hµ (f ) is the measure-theoretic entropy of f with respect to µ, and Pf |Λ (ϕ) is the topological pressure of ϕ with respect to f |Λ (see, for example, [5] for the definitions). We use the notation def

Df,ϕ (α) = Dµf,ϕ (α).

(3)

Define a function Tf,ϕ : R → R by def

Tf,ϕ (q) = Ts (q) + Tu (q),

(4)

where Ts (q) and Tu (q) are the unique real numbers satisfying Pf |Λ (Ts (q)ψs + q(ϕ − Pf |Λ (ϕ))) = 0 and Pf |Λ (Tu (q)ψu + q(ϕ − Pf |Λ (ϕ))) = 0. The existence and uniqueness of these numbers follow easily from the continuity properties of the topological pressure. We say that µ is a measure of maximal dimension for Λ if def

dimH µ = inf{dimH Z : µ(Z) = 1} = dimH Λ. The following statement establishes the multifractal properties of µ f,ϕ . Proposition 2. Let Λ be a basic set for a C 1+ε axiom A diffeomorphism f of a compact surface, for some ε > 0, and ϕ : Λ → R a H¨ older continuous function. Then the following properties hold: 1. The function Tf,ϕ is analytic. 2. The domain of the function Df,ϕ is a closed interval in [0, +∞) and 0 . coincides with the range of the function −T f,ϕ 3. If q ∈ R then 0 0 Df,ϕ (−Tf,ϕ (q)) = Tf,ϕ (q) − qTf,ϕ (q).

4. If µf,ϕ is not a measure of maximal dimension, then the functions Df,ϕ and Tf,ϕ are analytic and strictly convex. Proposition 2 is due to Simpelaere [14], with the expression “is a closed interval” replaced by “contains a closed interval”. In [13] Schmeling showed that one can interchange the two expressions. See [10] for more details and further references.

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Remark. In [8] McCluskey and Manning showed that there exists an invariant measure of maximal dimension if and only only if T s (0)ψs and Tu (0)ψu are cohomologous, i.e., if there exists a continuous function g : Λ → R such that Ts (0)ψs − Tu (0)ψu = g ◦ f − g. (5) By Livshitz’s theorem, (5) holds if and only if kdx f n |E s (x)kTs (0) kdx f n |E u (x)kTu (0) = 1 for every x ∈ Λ and n ∈ N such that f n x = x. If (5) holds, then the common equilibrium measure of Ts (0)ψs and Tu (0)ψu is the unique invariant measure of maximal dimension. Therefore, the assumption “µ f,ϕ is not a measure of maximal dimension” in Statement 4 of Proposition 2 is equivalent to “ϕ − Pf |Λ (ϕ) is not simultaneously cohomologous to T s (0)ψs and to Tu (0)ψu ”. There exist versions of Proposition 2 for other multifractal spectra, such as entropy spectrum for local entropies, and dimension and entropy spectra for Lyapunov exponents; see [10] for details. One can also obtain versions of the results formulated below for these other multifractal spectra, using a very similar approach. 3. Variational properties of dimension spectra In this section we describe how the dimension spectrum D f,ϕ (see (1) and (3)), and its Legendre transform Tf,ϕ (see (4)) vary under perturbations of the dynamical system f and perturbations of the potential ϕ. 3.1. Perturbations of the potential. We first consider perturbations of the potential. Let again Λ be a basic set for a C 1+ε axiom A diffeomorphism f : M → M for some ε > 0. Let C α (Λ) be the space of Hölder continuous functions ϕ : Λ → R with Hölder exponent α. We define the norm of a function ϕ ∈ C α (Λ) by |ϕ(x) − ϕ(y)| : x, y ∈ Λ and x 6= y . kϕkα = sup{|ϕ(x)| : x ∈ Λ} + sup d(x, y)α

Consider a parameterized family of potentials Φ = {ϕ η }η∈(−δ,δ) ⊂ C α (Λ) for some δ > 0. We shall say that Φ is a C k -family if the map def

(−δ, δ) 3 η 7→ Φ(η) = ϕη ∈ C α (Λ) is of class C k . For each η ∈ (−δ, δ) let τs (η, q) and τu (η, q) be the unique real numbers such that Pf |Λ (τs (η, q)ψs + q(ϕη − Pf |Λ (ϕη ))) = 0 and Pf |Λ (τu (η, q)ψu + q(ϕη − Pf |Λ (ϕη ))) = 0. s We denote by νq and νqu the unique equilibrium measures, respectively, of the potentials τs (0, q)ψs + q(ϕ0 − Pf |Λ (ϕ0 ))

and

τu (0, q)ψu + q(ϕ0 − Pf |Λ (ϕ0 )),

where ψs and ψu are defined in (2). The following statement provides a description of the variational properties of the functions τs and τu .


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Theorem 3. Let Λ be a basic set for a C 1+ε axiom A diffeomorphism f of a compact surface, and Φ ⊂ C ε (Λ) a C k -family of functions, for some ε > 0 and k ≥ 1. Then the functions τs and τu are of class C k in η and analytic in q, with Z .Z ∂τs ψs dνqs (6) ζ dνqs (0, q) = −q ∂η Λ Λ and Z .Z ∂τu u ψu dνqu (7) ζ dνq (0, q) = −q ∂η Λ Λ for every q ∈ R, where Z dΦ dΦ def d ζ = − dµf,ϕ0 . (Φ(η) − Pf |Λ (Φ(η)))|η=0 = dη dη η=0 Λ dη η=0

Proof. We need the following lemma.

Lemma 4. If ξ, ψη ∈ C ε (Λ) for each η ∈ (−δ, δ), with η 7→ ψη continuous, then Z d ψ0 dµf,ξ . P (ξ + ηψη )|η=0 = dη f |Λ Λ Proof of the lemma. Since d P (ξ + ηψ0 )|η=0 = dη f |Λ and

Z

ψ0 dµf,ξ Λ

|Pf |Λ (ξ + ηψη ) − Pf |Λ (ξ + ηψ0 )| ≤ |η| sup|ψη (x) − ψ0 (x)|, x∈Λ

we obtain Z Pf |Λ (ξ + ηψη ) − Pf |Λ (ξ) ψ0 dµf,ξ ≤ sup|ψη (x) − ψ0 (x)| + o(η). − η x∈Λ Λ

This implies the desired result.

Consider the function p(t, q, η) = Pf |Λ (tψs + q(ϕη − Pf |Λ (ϕη ))). Using Lemma 4, Birkhoff’s ergodic theorem, and the fact that the distribution E s is one-dimensional, we obtain Z ∂p (Ts (q), q, 0) = ψs dνqs ∂t Λ Z n−1 1X ψs (f k x) dνqs (x) lim = Λ n→∞ n k=0 Z 1 lim logkdx f n |E s (x)k dνqs (x) ≤ log λ < 0. = Λ n→∞ n

By the implicit function theorem, the function τ s : (−δ 0 , δ 0 ) × R → R is welldefined for some δ 0 ∈ (0, δ], with the desired regularity properties. A similar argument applies to τu .

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We now proceed in a similar way to that in [16]. We have τs (η, q)ψs + q(ϕη − Pf |Λ (ϕη )) = Ts (q)ψs +

∂τs (0, q)ψs η + q(ϕ0 − Pf |Λ (ϕ0 )) ∂η

d (ϕη − Pf |Λ (ϕη ))|η=0 η + o(η) dη = Ts (q)ψs + q(ϕ0 − Pf |Λ (ϕ0 )) ∂τs + (0, q)ψs + qζ η + o(η). ∂η +q

By Lemma 4, we obtain ∂ 0= P (τs (η, q)ψs + q(ϕη − Pf |Λ (ϕη )))|η=0 ∂η f |Λ Z ∂τs (0, q)ψs + qζ dνqs . = ∂η Λ

Hence (6) holds. Furthermore, again by Lemma 4, d dΦ ζ= ϕη − Pf |Λ ϕ0 + η + o(η) dη dη η=0 η=0 Z dΦ dΦ − dµf,ϕ0 . = dη η=0 Λ dη η=0

One can obtain (7) using similar arguments. This completes the proof of the theorem. Consider now the function τ (η, q) = τs (η, q) + τu (η, q). By Theorem 3 the function τ is of class C k in η and analytic in q. The following statement describes the variational properties of the dimension spectrum. Theorem 5. Let Λ be a basic set for a C 1+ε axiom A diffeomorphism f of a compact surface, and Φ ⊂ C ε (Λ) a C k -family of functions, for some ε > 0 and k ≥ 1. If µf,ϕ0 is not a measure of maximal dimension, then the function D : η 7→ Df,ϕη is of class C k in a neighborhood of η = 0, and . ∂2τ ∂τ ∂τ dD − (0, q) = q − (0, q) (0, q) (8) dη η=0 ∂q ∂η ∂η∂q for every q ∈ R.

Proof. We shall first prove that µf,ϕη is not of maximal dimension for all sufficiently small |η|. Since µf,ϕ0 is not of maximal dimension, either ϕ 0 − Pf |Λ (ϕ0 ) is not cohomologous to Ts (0)ψs or to Tu (0)ψu (see the remark after Proposition 2). Without loss of generality we assume that ϕ 0 − Pf |Λ (ϕ0 ) is not cohomologous to Ts (0)ψs . By Livschitz’s theorem (see, for example, [5, Theorem 19.2.1]), there exist two periodic points x i = f mi xi for i = 0, 1 such that m0 m1 1 X 1 X k k (ϕ0 − Ts (0)ψs )(f x0 ) − (ϕ0 − Ts (0)ψs )(f x1 ) 6= 0. δ= m0 m1 k=1

k=1


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For any function ϕ ∈ C ε (Λ) satisfying kϕ − ϕ0 kε < δ/2 we have mi 1 X δ (ϕ − ϕ0 )(f k xi ) ≤ sup{|ϕ(x) − ϕ0 (x)| : x ∈ Λ} ≤ kϕ − ϕ0 kε < , mi 2 k=1

for i = 0, 1, and hence, m0 m1 1 X 1 X (ϕ − Ts (0)ψs )(f k x0 ) 6= (ϕ − Ts (0)ψs )(f k x1 ). m0 m1 k=1

k=1

This implies that ϕ − Pf |Λ (ϕ) is not cohomologous to Ts (0)ψs , and hence, µf,ϕ is not a measure of maximal dimension. Since kϕ η − ϕ0 kε ≤ κ|η| for some constant κ > 0, and all sufficiently small |η|, we conclude that µ f,ϕη is not of maximal dimension for all sufficiently small |η|. By Proposition 2 and Theorem 3, the function q 7→ τ (η, q) is analytic and 2 strictly convex. Therefore, ∂∂qτ2 (η, q) 6= 0 for every q ∈ R, and q 7→ ∂τ ∂q (η, q) is strictly increasing with range equal to R, for all sufficiently small |η|. Hence, there exists a function uη : R → R of class C k such that − ∂τ ∂q (η, uη (p)) = p for every p ∈ R. Using Proposition 2 we obtain ∂τ ∂τ Df,ϕη − (η, q) = τ (η, q) − q (η, q). (9) ∂q ∂q By Theorem 3, the right-hand side of (9) is of class C k in η. Furthermore Df,ϕη (p) = τ (η, uη (p)) + uη (p)p, and the above discussion shows that the function η 7→ D f,ϕη is of class C k in η. Taking derivatives in (9) with respect to η and setting η = 0 we obtain the desired identity (8). Theorem 5 indicates that under C k -perturbations of the potential, the corresponding dimension spectra, for some fixed basic set of a diffeomorphism, vary in a C k -differentiable way. Remark. Theorems 3 and 5 can be generalized in a straightforward fashion to conformal axiom A diffeomorphisms on higher dimensional manifolds, i.e., axiom A diffeomorphisms such that dx f |E s (x) and dx f |E u (x) are multiples of isometries for every x ∈ Λ. A similar remark applies to all the results formulated below. 3.2. Measure preserving perturbations of the dynamics. Consider now a parameterized family F = {fη }η∈(−δ,δ) of C k diffeomorphisms of the compact manifold M for some k ≥ 1. We shall say that F is a C k -family if the function η 7→ fη is of class C k . Assume that f = f0 is an axiom A diffeomorphism, and that Λ = Λ0 ⊂ M is a basic set for f . By structural stability, there exists δ 0 ∈ (0, δ] such that for each η ∈ (−δ 0 , δ 0 ) there exists a basic set Λη ⊂ M for fη , i.e., an fη -invariant set which is a locally maximal hyperbolic set for fη , with fη |Λη topologically mixing, and such that the periodic points of fη |Λη are dense in Λη . Furthermore, there exists ε > 0 such that for each η ∈ (−δ 0 , δ 0 ) there exists a ε-Hölder homeomorphism hη : Λ → Λη satisfying fη ◦ hη = hη ◦ f on Λ.

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Let TΛη M = Eηs ⊕ Eηu be the dfη -invariant decomposition into stable and unstable subbundles, and consider the ε-Hölder continuous functions ψs,η : Λη → R and ψu,η : Λη → R defined by ψs,η (x) = logkdx fη |Eηs (x)k

and

ψu,η (x) = − logkdx fη |Eηu (x)k.

(10)

Define functions Ψs : (−δ, δ) → C ε (Λ) and Ψu : (−δ, δ) → C ε (Λ) by Ψs (η) = ψs,η ◦ hη

and

Ψu (η) = ψu,η ◦ hη .

(11)

The following result is due to Ma˜ né [7]. Proposition 6. If {fη }η∈(−δ,δ) is a C k -family of C k diffeomorphisms of a compact surface for some k ≥ 2, and Λ is a basic set for f 0 , then the maps Ψs and Ψu are of class C k−1 . Let ϕ : Λ → R be a Hölder continuous potential. For each η ∈ (−δ 0 , δ 0 ) we consider another Hölder continuous potential ϕ η : Λ → R defined by ϕη = ϕ ◦ h−1 η .

(12)

h−1 η .

When {fη }η∈(−δ,δ) has assoOne can easily verify that µfη ,ϕη = µf,ϕ ◦ ciated a parameterized family of functions {ϕ η }η∈(−δ,δ) such that (12) holds for some Hölder continuous function ϕ : Λ → R and every η, we say that {fη }η∈(−δ,δ) is a measure-preserving perturbation. At the level of symbolic dynamics (associated to some Markov partition of the unperturbed basic set Λ) all the measures induced by each measure µ fη ,ϕη on the symbolic space (i.e., the associated Gibbs measures) are indeed equal. Let now Ts (η, q) and Tu (η, q) be the unique real numbers satisfying Pfη |Λη (Ts (η, q)ψs,η + q(ϕη − Pfη |Λη (ϕη ))) = 0

(13)

Pfη |Λη (Tu (η, q)ψu,η + q(ϕη − Pfη |Λη (ϕη ))) = 0. We denote by µsq and µuq the unique equilibrium measures of

(14)

and

Ts (0, q)ψs,0 + q(ϕ − Pf |Λ (ϕ))

and

Tu (0, q)ψu,0 + q(ϕ − Pf |Λ (ϕ)),

respectively. Theorem 7. If {fη }η∈(−δ,δ) is a C k -family of C k diffeomorphisms on a compact surface for some k ≥ 2, Λ is a basic set for f 0 , and ϕ : Λ → R is a H¨ older continuous function, then the functions T s and Tu are of class C k−1 in η and analytic in q, with Z .Z dΨs ∂Ts s (0, q) = −Ts (0, q) ψs,0 dµsq (15) dµq ∂η Λ dη η=0 Λ and Z .Z ∂Tu dΨu u dµ ψu,0 dµuq , (16) (0, q) = −Tu (0, q) q ∂η dη η=0 Λ Λ for every q ∈ R. Proof. Consider the function

p(t, q, η) = Pfη |Λη (tψs,η + q(ϕη − Pfη |Λη (ϕη ))). Since Pfη |Λη (ϕ) = Pf |Λ (ϕ ◦ hη ) for every potential ϕ, we obtain p(t, q, η) = Pf |Λ (tΨs (η) + q(ϕ − Pf |Λ (ϕ))).


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Proceeding in a similar way to that in the proof of Theorem 3, and using Lemma 4, we conclude that Z ∂p ψs,0 dµsq < 0. (Ts (0, q), q, 0) = ∂t Λ

The implicit function theorem implies that the function T s has the desired regularity properties. Furthermore ∂Ts Ts (η, q)Ψs (η) + q(ϕ − Pf |Λ (ϕ)) = Ts (0, q)ψs,0 + (0, q)ψs,0 η ∂η dΨs + Ts (0, q) η dη η=0 + q(ϕ − Pf |Λ (ϕ)) + o(η). By Lemma 4, we obtain ∂ P (Ts (η, q)Ψs (η) + q(ϕ − Pf |Λ (ϕ)))|η=0 ∂η f |Λ Z ∂Ts dΨs = (0, q)ψs,0 + Ts (0, q) dµsq . ∂η dη η=0 Λ

0=

This establishes (15). With similar arguments one can establish (16) and the regularity properties of the function T u . 3.3. General perturbations. One can also consider a combination of the approaches in Sections 3.1 and 3.2. Namely, let F = {f η }η∈(−δ,δ) be a C k family of C k diffeomorphisms of the compact manifold M for some k ≥ 2, such that f = f0 is an axiom A diffeomorphism with some basic set Λ = Λ0 ⊂ M . Let also {ϕη }η∈(−δ,δ) ⊂ C α (Λ) be a parameterized family of εHölder continuous functions. We emphasize that the identity (12) need not longer hold in this situation. We use the same notations as in Section 3.2. In particular we define functions Ψs and Ψu as in (11), and functions Ts and Tu as in (13) and (14). Theorem 8. On a compact surface, if η 7→ f η and η 7→ ϕη ◦ hη are of class C k for some k ≥ 2, then the functions Ts and Tu are of class C k−1 in η and analytic in q, with Z Z dΨs ∂Ts s dµsq (0, q) ψs,0 dµq = −Ts (0, q) ∂η dη η=0 Λ Λ Z d −q (ϕη ◦ hη − Pfη |Λη (ϕη )) dµsq dη η=0 Λ

and

∂Tu (0, q) ∂η

Z

Λ

ψu,0 dµuq

Z

dΨu = −Tu (0, q) dµuq dη η=0 Λ Z d dµuq . −q (ϕη ◦ hη − Pfη |Λη (ϕη )) dη η=0 Λ

Proof. The proof can be obtained by a straightforward modification of the arguments in the proof of Theorem 7.

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By Proposition 6, the map η 7→ ϕη ◦ hη possesses the regularity required in Theorem 8 when ϕη = ψs,η for every η, or when ϕη = ψu,η for every η (see (10) for the definition of ψs,η and ψu,η ). We emphasize that our results in general do not extend to non-conformal hyperbolic diffeomorphisms on manifolds of dimension greater than two (see the discussion after Theorem 10 below). 4. Variational properties of Hausdorff and box dimensions In this section we consider related problems to those of the former sections for the Hausdorff and box dimensions. Given a set Z ⊂ M and a number s ≥ 0, the s-dimensional Hausdorff measure of Z is given by X mH (Z, s) = lim inf (diam U )s , δ→0 U

U ∈U

where the infimum is taken over all finite or countable covers U of Z by sets of diameter at most δ. The Hausdorff dimension of Z is defined by dimH Z = inf{s : mH (Z, s) = 0}.

Let now Nδ (Z) denote the minimum number of sets of diameter at most δ needed to cover Z. The lower and upper box dimensions of Z are defined by dimB Z = lim inf δ→0

log Nδ (Z) − log δ

and

dimB Z = lim sup δ→0

log Nδ (Z) . − log δ

One can easily show that dimH Z ≤ dimB Z ≤ dimB Z. Let again {fη }η∈(−δ,δ) be a parameterized family of C k diffeomorphisms as in Section 3.2. We assume that f = f 0 is an axiom A diffeomorphism with basic set Λ = Λ0 ⊂ M . Let V s (x) and V u (x) be the local stable and unstable manifolds of size ε, defined by V s (x) = {y ∈ M : d(f n y, f n x) ≤ ε for all n ≥ 0} and V u (x) = {y ∈ M : d(f n y, f n x) ≤ ε for all n ≤ 0}. They satisfy Tx V s (x) = E s (x) and Tx V u (x) = E u (x) for every x ∈ Λ. Consider the Hölder continuous functions ψ s : Λ → R and ψu : Λ → R defined by (2), and let ts and tu be the unique real numbers such that Pf |Λ (ts ψs ) = 0

and

Pf |Λ (tu ψu ) = 0.

The following result concerns the Hausdorff dimension, and the lower and upper box dimensions of basic sets. Proposition 9. Let f be a C 1 axiom A diffeomorphism on a compact surface, and Λ a basic set for f . Then the following properties hold: 1. If x ∈ Λ then dimH (V s (x) ∩ Λ) = dimB (V s (x) ∩ Λ) = dimB (V s (x) ∩ Λ) = ts and dimH (V u (x) ∩ Λ) = dimB (V u (x) ∩ Λ) = dimB (V u (x) ∩ Λ) = tu . 2. dimH Λ = dimB Λ = dimB Λ = ts + tu .


11

Proposition 9 is a combination of work of several people. McCluskey and Manning [8] proved the statements concerning the Hausdorff dimension. The coincidence of the Hausdorff dimension with the lower and upper box dimensions was established by Takens [15] for C 2 diffeomorphisms, and by Palis and Viana [9] for arbitrary C 1 diffeomorphisms. See [10] for more details and further references. For each η we denote the local stable and unstable manifolds of f η at the point x ∈ Λη respectively by Vηs (x) and Vηu (x). Let Ψs and Ψu be as in (11). Setting q = 0 in Theorem 7 and applying Proposition 9 we obtain the following statement. Theorem 10. If {fη }η∈(−δ,δ) is a C k -family of C k diffeomorphisms on a compact surface for some k ≥ 2, and Λ is a basic set for f 0 , then: 1. The maps η 7→ dimH (Vηs (x) ∩ Λη ) = dimB (Vηs (x) ∩ Λη ) = dimB (Vηs (x) ∩ Λη ) and η 7→ dimH (Vηu (x) ∩ Λη ) = dimB (Vηu (x) ∩ Λη ) = dimB (Vηu (x) ∩ Λη ) are independent of x ∈ Λ, and are of class C k−1 . 2. If x ∈ Λ then d d d dimB Λη η=0 dimH Λη η=0 = dimB Λη η=0 = dη dη dη Z Z .Z .Z dΨu dΨs s u s dµ0 dµ0 ψs dµ0 − tu ψu dµu0 . = −ts η=0 η=0 dη dη Λ Λ Λ Λ

In the case of the Hausdorff dimension, statement 1 is due to Ma˜ né [7], and statement 2 is due to Weiss [16]. McCluskey and Manning [8] (in the case of the Hausdorff dimension), and Palis and Viana [9] proved earlier that each of the maps in Statement 1 is continuous for C 1 surface diffeomorphisms. More recently Bonatti, D´ıaz, and Viana [4] gave an example which shows that the Hausdorff dimension of basic sets may vary discontinuously with the dynamics in manifolds with dimension bigger than two. Ruelle [12] obtained a version of Statement 1 in the case of repellers of conformal expanding maps. 5. Local multifractal rigidity 5.1. Rigidity of horseshoes. In this section we consider the same setup as in Section 3.1. Namely, let Λ be a basic set for a C 1+ε axiom A surface diffeomorphism f , and let Φ = {ϕη }η∈(−δ,δ) ⊂ C ε (Λ) be a parameterized family of potentials, for some ε > 0. Consider also the associated family of dimension spectra {Df,ϕη }η∈(−δ,δ) . We want to address the following problem: A. show that if the function η 7→ Df,ϕη is constant in a neighborhood of η = 0, then µf,ϕη1 = µf,ϕη2 for every η1 and η2 in a neighborhood of η = 0. We say that this is a problem of local multifractal rigidity. Problem A is related to the problem of multifractal rigidity described in Section 1: B. show that if Df,ϕ = Df,ψ , then µf,ϕ = µf,ψ .

12

LUIS BARREIRA

Clearly, an affirmative solution to Problem B implies an affirmative solution to Problem A. If ϕ and ψ are cohomologous, then D f,ϕ = Df,ψ . Therefore, what Problem B asks is if the cohomology of functions can be characterized by the dimension spectra of the corresponding equilibrium measures. We believe that this multifractal classification holds for a large class of hyperbolic dynamical systems. Partial results in this direction were given by Barreira, Pesin, and Schmeling [1, 2], when µ ϕ and µψ are Bernoulli measures on two symbols. It seems impossible to extend their technique for arbitrary Gibbs measures, and in particular even for Bernoulli measures on an arbitrary number of symbols. In contrast with the approach in [1, 2], we shall obtain an affirmative solution to a version of Problem A for arbitrary equilibrium measures, applying the results of Section 3. Consider negative functions ψbs : Λ → R and ψbu : Λ → R. In a similar way to that in Section 3.1, let τbs (η, q) and τbu (η, q) be the unique numbers satisfying Pf |Λ (b τs (η, q)ψbs + q(ϕη − Pf |Λ (ϕη ))) = 0 and Pf |Λ (b τu (η, q)ψbu + q(ϕη − Pf |Λ (ϕη ))) = 0. b s and For each fixed η ∈ (−δ, δ), we consider the Legendre transforms D f,η

b u of q 7→ τbs (η, q) and q 7→ τbu (η, q), given by D f,η and

b s (α) = sup{b D τs (η, q) + qα : q ∈ R} f,η

b u (α) = sup{b D τu (η, q) + qα : q ∈ R}. f,η Using these dimension spectra we obtain the following version of local multifractal rigidity.

Theorem 11. Let Λ be a basic set for a C 1+ε axiom A diffeomorphism f of a compact surface, and Φ ⊂ C ε (Λ) a C 1 -family of functions, for some ε > 0 such that µf,ϕ0 is not a measure of maximal dimension. If for each ( ψbs , ψbu ) b s and η 7→ D bu C ε (Λ)-sufficiently close to (ψs , ψu ) the functions η 7→ D f,η f,η are constant in a neighborhood of η = 0, then µ f,ϕη1 = µf,ϕη2 for every η1 and η2 in a neighborhood of η = 0. Proof. Since µf,ϕ0 is not a measure of maximal dimension, we can assume, without loss of generality, that for all sufficiently small |η| and all ( ψbs , ψbu ) C ε (Λ)-sufficiently close to (ψs , ψu ), the function τs (η, 0)ψbs is not cohomologous to ϕη − Pf |Λ (ϕη ) (see the remark after Proposition 2 and the proof of s be the equilibrium measure of the function Theorem 5). Let νη,q ϕη,q = τbs (η, q)ψbs + q(ϕη − Pf |Λ (ϕη ))

It follows from the general theory of multifractal analysis of Birkhoff averages b s is analytic and strictly convex, and that for (see [3] for details) that D f,η s are distinct for different q in some each fixed η the equilibrium measures ν η,q open interval. Since Pf |Λ (ϕη,q ) = 0 for every η and q, we conclude that for each fixed η the functions ϕη,q are pairwise non-cohomologous for different q in some open interval.


13

b s is constant in a neighborhood of η = 0, the Since the function η 7→ D f,η function η 7→ τbs (η, ·) is also constant in a neighborhood of η = 0. By a straightforward modification of Theorem 3 (with the functions ψ s and ψu R s = 0 for every q ∈ R, replaced by ψbs and ψbu ), we conclude that Λ ζη dνη,q every sufficiently small |η|, and every ( ψbs , ψbu ) C ε (Λ)-sufficiently close to (ψs , ψu ), where d ζη = [Φ(η) − Pf |Λ (Φ(η))]. dη Since the topological pressure is analytic in the space of Hölder continuous functions C ε (Λ) (see [11, Corollary 5.27]), the map Z ∂ F : C ε (Λ) 3 ϕ 7→ Pf |Λ (ϕ + rζη ) , ζη dµf,ϕ = ∂r r=0 Λ

is also analytic. The above discussion shows that for all sufficiently small |η| we have F (ϕη,q ) = 0 for every q in some open interval. Furthermore, for each fixed η and q0 the functions ϕη,q cover some open subset of C ε (Λ) as (ψbs , ψbu , q) varies in a C ε (Λ) × R-open neighborhood of (ψs , ψu , q0 ). It follows immediately from the analyticity of F that Z d [Φ(η) − Pf |Λ (Φ(η))] dµ = 0 (17) Λ dη

for every equilibrium measure µ with a potential in C ε (Λ). We denote by C(Λ) the space of continuous functions on Λ, and by I(ϕ) the family of equilibrium measures of a function ϕ ∈ C(Λ). If (ϕ n )n ⊂ C ε (Λ) is a sequence of Hölder continuous functions converging uniformly to a function ϕ ∈ C(Λ) (not necessarily Hölder continuous), then any weak sublimit of the sequence (µf,ϕn )n belongs to I(ϕ) (see [6, Theorem 4.2.11]). In particular, when ϕ has a unique equilibrium measureS the sequence (µ f,ϕn )n converges weakly to µf,ϕ . Therefore, the family ϕ∈C ε (Λ) I(ϕ) is weakly dense in def J = {µf,ϕ : ϕ ∈ C(Λ) and card I(ϕ) = 1}. Furthermore, the closed convex hull of J coincides with [ def IC(Λ) = I(ϕ) ϕ∈C(Λ)

S (see [11, Appendix A.3.7]). This shows that ϕ∈C ε (Λ) I(ϕ) is weakly dense in IC(Λ) , and thus (17) holds for every measure µ ∈ I C(Λ) . Since IC(Λ) is weakly dense in the family of invariant measures (see [11, Corollary 3.17]), we conclude that the family of equilibrium measures of Hölder continuous functions is weakly dense in the family of invariant measures, and thus (17) holds for every invariant measure µ. In particular, considering the identity in (17) for all invariant measures supported on periodic orbits we conclude from Livshitz theorem (see, for example, [5, Theorem 19.2.1]) that for all η in a sufficiently small open neighborhood of η = 0, there exists a continuous function g η : Λ → R such that d [Φ(η) − Pf |Λ (Φ(η))] = gη ◦ f − gη . dη

(18)

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LUIS BARREIRA

Furthermore, it follows from the proof of Theorem 19.2.1 in [5] that there exists a constant C > 0 such that sup{|gη (x)| : x ∈ Λ} ≤ Ckζη kε

(19)

for all η. Integrating (18) from η1 to η2 we obtain ϕη1 − Pf |Λ (ϕη1 ) = ϕη2 − Pf |Λ (ϕη2 ) + Gη1 ,η2 ◦ f − Gη1 ,η2 , where Gη1 ,η2 : Λ → R is the bounded function defined by Z η2 Gη1 ,η2 (x) = gη (x) dη. η1

It follows from (19) and the continuity of η 7→ ζ η ∈ C ε (Λ) that Gη1 ,η2 is indeed well-defined. We conclude that the functions ϕ η1 − Pf |Λ (ϕη1 ) and ϕη2 − Pf |Λ (ϕη2 ) are cohomologous for every η1 and η2 in a sufficiently small neighborhood of η = 0. This completes the proof of the theorem. 5.2. Linear horseshoes. We shall now illustrate the local multifractal rigidity with a special class of horseshoes. Let f be a “linear horseshoe map”, i.e., a piecewise linear map f : [0, 1] 2 → [0, 1]2 for which there exist disjoint horizontal strips H1 , . . ., Hp , and disjoint vertical strips V1 , . . ., Vp such that f |Hi : Hi → Vi is a linear bijection for all i. We denote the constant values of the partial derivatives of f on H i by ci = |∂2 f |H1 | and di = |∂1 f |H1 |−1 . Let Λ be the horseshoe defined by f , i.e., the set \ Λ= f k (H1 ∪ · · · ∪ Hp ). k∈Z

The restriction of f to Λ is topologically conjugated to the shift map σ on Σ = {1, . . . , p}Z . Let now {ϕη }η∈(−δ,δ) be the parameterized family of functions on Σ defined by ϕη |Hi = log rη,i , (20) for some probability vector def

(rη,1 , . . . , rη,p ) ∈ X =

(

(ρ1 , . . . , ρp ) ∈ (0, 1)p :

p X

)

ρi = 1 .

i=1

For simplicity we shall write ri = r0,i for each i. We have Pσ|Σ (ϕη ) = log

p X

rη,i = 0

i=1

for every η. Define maps ψs : Σ → R and ψu : Σ → R by ψs (x) = log di if x ∈ Vi , and ψu (x) = − log ci if x ∈ Hi . In a similar way to that in Section 5.1, let τs (η, q) and τu (η, q) be the unique numbers such that Pσ|Σ (τs (η, q)ψs + qϕη ) = Pσ|Σ (τu (η, q)ψu + qϕη ) = 0. Denote by νqs and νqu the equilibrium measures (with respect to σ), respectively, of the potentials τs (0, q)ψs + qϕ0

and

τu (0, q)ψu + qϕ0 .


15

For each fixed η ∈ (−δ, δ), we consider the Legendre transforms D sf,η and Duf,η of q 7→ τs (η, q) and q 7→ τu (η, q), given by Dsf,η (α) = sup{τs (η, q) + qα : q ∈ R} and Duf,η (α) = sup{τu (η, q) + qα : q ∈ R}. These functions are called respectively the stable and unstable dimension spectra for the measure µf,ϕη . It follows from Proposition 2 that Df,η (α) = sup{τs (η, q) + τu (η, q) + qα : q ∈ R} = sup inf (Dsf,η (β) − qβ) + inf (Duf,η (β) − qβ) + qα : q ∈ R β∈R

β∈R

for each α. The following statement establishes a stronger version of local multifractal rigidity for linear horseshoe maps. Theorem 12. For every (r1 , . . . , rp ) in an open dense set of X, if Φ is of class C 1 , and at least one of the functions η 7→ D sf,η and η 7→ Duf,η is constant in a neighborhood of η = 0, then Φ is constant in a neighborhood of η = 0. Proof. Without loss of generality we shall assume that η 7→ D sf,η is constant in a neighborhood of η = 0. This implies that the function η 7→ τ s (η, ·) R is constant. By Theorem 3, we have Σ ζ dνqs = 0 for every q ∈ R, where

τ (0,q) q ri .

s s ζ = dΦ dη |η=0 . Note that νq is a Bernoulli measure with probabilities d i Therefore, if q ∈ R then Z p X τ (0,q) q s ζi di s ri = 0, ζ dνq =

Σ

where ζi =

d log rη,i |η=0 . dη

(21)

i=1

One can show that (see, for example, [10])

def

βη = lim

q→+∞

∂τs τs (η, q) = (η, +∞) < 0. q ∂q

For every (r1 , . . . , rp ) in an open dense set of X, the numbers d βi 0 ri are all distinct. Choose k such that dβk 0 rk > dβi 0 ri for every i 6= k. By (21), we obtain Pp τs (0,q) q ri i=1 ζi di 0 = lim = ζk . τ (0,q) q q→+∞ dks rk We can now apply a similar argument to the remaining sum X τ (0,q) q ζi di s ri = 0. i:i6=k

After a finite number of steps we conclude that ζ i = 0 for i = 1, . . ., p. By Theorem 3, for all η in a sufficiently small neighborhood of η = 0, β the numbers di η rη,i are all distinct provided that the numbers d βi 0 ri are all d log r distinct. Hence, one can repeat the above argument to show that dη η,i = 0 for all all i = 1, . . ., p, and all η in a neighborhood of η = 0. This completes the proof of the theorem.

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LUIS BARREIRA

Remark. One can easily verify that for the special type of functions considered in (20), i.e., for functions ψ 1 and ψ2 such that ψk (· · · i0 · · · ) = ψk (· · · j0 · · · ) whenever i0 = j0 , the following property holds: ψ1 = ψ2 if and only if ψ1 is cohomologous to ψ2 . Theorem 1 in the introduction can be obtained from Theorem 12 in the following manner. Under the assumptions of Theorem 1 we have Pf |Λ (q(ϕη − Pf |Λ (ϕη ))) = τs (η, q) log 3 = τu (η, q) log 3. Therefore τs (η, q) + τu (η, q) =

2 P (q(ϕη − Pf |Λ (ϕη ))), log 3 f |Λ

and the function η 7→ τs (η, ·) + τu (η, ·) is constant if and only if at least one of the functions η 7→ τs (η, ·) and η 7→ τu (η, ·) is constant. This shows that in the case of Theorem 1 the assumption in Theorem 12 concerning the maps η 7→ Dsf,η and η 7→ Duf,η can be replaced by the requirement that η 7→ D f,η is constant. We emphasize that the related (and presumably more difficult) Problem B remains open. In the related case of entropy spectra, Schmeling announced recently a version of multifractal rigidity (roughly speaking entropy spectra correspond to the special case of dimension spectra with constant functions ψs and ψu ; see [1] for details). References 1. L. Barreira, Ya. Pesin and J. Schmeling, On a general concept of multifractality: multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity, Chaos 7 (1997), 27–38. 2. L. Barreira, Ya. Pesin and J. Schmeling, Multifractal spectra and multifractal rigidity for horseshoes, J. Dynam. Control Systems 3 (1997), 33–49. 3. L. Barreira and J. Schmeling, Sets of “non-typical” points have full topological entropy and full Hausdorff dimension, Israel J. Math. 116 (2000), 29–70. 4. C. Bonatti, L. D´ıaz and M. Viana, Discontinuity of the Hausdorff dimension of hyperbolic sets, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 713–718. 5. A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of mathematics and its applications 54, Cambridge University Press, 1995. 6. G. Keller, Equilibrium states in ergodic theory, London Mathematical Society Student Texts 42, Cambridge University Press, 1998. 7. R. Ma˜ né, The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.) 20 (1990), 1–24. 8. H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergodic Theory Dynam. Systems 3 (1983), 251–260. 9. J. Palis and M. Viana, On the continuity of Hausdorff dimension and limit capacity for horseshoes, Dynamical Systems, Valparaiso 1986 (R. Bam´ on, R. Labarca and J. Palis Jr., ed.), Lect. Notes in Math. 1331, Springer, 1988, pp. 150–160. 10. Ya. Pesin, Dimension theory in dynamical systems: contemporary views and applications, Chicago Lectures in Mathematics, Chicago University Press, 1997. 11. D. Ruelle, Thermodynamic formalism, Encyclopedia of mathematics and its applications 5, Addison-Wesley, 1978. 12. D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), 99–107. 13. J. Schmeling, On the completeness of multifractal spectra, Ergodic Theory Dynam. Systems 19 (1999), 1595–1616.


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14. D. Simpelaere, Dimension spectrum of axiom A diffeomorphisms. II. Gibbs measures, J. Statist. Phys. 76 (1994), 1359–1375. 15. F. Takens, Limit capacity and Hausdorff dimension of dynamically defined Cantor sets, Dynamical Systems, Valparaiso 1986 (R. Bam´ on, R. Labarca and J. Palis Jr., ed.), Lect. Notes in Math. 1331, Springer, 1988, pp. 196–212. 16. H. Weiss, Some variational formulas for Hausdorff dimension, topological entropy, and SRB entropy for hyperbolic dynamical systems, J. Statist. Phys. 69 (1992), 879–886. ´ tica, Instituto Superior T´ Departamento de Matema ecnico, 1049-001 Lisboa, Portugal E-mail address: [email protected] URL: http://www.math.ist.utl.pt/~barreira/