1Introduction Sinai-Ruelle-BowenMeasuresforPiecewiseHyperbolicTransformations

Sinai-Ruelle-Bowen Measures for Piecewise Hyperbolic Transformations

Medidas de Sinai-Ruelle-Bowen para Transformaciones Hiperb´olicas a Trozos

Fernando Jos´e S´anchez-Salas (fsanchez@luz.ve)

Departamento de Matem´atica y Computaci´on Facultad de Ciencias, Universidad del Zulia

Maracaibo, Venezuela.

Abstract

In this work we give sufficient conditions for the existence of an er- godic Sinai-Ruelle-Bowen measure preserved by transformations with infinitely many hyperbolic branches.

Key words and phrases: invariant measures, SRB measures, equilib- rium states, fractal dimensions, horseshoes, piecewise hyperbolic trans- formations.

Resumen

En este trabajo damos condiciones suficientes para que exista una medida de Sinai-Bowen-Ruelle erg´odica preservada por transformacio- nes con infinitas ramas hiperb´olicas.

Palabras y frases clave: medidas invariantes, medidas SRB, esta- dos de equilibrio , dimensi´on fractal, herraduras, transformaciones hi- perb´olicas a trozos.

1 Introduction

We say that a Borel probability measure µ is a Sinai-Ruelle-Bowen (SRB) measure if it is smooth along unstable leaves. An SRB measure preserved by a C²diffeomorphism of a compact riemannian manifold isphysically observable since it reflects the asymptotic behaviour of a set of positive volume, that

Recibido 2000/10/08. Revisado 2001/02/23. Aceptado 2001/02/25.

MSC (2000): Primary 37C40, 37C45. Secondary 37D20, 37D35.

is, for every continuous function φ it holds limn→+∞1/ nP_n−1

k=0φ(T^k(x)) = R φ(z)dµ(z) for a set of pointsxof positive volume. It is a most interesting fact that the relevant observable measures of mappings which are regular on their expanding directions are smooth along unstable leaves and they are extremals of certain variational principle. Cf. [8] and [12].

The aim of this paper is to prove the existence of SRB measures for cer- tain generalized baker’s transformations defined by infinitely many hyperbolic branches.

Theorem A Let R = B^u×B^s be a rectangle in R^m (m = s+u) and T : Ωªbe aC² horseshoe constructed inRdefined byC² hyperbolic branches Ti : Si −→ Ui where {Si} is a countable (possibly infinite) collection of non overlapping stable cylinders which cover R up to a subset of zero Lebesgue measure. Suppose in addition that the non-linear expansion along invariant manifolds is bounded from below by someλ >1and that the followinga priori bounded distortion condition holds:

sup

i>0

sup_z∈Sisup_ξ,η∈Ku(w),kξk≤1kD²Ti(z)(ξ, η)k

(inf_ξ∈K^u_(w),kξk=1kDTi(w)ξk)² <+∞ (1) fori >0, whereK^u(w)is the unstable cone atw. Then T preserves a unique ergodic SRB measureµ=µΩsupported onΩwhose ergodic basin coversRup to a Lebesgue measure zero set.

We refer to next Section for definitions. Here, as for Markov piecewise expanding endomorphisms, there is an essential difference between finite and infinite hyperbolic branches since derivatives grow with i and a priori rela- tions between the first and second derivatives are key if we want to get some bounded distortion estimates.

To the best of my knowledge these type of problems were considered for the first time by Jakobson and Newhouse in [3]. However, the present approach not only provides a higher dimensional generalization of [3, Theorem 1] but it also gives a new proof of that result in dimension two.

One possible source of interest in these models comes from a program out- lined at [4] aimed at describing H´enon attractors by an inducing approach similar to that used in one dimensional dynamics (cf. [2, Chapter 5]). In fact, generalized baker’s transformations as described in Theorem A are natural higher dimensional generalizations of piecewise expanding interval transforma- tions. However, as pointed to me by Marcelo Viana, in the case of H´enon maps we should not expect a reduction to a model like one described in Theorem A. Instead, we would expect to induce some sort of generalized baker’s trans- formations having a maximal invariant compact set with hyperbolic product

structure and non trivial unstable Cantor sets of positive Lebesgue measure.

This is due to the presence of homoclinic tangencies. Unfortunely, as far as I know, no complete exposition of these constructions seems to be yet available.

However, we can see an outline in [15].

Indeed, the objects that we shall consider are similar to the generalized horseshoes introduced in Young’s paper [15]. Actually, under the a priori bounded distortion condition (1) we can prove that they support an SRB measure iff their unstable Cantor sets have positive Lebesgue measure. That gives an independent proof and a sort of converse to [15, Theorem 1.1].

Condition (1) seems to be a natural higher dimensional analogous of bounded distortion condition (D1) in [3] and permits to treat generalized horseshoes with non trivial unstable Cantor sets as well, improving Jakobson- Newhouse’s results, even in dimension two.

Horseshoes with infinitely many branches and bounded distortion also ap- pear when inducing hyperbolicity in non uniformly hyperbolic systems. Ac- tually, we proved in [14], using ideas and methods of Pesin theory as exposed in [6], that given a point pin the support of an ergodic hyperbolic measure µ with positive entropy and 0 < δ <1, we can find a regular neighborhood R of p and a subset Ω ⊂ R with hyperbolic product structure, such that µ(Ω) ≥(1−δ)µ(R). Ω has the same structure of the generalized horseshoes introduced in the present research. This result seems to sustain our claim that the study of these models can give us a better understanding of some geometrical and statistical properties of chaotical dynamical systems.

2 Generalized horseshoes: definitions and statement of results

We shall first recall some definitions and terminology needed to state our main results.

Definition 2.1. Let Ω⊆ R^m be a compact subset which is invariant by a piecewise smooth invertible transformation T : D −→ R^m from a domain D ⊂R^m. We say that Ω has ahyperbolic product structure if there are two continuous laminations by discs of complementary dimensionF^sandF^usuch that:

1. eachW ∈ F^s is an stable invariant manifold, i.e., distances of positive iterates of pointsx, y∈W contract exponentially; similarly for negative iterates of points on leaves ofF^u;

2. any pair of invariant submanifolds W^s ∈ F^s and W^u ∈ F^u intersect transversally at an angle bounded from below;

3. Ω =S

F^s ∩ S F^u.

A generalized baker’s transformation defined by countable (possibly in- finitely many) hyperbolic branches defines a maximal invariant subset with hyperbolic product structure. We shall refer to these sets asgeneralized horse- shoes.

For this we let R = B^s×B^u ⊆ R^m, where B^s ⊆ R^s and B^u ⊆ R^u are unit closed balls. We shall suppose that R is endowed with two cone fields K^u = K^u(x) and K^s = K^s(x) defined at every point in x ∈ R. We also suppose that these cones families extend continuously to a slightly larger neighborhood ˆRcontainingR. We associate with these cone fieldsadmissible stable and unstable submanifolds Γ^s(R) = {γ^s} and Γ^u(R) = {γ^u}. Each γ^s∈Γ^s(R) is the graph of aC¹ mapφ:B^s−→B^u such that TxW ⊆K^s(x) for every x∈γ^s. Likewise for admissible unstable submanifolds. Also stable and unstable cylinders can be defined. A compact connected subset S ⊆ R is an admissible stable cylinder if it admits a foliation by admissible stable submanifolds and if its unstable sections S ∩ γ^u are convex sets for every γ^u∈Γ^u(R). Unstable cylinders are defined similarly.

Admissible manifoldsγ^u∈Γ^uandγ^s∈Γ^sintersect transversally with an- gle ∠(Txγ^u, Txγ^s) bounded from below. Admissible manifolds have bounded geometry, that is, we can find a constant C = C(Γ) > 1 depending only on γ and the diameter of R such that Vol (γ^u) and diam (γ^u) are bounded in [C⁻¹, C]. Further, C⁻¹ ≤ distγ^u(x, y)/kx−yk ≤ C for every x, y in an unstable admissible submanifold γ^u ∈ Γ^u where distγ^u (resp. diamγ^u) is the distance (resp. diameter) on the submanifold γ^u defined with the in- trinsic riemannian metric. Similarly for Volγ^u. Likewise for the admissible s-submanifolds. This fact will be used throughout.

The set Γ^u (resp. Γ^s) is endowed with an structure of Banach space given by the identification with C¹(B^u, B^s) equipped with theC¹ norm. In particular it is a complete metric space. Moreover, there is a well definedgraph transform ΓT : Γ^u −→ Γ^u defined by ΓT(γ^u) = T(γ^u)∩R. The following result, due to Aleeksev and Moser, will be used elsewhere without further comments: The graph transform is a contraction, i.e., there exists0< θ <1 such that distC¹(ΓT(γ₁^u),ΓT(γ₂^u))≤θdistC¹(γ₁^u, γ₂^u). In addition, (γ^u, γ^s)7−→

γ^u∩γ^sis a Lipschitz map from Γ^u×Γ^sto R.

We will consider maps{Ti:Si−→Ui} where{Si}(resp.{Ui}) are count- able collections of non overlapping stable (resp. unstable) cylinders. Each map has a C² extension ˆTi : ˆSi −→ Uˆi to neighborhoods of Si and Ui

which are stable and unstable cylinders in ˆR and such that ˆTi maps hyper- bolically ˆSi onto ˆUi, that is, it preserves strictly the cone families K^s and K^u: K^s(Ti(x)) ⊆ intDTi(x)K^s(x) and DTi(x)K^u(x) ⊆ intK^u(Ti(x)). Each Ti:Si−→Ui will be called anhyperbolic branch.

We can use the hyperbolic branches Ti to define a piecewise hyperbolic map T :S

i intSi −→S

i intUi, by settingT | intSi=Ti and extendsT to some well defined measurable transformation ˆT :S

iSi−→S

i Ui, which isC² smooth in a dense subset of its domain and preserves the admissible manifolds.

This extensions are non unique. However, as long as their singular set is negligible for all the purposes of ergodic theory, we can omit this arbitrariness, so we will continue denoting extensions ˆT byT to avoid messy notations.

Now, given a family of hyperbolic branchesTi:Si−→Uiwe define nested sequences of stable and unstable cylinders converging to two laminations of stable and unstable admissible manifolds F^s and F^u, respectively. In fact, given a sequence (i0,· · · , in,· · ·) =i∈N^Nwe define

Ui0···in−1 =

n−1\

k=0

Tin−1 ◦ · · · ◦ Ti0Ui0, Si0···in−1 =

n−1\

k=0

T_i⁻¹_n−1 ◦ · · · ◦ T_i⁻¹₀ Si0

where, by abuse of language, we omit the domain of compositions. We will call thesestable and unstable cylinders of leveln. The sequence{Si0···in−1}n≥0

is a nested sequence of stable cylinders. Similarly so {Ui0···in−1}n≥0. In fact, graph transform contraction properties imply that there is a unique admissible manifold γ^s =γ^s(i) such that d_C1(S_{i0···in−1}, γ^s) converges to zero as n−→

+∞and likewise for the unstable cylinders.

We shall suppose that non linear expansion along unstable admissible man- ifolds is bounded from below in the following sense: there is a constantC >1 such that, for every pair of points xand y contained in γ⁰ = γ^u ∩ P (the intersection of an admissible unstable manifold γ^u and an stable cylinder of level n, P ∈ ℘n), d_γ^k(Tⁿ(x), Tⁿ(y)) ≥ Cλ^n−kd_γ⁰(T^k(x), T^k(y)) holds for k = 0,· · ·, n−1, uniformly in x and y, where γ^k = T^k(γ⁰). Due to the bounded geometry of admissible submanifolds these conditions are equivalent to kTⁿ(x)−Tⁿ(y)k ≥Cλ^n−kkT^k(x)−T^k(y)k, for a suitable constantC >0 depending only on the bounded geometry of Γ^u . A similar statement holds true for the inverseT⁻¹.

Now we defineF^s={γ^s(i) :i∈N^N}(resp. F^u={γ^u(i) :i∈N^N}). These laminations are clearly T-invariant. Moreover, due to non-linear expansion properties along admissible manifolds, we conclude thatW^s(x,R) =γ^s(i) and W^u(x,R) =γ^u(j) are the local stable manifold ofxwhere{x}=γ^s(i)∩γ^u(j).

Indeed, W^s(x,R) = {y ∈ R : kTⁿ(x)−Tⁿ(y)k ≤ Cλ⁻ⁿ, ∀n ≥ 0} and

similarly so for the unstable local manifold for backward orbits.

We also recall for further use that there are two continuous subbundlesE^s (resp. E^u) of stable (resp. unstable) subspaces which are the tangent spaces to invariant leaves. These families of subspaces satisfies a H¨older condition.

This is standard. See for example [5].

Let Ω =S

F^s ∩ S

F^u. Ω is a compact, perfect subset ofR^m contained in the cube R. Topologically it might be a Cantor set times an interval or a product of two Cantor sets or even it might fill R up to a measure zero set. Ω is the maximal invariant subset of ˆT. Therefore, Ωis endowed with a hyperbolic product structure.

Definition 2.2. A set Ω with a hyperbolic product structure and a dynamics T : Ω ª given by a collection of C² hyperbolic branches Ti : Si −→ Ui as defined above shall be called a horseshoe with infinitely many branches or, shortly, ageneralized horseshoe.

Ordinary horsehoes are simply those defined by finitely many disjoint hy- perbolic branches. The following is our first main result

Theorem B Let T : Ωªbe aC² generalized horseshoe with bounded distor- tion and expansion coefficient bounded from below. Then there exists a unique ergodic measure µ=µΩsuch that:

1. dim_u(Ω) =h_µ(T)/R

logJ^uT(x)dµ(x), where dim_u(Ω) denotes the un- stable dynamical dimension ofΩ;

2. µ_F^u_(x), the projection of µonto the unstable leave F^u(x), is equivalent to the dynamical measure D_α,F^u_(x) in dimension α = dimu(Ω). Fur- thermore, there is a constant C >1 such that

C⁻¹≤ µF^u(x)(P)

(VolF^u(x)P)^dimu(Ω) ≤C for every P ∈℘, (2) where Vol_F^u_(x)is the volume defined by the intrinsic riemannian metric of the unstable submanifold F^u(x); in particular, there is a constant C >1 such that

C⁻¹≤ µ(P)

Vol(P)^dimu(Ω)0 ≤C for every geometrical cylinder;

3. the stable lamination F^s is absolutely continuous with respect to the dynamical measure classes {Dα,γ^u}γ^u∈Γ^u and it has a continuous and

bounded Jacobian:

d H^∗Dα,γ^u

dDα,γ^u (x) =

"_+∞

Y

i=1

J(T |γ^u)(Tⁱ(x)) J(T |γ^u)(Tⁱ(H(x)))

#_α

. (3)

Here H^∗Dα,γ^u is the pullback measure under the holonomy mapH. Notation: throughout this paper we will adopt the following convention: given two positive functionsf andgwe will writef ³g if there is a constantC >1 such that C⁻¹≤f /g≤C uniformly in their domain.

HereHγ^u,γ^u is the holonomy map of the local stable laminationF^sdefined by admissible unstable sectionsγ^u, γ^u∈Γ^u(R): Hγ^u,γ^u(x) =F^s(x)∩γ^u, for every x ∈ γ^u ∩ S

x∈ΩF^s(x). J(T | γ^u) will denote throughout this work the Jacobian of T restricted to γ^u with respect to the intrinsic riemannian volume. J^uT(x) =J(T | F^s(x))(x) is the unstable Jacobian ofT with respect to to the intrinsic riemannian volume of the local unstable manifold F^s(x).

The measureµF^u(x) is defined on the unstable Cantor sets cut by a local unstable manifold of Ω:

µF^u(x)(A) =µµ [

z∈A

F^s(z)

¶

, for every Borel subsetA⊂ F^u(x)∩Ω.

We proved the above result for ordinary horseshoes. Cf. [13].

To recall what is the dynamical measure we first introduce the dynamically defined generating net of stable cylinders. Namely, let ℘n ={Si0···in−1 : i∈ N^N} be the stable cylinders of level n and ℘0 = {Si}i>0. Clearly ℘n = T⁻ⁿ℘0. Further W

n∈ZT⁻ⁿ℘0 generates the σ-algebra of Borelian sets of Ω and W

n≥0T⁻ⁿ℘0 generates the σ-algebra B^s of Borel subsets which are a reunion of stable leaves, that is, ifB∈ B^sthenF^s(x)∩B =F^s(x), for every x∈B. We denote by ℘the reunion of all cylinders in℘n forn≥0.

Let γ^u ∈ Γ^u and define a generating net of subsets on γ^u taking the intersections of γ^u with cylinders in℘:

℘(γ^u) ={P ∩ γ^u:P ∈℘n is an stable n-cylinder for some n≥0}.

Then we define an outer measure inγ^u, using the Carath´eodory construction, by taking coverings by℘(γ^u)-sets and using a riemannian volume Vol^u_γ as set function settingDa,℘,γ^u(X) = limδ→0⁺infU

P_+∞

i=1Volγ^u(Ui)^a X⊆γ^u, where the infimum is taken over all the δ-coverings by℘(γ^u)-sets.

A main point in the proof of Theorem B is to show that this construction defines a non-trivial measure class which shall be denoted Dα,W. Associated

to this fractal measure there is a Carath´eodory’s dimensional characteristic, the dynamical dimension: dimD,W(X) = inf{a > 0 :Da,W(X) = 0} Cf. [13]

and [11]. Absolute continuity implies that there is a well defined unstable dimension, dimu(Ω) = dimD,W(Ω), which does not depend on the particular local unstable manifoldW ∈ F^u.

The first statement in Theorem B follows from inequalities (2) using Billings- ley’s [1, Theorem 14.1] and bounded distortion estimates. Indeed, (2) implies that µ(℘n(x))³Vol (℘n(x))^dimu(Ω), by the definition of transversal measure and using bounded distortion estimates (see next Section). So, it holds that

dimu(Ω) = lim

n→+∞

lnµ(℘n(x) ln Vol (℘n(x)).

Now bounded distortion of the volume implies that Vol (℘n(x))³[J^uTⁿ(x)]⁻¹, uniformly bounded by some universal constant. Then, using the Pointwise Er- godic Theorem and Shannon-McMillan-Breiman’s property we get the claimed identity. This is exactly what Billingsley did in [1] in a simpler scenario.

For horseshoes in the plane we have the following result, which gener- alizes [3, Theorem 1] for bidimensional generalized baker’s transformations producing horseshoes with non trivial unstable Cantor sets.

Theorem C Let T : Ωª be aC² generalized horseshoe with bounded dis- tortion and µthe equilibrium state given by Theorem A. If dimH denotes the dimension of a set or of a measure, then :

1. the unstable dimension ofΩ is the Hausdorff dimension of its unstable Cantor sets, that is,dimu(Ω) = dimH(Ω∩ F^u(x))for every x∈Ω;

2. the stable laminationF^sof Ωis Lipschitz;

3. transversal measures µ_F^u_(x) are equivalent to the Hausdorff measure bounded by uniform constants; indeed, there is a constant C >1 such that

µF^u(x)(B(z, r))³r^dimu(Ω) (4) for every x∈Ω andz∈Ω∩ F^u(x), bounded in [C⁻¹, C]In particular, the transversal measures are dimensionally exact.

This is a consequence of the equivalence between the dynamical measure and Hausdorff measure for conformal dynamically defined Cantor sets and it gen- eralizes [3, Theorem 1.1]. Compare [10, Chapter 4, Proposition 3, pp. 72].

Theorem D Let T : Ωª be a C² generalized horseshoe satisfying the hy- potheses in Theorem C. The following statements are equivalent:

1. µsatisfies the Pesin entropy formula hµ(T) =R

logJ^uT(x)dµ(x);

2. VolW(Ω∩W)>0 for some local unstable manifold W ∈ F^u;

3. the volume VolW(Ω) of the unstable Cantor sets Ω ∩ W^u(x,R)is uni- formly bounded away from zero and F^s is absolutely continuous with respect to Lebesgue measure with uniformly bounded Jacobians;

4. µ_Ωis absolutely continuous with respect to the riemannian volume along the local unstable manifolds.

In addition, if any of the above conditions hold the stable invariant lamination F^s is absolutely continuous with respect to Lebesgue measure and it has a bounded Jacobian, namely

d H^∗Lγ^u

dLγ^u

(x) =

+∞Y

i=1

J(T |γ^u)(Tⁱ(x))

J(T |γ^u)(Tⁱ(H(x))) (5) for every pair of admissible unstable manifolds γ^u, γ^u∈Γ^u. HereLγ^u is the Lebesgue measure class of γ^u. Therefore, the ergodic basin of the asymptotic measure contains a set of positive volume, so µΩis physically observable.

Theorem A at the Introduction is simply a particular case of Theorem D.

As we can see, arguments in [13] extend straighforwardly to the present set up once we check that the dynamical measure class is non trivial and that τ : Ω/F^s−→Ω/F^ssatisfies an standard bounded distortion condition.

3 Bounded distortion and bounded geometry estimates

Let T : Ω ª be a C² hyperbolic horseshoe defined by countably (possibly infinite) many hyperbolic branches Ti : Si −→ Ui. We suppose in addition that the collection Ti have non-linear expansion bounded from below and bounded distortion. We introduce for further use the following

Definition 3.1. We define the unstable infimum norm as m^u(DT(z)) = inf

ξ∈K^u(z),kξk=1 kDT(z)ξk.

Lemma 3.1. There is constant C >1, only depending on the distortion, the expansion coefficient and the bounded geometry of the admissible manifolds, such that for everyγ^u∈Γ^u andn≥0

J(Tⁿ |γ^u)(x)

J(Tⁿ|γ^u)(y) ≤exp(Cdu(Tⁿ(x), Tⁿ(y))) (6) holds whenever ℘n(x) =℘n(y)andx, y∈γ^u.

In particular, we can findC >0 such that for everyn >0

¯¯

¯¯J^uTⁿ(x) J^uTⁿ(y)−1

¯¯

¯¯≤C·du(Tⁿ(x), Tⁿ(y)), wheny∈℘n(x)∩ W^u(x,R).

Proof. Using bounded distortion condition (1) we prove that for everyγ^u∈Γ^u andi >0 the following estimate holds:

sup_z∈γu∩S_i k∇log J(Ti|γ^u)(z)k

infw∈γ^u∩Si m^u(DTi(w)) ≤∆<+∞. (7) This is a straightforward computation. Now, we use (7) and a standard argu- ment to get (6). Letγ^u∈Γ^u an unstable admissible submanifold and denote γ⁰=γ^u∩℘n(x),γⁱ=Tⁱ(γ⁰):

log J(Tⁿ|γ^u)(x) J(Tⁿ|γ^u)(y) ≤

n−1X

i=0

|logJ(T |γⁱ)(Tⁱ(x))−log J(T |γⁱ)(Tⁱ(y))|

≤

n−1X

i=0

sup

z∈γⁱ

k∇logJ(T |γⁱ)(z)k · kTⁱ(x)−Tⁱ(y)k.

Now,

lenght (T(γ))≥ inf

γ⁰(t)kDT(γ(t))γ⁰(t)klenght (γ)

≥ inf

t∈[0,1]m^u(DT(γ(t))) lenght(γ)

for every C¹ smooth curve γ =γ(t), t ∈[0,1], contained inside an unstable admissible submanifold, in particular,

du(T(x), T(y))≥ inf

w∈γⁱ m^u(DT(w))du(x, y), (8)

so kT(x)−T(y)k ≥ C⁻²inf_w∈γⁱm^u(DT(w))kx−yk, for every x, y ∈ γⁱ. Therefore

logJ(Tⁿ|γ^u)(x)/J(Tⁿ |γ^u)(y)

≤C²

n−1X

i=0

sup_z∈γik∇log J(T |γⁱ)(z)k · kTⁿ(x)−Tⁿ(y)k inf_w∈γim^u(DTⁿ⁻ⁱ(w))

≤C²

n−1X

i=0

sup_z∈γik∇log J(T |γⁱ)(z)k · kTⁿ(x)−Tⁿ(y)k inf_w∈γi+1m^u(DTⁿ⁻ⁱ⁻¹(w)) inf_w∈γim^u(DT(w)ξ)

≤C²∆

n−1X

i=0

λ^{−(n−i−1)}· kTⁿ(x)−Tⁿ(y)k

≤C²∆

+∞X

n=0

λ⁻ⁿ· kTⁿ(x−Tⁿ(y)k,

which is bounded byC³∆(1−λ⁻¹)du(Tⁿ(x), Tⁿ(y)), using again the bounded geometry of admissible manifolds and condition (7).

Corollary 3.1. There is a constant C=C(∆, λ,Γ)>1 such that, for every admissible unstable manifold γ^u∈Γ^u and every n >0 it holds

C⁻¹ inf

γ^u∈Γ^u Vol(γ^u)≤J(Tⁿ|γ^u)Volγ^u(℘n(x))≤C sup

γ^u∈Γ^u Vol(γ^u).

Lemma 3.2. Let γ^u = γ^u(x) and γ^u = γ^u(y) be two admissible unstable manifolds passing by xand y in Ω, respectively. Suppose thaty ∈ W^s(x,R) and denote

h(x, y) =

"_+∞

Y

i=1

J(T |γ^u)(Tⁱ(x)) J(T |γ^u)(Tⁱ(y))

#α

.

Then h(x, y) ≤ exp(Cds(x, y)) for a constant C = C(∆, λ,Γ) > 1 where ds(x, y) =dW^s(x,R)(x, y). We have also

h(x, y)

h(x⁰, y⁰) ≤max{e^Cdu(x,x0)), e^Cdu(y,y0))},

for every x⁰ ∈γ^u(x) and y⁰ ∈ γ^u(y) with y⁰ ∈W^s(x⁰,R), where du(x, x⁰) = d_γ^u_(x)(x, x⁰)and likewisedu(y, y⁰).

Proof. We can find a C² foliation of R by admissible unstable leaves, say γ^u=γ^u(z) passing by γ^u(x) andγ^u(y). Using bounded distortion condition we get

sup_w∈γs(z)k∇log J(Ti|γ^u(w))(w)k

inf_w∈γs(z)m^u(DT_i(w)) ≤∆<+∞, (9) for every leave γ^s(z) contained in Si, every z ∈Rand i >0. Let us denote Ji(z) =J(Ti|γ^u(z))(z). Arguing similarly as we did before we get

log Ji(T^j(x))

Ji(T^j(y)) ≤ sup

w∈W^s(T^j(x),R)

k∇logJi(w)k · kT^j(x)−T^j(y)k

≤ C²sup_w∈Ws(T^j(x),R)k∇logJi(w)k

infw∈W^s(T^j(x),R) m^u(DTi(w)) kT^j+1(x)−T^j+1(y)k.

Thus log(Ji(T^j(x))/ Ji(T^j(y)))≤∆C⁴λ^−(j+1)ds(x, y) and then log h(x, y)≤

+∞X

j=1

∆C⁴λ^−(j+1)ds(x, y) = ∆C⁴(1−λ⁻¹)λ⁻²ds(x, y).

Further, h(x, y)/ h(x⁰, y⁰)≤exp (C(ds(x, y)−ds(x⁰, y⁰)). Therefore, h(x, y)

h(x⁰, y⁰)≤max{exp(Cdu(x, x⁰)),exp(Cdu(y, y⁰))}, for some C=C(∆, λ,Γ)>1 as claimed.

Corollary 3.2. There is a constant C =C(∆, λ,Γ) >1 such that, for any two admissible manifolds γ^u andγ^u inΓ^u it holds Volγ^u(P)³Volγ^u(P)for every P ∈℘.

In particular, the volume of the cylinder Vol (P) is comparable with the volume of its unstable sectionsγ^u ∩P, i.e., Volγ^u(P)³Vol (P), bounded by uniform constants which do not depend onpneitherγ^u∈Γ^u.

Proof. It follows from Corollary 3.1 and Lemma 3.2 that for everyn >0 Volγ^u(℘n(x))

Volγ^u(℘n(x))≤Csup_γu∈Γ^u Vol (γ^u) infγ^u∈Γ^u Vol (γ^u), and similarly for the lower bound.

Corollary 3.3. The holonomy of the stable lamination is absolutely contin- uous with respect to the dynamical class {Dα,γ^u}γ^u∈Γ^u and it has a bounded Jacobian.

This follows from Dα,γ^u(B) ³ Dα,γ^u(H(B)), which is bounded by con- stants C = C(∆, λ,Γ) >1, for every γ^u and γ^u, where H = H_γ^u_,γ^u is the holonomy transformation defined by these unstable admissible manifolds.

We shall prove later that h = hγ^u,γ^u below is the Jacobian of H with respect to the dynamical measure class:

h(x) =

"_+∞

Y

i=1

J(T |γ^u)(Tⁱ(x)) J(T |γ^u)(Tⁱ(H(x)))

#_α .

The following result is a straightforward consequence of Lemma 3.2.

Lemma 3.3. For everyγ^u andγ^u in Γ^u it holds

h(x)≤exp(Cds(x, H(x))) (10)

for a constant C=C(∆, λ,Γ)>1. Also, for everyxandy in γ^u h(x)

h(y) ≤max{e^Cdu(x,y)), e^Cdu(H(x),H(y)))}. (11) As a consequence of the previous discussion we get a constant C > 1, depending only on the distortion and the bounded geometry of admissible manifolds, such that:

1. C⁻¹≤Volu(P∩γ^u)J(Tⁿ|γ^u)(x)≤C for everyx∈P∩γ^u; 2. C⁻¹≤Volu(P∩γ^u)/Volu(P∩γ^u)≤C;

3. C⁻¹≤Vol (P)/Volu(P∩γ^u)≤C and

4. C⁻¹≤Vol (P)J(Tⁿ |γ^u)(x)≤C, for everyx∈P∩γ^u,

for any pair of admissible unstable sections γ^u, γ^u ∈ Γ^u, n > 0 and every stable cylinder P ∈℘. We shall refer to all these properties as the volume lemma.

4 Proofs of the main results

Lemma 4.1. The dynamical measure class is non trivial.

Proof. LetT : Ωªbe aC²hyperbolic horseshoe with infinitely many branches and bounded distortion. We claim that there is a sequence Ωn ⊂Ω of ordi- nary horseshoes with finitely many branches and ergodic measures µn such that

1. Ω =S

nΩn;

2. there is a constant C > 1 such that, for every local unstable mani- folds W = W^u(x,R) and for every P ∈ ℘(Ωn), the family of stable cylinders generating the stable subsets of Ωn, it holds thatµn,W(P)³ VolW(P)^dimu(Ωn) bounded by C. Here µn,W denotes the projection of µn the natural equilibrium state ofµn ontoW along the stable lamina- tion.

Indeed, letµα the equilibrium state of the potential−αlnJ^uT of an or- dinaryC² horseshoe andµW the projection ofµαontoW =W^u(x,R) along the stable leaves. We proved in [13, Theorem 2.4] that µW is equivalent to the dynamical measureDα,W. Moreover, we found a constantC >1 only de- pending on bounds of the non linear distortion of the volume along unstable leaves, the expansion coefficient of Ω and the bounded geometry of admissible manifolds such that

C⁻¹ inf

W∈F^uVol (W) ≤ µW(P)

(VolWP)^dimu(Ω) ≤ C sup

W∈F^uVol (W), (12) for every stable cylinderP ∈℘.

Now, let Ωnbe the horseshoe generated byTi:Si−→Ui, fori= 1,· · · , n.

Ω is the topological limit of these Ωn, that is Ω = S

nΩn. By distortion estimates in Corollary 3.1 and Lemma 3.3 we can see that the bounds for the non linear volume distortion of Ωn are independent of n > 0. In particular we can find d >0 such that, for everym >0 andn >0 it holds that

J^uT^m(x)

J^uT^m(y) ∈[e^−d, e^d] whenever ℘ⁿ_m(x) =℘ⁿ_m(y),

where wpⁿ_m=wpm(Ωn) are the generating stable cylinders of orderm >0 of the horseshoe Ωn.

Let µ_n denotes the Gibbs measure of φ = −dim_u(Ω_n) lnJ^uT. Then it is the equilibrium state which maximizes the unstable dimension for Ωn and

µn,W(P) ³VolW(P)^dimu(Ωn) for everyP ∈℘(Ωn), bounded in [C⁻¹, C] by some constantC=C(∆, λ,Γ)>1. Notice also that℘(Ωn)⊂℘(Ωn+1).

Now let α= limn→+∞dimD,W(Ωn). This limit exists since dimD,W(Ωn) is monotone and bounded. Let µ^∗ be a limit point ofµn. µ^∗ is ergodic since it is a limit of ergodic measures and for everyP ∈℘it holds

µ^∗(P)≥lim sup

n→+∞µn(P)≥C⁻¹lim sup

n→+∞ VolW(P)^dimD,W(Ωn)

sinceP ∈℘is closed. Thusµ^∗(P)≥C⁻¹VolW(P)^αfor everyP ∈℘, conclud- ing that Dα,W(Ω) <+∞, using Frostman’s lemma argument. Furthermore, µ^∗(B) ≥ C⁻¹· Dα,W(B) for every Borel subset B ⊆ Ω so it is absolutely continuous respect to the dynamical measure class. Similarly so,

µ^∗(intP)≤lim inf

n→+∞µn(intP)≤C lim inf

n→+∞VolW(P)^dimD,W(Ωn),

since Vol (∂ P) = 0 for everyP ∈℘. Then,µ^∗(intP)≤CVolW(P)^αand this implyDα,W(Ω)>0.

Now, we recall thatτ(x) =H_W(T(x)),W◦T(x), is the projection ofT along the stable leaves. HereW =W^u(x,R) andW(T(x)) =W^u(T(x),R). Now,τ has a Jacobian with respect toDα,W. For this we use that the dynamical mea- sure class is conformal and the absolute continuity of the stable lamination.

Indeed, using conformality we check easily that Jατ(x) = d H^∗Dα,W

dDα,W(T(x))(T(x))[J^uT(x)]^α, for every local unstable manifold W =W^u(x,R).

Lemma 4.2. There exists a constantC=C(∆, λ,Γ)>1 such that

¯¯

¯¯J_ατⁿ(x) Jατⁿ(y) −1

¯¯

¯¯≤C d^∗(τⁿ(x), τⁿ(y)),

whenever ℘(n, W)(x) =℘(n, W)(y), where℘(n, W)(x) denotes the connected component of the intersection℘n(x) ∩ W in W =W^u(x,R) containingx.

Proof. First notice that there is a constant C > 1, depending only on the bounded geometry of the admissible manifolds, such that

ln

"_n−1 Y

k=0

J^u(τ^k(x)) J^u(τ^k(y))

#

≤Cd^∗(τⁿ(x), τⁿ(y))^θ.

This follows from the Bounded Distortion Property (7) andkτ^k(x)−τ^k(y)k ³ d^∗(τ^k(x), τ^k(y)) =d^∗(T^k(x), T^k(y)).

Also, using the Bounded Distortion Property (11) we conclude that lnh(T(τ^k(x)))

h(T(τ^k(y))) ≤Cmax{du(T(τ^k(x)), T(τ^k(y)))^θ, du(τ^k+1(x), τ^k+1(y))^θ}, which is in turn no greater than 2Cd^∗(τ^k+1(x), τ^k+1(y)^θ), for some constant C >1, depending only on the bounded geometry, sinceduis comparable with d^∗. Thus, for everyn >0 we have

ln

"_n−1 Y

k=0

h(T(τ^k(x))) h(T(τ^k(y)))

#

≤2C

n−1X

k=0

λ^−n−k−1d^∗(τⁿ(x), τⁿ(y))^θ

≤C(1−λ⁻¹)d^∗(τⁿ(x), τⁿ(y))^θ, absorbing the various constants into someC=C(Γ) we get

Jατⁿ(x)

Jατⁿ(y) ≤exp(Cd^∗(τⁿ(x), τⁿ(y))^θ).

Then, an easy argument concludes the proof.

Lemma 4.2 above shows that the endomorphismτ of the unstable Cantor set Λ =W∩Ω endowed with the Borel measureDα,W satisfies the hypotheses of [9, Chapter III, Theorem 1.3]. Therefore, there is a unique ergodic mea- sureµ_F^u_(x) which is equivalent toD_α,F^u_(x)and which maximizes dimension.

Compare also [2, Chapter V, Theorem 2.2], [7, Chapter 6] and [16].

This defines an ergodic Borel probability ˜µ defined over the stable Borel subsetsB^s simply by setting

µW(B) = ˜µ Ã[

z∈B

F^s(z)

!

for every Borel subset B⊂ F^u(x)∩Ω.

By the absolute continuity of the stable lamination this measure ˜µ does not depend on the unstable leave W = W^u(x,R) choosen. Now, W

n≥0TⁿB^s generatesB(Ω), the Borel subsets of Ω. This permits to extend ˜µ to a Borel probability µ=µΩ defined overB(Ω). µW is precisely the projection ofµΩ

ontoW. This concludes the proof of Theorem A.

On the other hand it can be proved, using bounded distortion estimates, thatDα,W(P)³(VolWP)^αfor every local unstable manifoldW ∈ F^u, where

α = dimu(Ω). Compare for example [10, Chapter IV]. As the transver- sal measure µW is equivalent to the dynamical measure, we conclude that µW(P)³(VolWP)^dimu(Ω). This concludes the proof of Theorem B.

Now we are ready to prove Theorems C and D.

Proof of Theorem C

Using the Bounded Distortion Lemma and volume estimates we show that for every n≥0 andP ∈ ℘^∗ such thatTⁿ(P) is an unstable cylinder (i.e., such thatTⁿ(P) has full width) diam (P∩W^u(x,R))³ kJ^uTⁿ(x)k⁻¹bounded by a constant not depending onx,Porn. Now, givenr >0 andx∈Ω^∗we define n=n(x, r)>0 as the minimum positive integer satisfying (J^uTⁿ(x))⁻¹≤r and (J^uTⁿ⁻¹f(x))⁻¹> r. Therefore,

B(x, C⁻¹r)∩γ^u⊆Sn(x)∩γ^u∩γ^u⊆B(x, C r)∩γ^u (13) for every 0< r <1 andx∈Ω and some universal constantC >1, depending only on the distortion and the bounded geometry of admissible manifolds, where W^s(x,R) = T

n≥0Sn(x) is a nested sequence of stable cylinders con- verging to the local stable manifold, n = n(x, r) and Tⁿ(S_n(x)∩γ^u) is an admissible unstable manifold.

We use this to define Moran’s covers ℘r for every 0 < r < 1 associated to ℘according to Cf. [11, Chapter 7]. Moran’s covers satisfy the following finite multiplicity property: there exists a universal constant M > 1 such thatB(x, r)∩γ^u intersects at mostM atoms in the family℘r(γ^u), for every x ∈ Λ, 0 < r < 1 and admissible unstable manifold γ^u. M should depend in principle on γ^u, however, by geometry of admissible manifolds shows that this dependence can be dropped out.

By Theorem C,µF^u(x)(P)³diam (P∩W^u(x,R))^dimu(Ω). So for everyP ∈

℘, henceµF^u(x)(B(x, r))³diam (B(x, r) ∩ γ^u))^dimu(Ω) which is comparable with r^dimu(Ω) for every x ∈ Ω, 0 < r < 1, using the bounded geometry of admissible manifolds. So, dimH(Ω∩W^u(x,R) = dimu(Ω), and the dynamical measure, the Hausdorff measure and the transversal measure are in the same measure class and the transversal measure µF^u(x) has the strong uniform distribution property µF^u(x)(B(x, r)) ³ r^dimu(Ω), for every x∈ Ω and 0 <

r <1, bounded by a universal constant.

In particular, the holonomies have a bounded Jacobian respect to the Hausdorff measure. Finally, by Frostman’s uniform distribution property and the existence of a bounded Jacobian with respect to the Hausdorff measure for the holonomies of the stable laminationF^s we see that

kx−yk^α³ Hα,γ^u([x, y])³ Hα,γ^u([H(x), H(y)])³ kH(x)−H(y)k^α,