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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

C2_{diffeomorphism of a compact riemannian manifold isphysically observable}

since it reflects the asymptotic behaviour of a set of positive volume, that

is, for every continuous function φ it holds limn→+∞1/ nP_kn=0−1φ(Tk(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 = Bu_×Bs _{be a rectangle in R}m _{(m = s+u) and}

T : Ωª_{be aC}2 _{horseshoe constructed inRdefined byC}2 _{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

supz∈Sisupξ,η∈Ku(w),kξk≤1kD

2_{Ti(z)(ξ, η)k}

(infξ∈Ku_(w),kξk=1kDTi(w)ξk)2

<+∞ (1)

fori >0, whereKu_{(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.

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⊆ Rm _{be a compact subset which is invariant by a}

piecewise smooth invertible transformation T : D −→ Rm _{from a domain}

D ⊂Rm_{. We say that Ω has ahyperbolic product structure if there are two}

continuous laminations by discs of complementary dimensionFs_andFu_such that:

2. any pair of invariant submanifoldsWs _{∈ F}s _{and W}u _{∈ F}u _intersect transversally at an angle bounded from below;

3. Ω =S_Fs _∩ S_Fu_.

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 = Bs_×Bu _{⊆ 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 Ku = Ku(x) and Ks = Ks(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 Γs_{(R) = {γ}s_{} and Γ}u_{(R) = {γ}u_{}. Each}

γs_∈Γs_{(R) is the graph of aC}1 _mapφ:Bs_−→Bu _{such that T}

xW ⊆Ks(x) for every x∈γs_{. 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 ∩ γu _{are convex sets for every}

γu_∈Γu_{(R). Unstable cylinders are defined similarly.}

Admissible manifoldsγu_∈Γu_andγs_∈Γs_{intersect transversally with} an-gle ∠(Txγu, Txγs) 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 (γu_{) and diam (γ}u_{) are bounded} in [C−1_{, C]. Further, C}−1 _{≤ dist}

γu(x, y)/kx−yk ≤ C for every x, y in

an unstable admissible submanifold γu _{∈ Γ}u _{where dist}

γu (resp. diam_γu)

is the distance (resp. diameter) on the submanifold γu _{defined with the} in-trinsic riemannian metric. Similarly for Volγu. Likewise for the admissible

s-submanifolds. This fact will be used throughout.

The set Γu _{(resp. Γ}s_{) is endowed with an structure of Banach space} given by the identification with C1(Bu_{, B}s_{) equipped with theC}1 _{norm. In}

particular it is a complete metric space. Moreover, there is a well definedgraph transform ΓT : Γu _{−→ 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(γ₁u),ΓT(γ₂u))≤θdist_C1(γ₁u, γ₂u). In addition, (γu, γs)7−→

γu_∩γs_{is a Lipschitz map from Γ}u_×Γs_to R.

We will consider maps{Ti:Si−→Ui} where{Si}(resp.{Ui}) are count-able collections of non overlapping stcount-able (resp. unstcount-able) cylinders. Each map has a C2 _{extension ˆT}

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 Ks and Ku_{: 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 −→Si intUi, by settingT | intSi=Ti and extendsT to some well defined measurable transformation ˆT :S

iSi−→

S

i Ui, which isC

2

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 Fs _{and F}u_{, respectively. In fact,} given a sequence (i0,· · · , in,· · ·) =i∈N

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 γs _=γs_{(i) such that d}

C1(S_i0···i

n−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 γu _{and an stable cylinder of} level n, P ∈ ℘n), dγk(Tn(x), Tn(y)) ≥ Cλn−kd_γ0(Tk(x), Tk(y)) holds for

k = 0,· · ·, n−1, uniformly in x and y, where γk _{= T}k_(γ0_{). Due to the}

bounded geometry of admissible submanifolds these conditions are equivalent to kTn_(x)−Tn_{(y)k ≥Cλ}n−k_kTk_(x)−Tk_{(y)k, for a suitable constantC >0} depending only on the bounded geometry of Γu _{. A similar statement holds} true for the inverseT−1_.

Now we defineFs_={γs_{(i) :i∈N}N

}(resp. Fu_={γu_{(i) :i∈N}N

}). These laminations are clearly T-invariant. Moreover, due to non-linear expansion properties along admissible manifolds, we conclude thatWs_(x,

R) =γs_{(i) and}

Wu_(x,

R) =γu_{(j) are the local stable manifold ofxwhere{x}=γ}s_(i)∩γu_(j). Indeed, Ws_(x,

similarly so for the unstable local manifold for backward orbits.

We also recall for further use that there are two continuous subbundlesEs (resp. Eu_{) 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

Fs _∩ S

Fu_{. 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 T}

i : 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 : Ωª_{be aC}2 _{generalized horseshoe with bounded}

distor-tion and expansion coefficient bounded from below. Then there exists a unique ergodic measure µ=µΩsuch that:

1. dimu(Ω) =hµ(T)/R

logJu_{T(x)dµ(x), where dimu(Ω) denotes the}

un-stable dynamical dimension ofΩ;

2. µFu_(x), the projection of µonto the unstable leave Fu(x), is equivalent

to the dynamical measure Dα,Fu_(x) in dimension α = dimu(Ω).

Fur-thermore, there is a constant C >1 such that

C−1≤ µFu(x)(P) (VolFu_(x)P)dimu(Ω)

≤C for every P ∈℘, (2)

where VolFu_(x)is the volume defined by the intrinsic riemannian metric

of the unstable submanifold Fu_{(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 Fs _{is absolutely continuous with respect to the}

bounded Jacobian:

Here H∗_{Dα,γu 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 ≍g if there is a constantC >1

such that C−1_{≤f /g≤C uniformly in their domain.}

HereHγu_,γu is the holonomy map of the local stable laminationFsdefined

by admissible unstable sectionsγu_{, γ}u_∈Γu_{(R): H}

γu_,γu(x) =Fs(x)∩γu, for

every x ∈ γu _∩ S

x∈ΩFs(x). J(T | γu) will denote throughout this work

the Jacobian of T restricted to γu _{with respect to the intrinsic riemannian} volume. Ju_{T(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 Fs_(x).

The measureµFu_(x) is defined on the unstable Cantor sets cut by a local

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∈ intersections of γu _{with cylinders in℘:}

℘(γu_{) ={P ∩ γ}u_{:P ∈℘}

n is an stable n-cylinder for some n≥0}.

Then we define an outer measure inγu_{, using the Carath´eodory construction,} by taking coverings by℘(γu_{)-sets and using a riemannian volume Vol}u

γ as set function settingDa,℘,γu(X) = lim_δ→0+inf_UP+∞

i=1Volγu(Ui)a X⊆γu, where

the infimum is taken over all theδ-coverings by℘(γu_)-sets.

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 ∈ Fu_.

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))≍Vol (℘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))≍[Ju_Tn_(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∩ Fu(x))for every x∈Ω;

2. the stable laminationFs_{of Ωis Lipschitz;}

3. transversal measures µFu_(x) are equivalent to the Hausdorff measure

bounded by uniform constants; indeed, there is a constant C >1 such that

µFu_(x)(B(z, r))≍rdimu(Ω) (4)

for every x∈Ω andz∈Ω∩ Fu_{(x), bounded in [C}−1_{, C]In particular,}

the transversal measures are dimensionally exact.

Theorem D Let T : Ωª _{be a C}2 _{generalized horseshoe satisfying the} hy-potheses in Theorem C. The following statements are equivalent:

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

logJu_T(x)dµ(x);

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

3. the volume VolW(Ω) of the unstable Cantor sets Ω ∩ Wu(x,R)is

uni-formly bounded away from zero and Fs _{is absolutely continuous with}

respect to Lebesgue measure with uniformly bounded Jacobians;

4. µΩis 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

Fs _{is absolutely continuous with respect to Lebesgue measure and it has a}

bounded Jacobian, namely

d H∗_Lγu

dLγu (x) =

+∞

Y

i=1

J(T |γu_)(Ti_(x))

J(T |γu_)(Ti_(H(x))) (5)

for every pair of admissible unstable manifolds γu_{, γ}u_∈Γu_{. HereLγu is the}

Lebesgue measure class of γu_{. 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/Fs_−→Ω/Fs_{satisfies 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

mu_{(DT(z)) = inf}

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 everyγu_∈Γu _andn≥0

J(Tn _|γu_)(x)

J(Tn_|γu_)(y) ≤exp(Cdu(T

n_{(x), T}n_{(y))) (6)}

holds whenever ℘n(x) =℘n(y)andx, y∈γu_.

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

¯

Proof. Using bounded distortion condition (1) we prove that for everyγu_∈Γu andi >0 the following estimate holds:

supz∈γu_∩S

i k∇log J(Ti|γ

u_)(z)k

infw∈γu_∩Si mu(DTi(w))

≤∆<+∞. (7)

This is a straightforward computation. Now, we use (7) and a standard argu-ment to get (6). Letγu_∈Γu _{an unstable admissible submanifold and denote}

so kT(x)−T(y)k ≥ C−2infw∈γimu(DT(w))kx−yk, for every x, y ∈ γi. 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 γu_∈Γu _{and every n >0 it holds}

Proof. We can find a C2 foliation of R by admissible unstable leaves, say

γu_=γu_{(z) passing by γ}u_{(x) andγ}u_{(y). Using bounded distortion condition} we get

supw∈γs_(z)k∇log J(T_i|γu(w))(w)k

infw∈γs_(z)mu(DTi(w))

≤∆<+∞, (9)

for every leave γs_{(z) contained in Si, every z ∈Rand i >0. Let us denote}

Ji(z) =J(Ti|γu(z))(z). Arguing similarly as we did before we get

log Ji(T j_(x))

Ji(Tj_(y)) ≤ sup w∈Ws_(Tj_(x),R)

k∇logJi(w)k · kTj_(x)−Tj_(y)k

≤ C2supw∈Ws(Tj(x),R)k∇logJi(w)k

infw∈Ws_(Tj_(x),R) mu(DTi(w))

kTj+1(x)−Tj+1(y)k.

Thus log(Ji(Tj_{(x))/ Ji(T}j_(y)))≤∆C4_λ−(j+1)_{ds(x, y) and then}

log h(x, y)≤

+∞

X

j=1

∆C4λ−(j+1)_{ds(x, y) = ∆C}4_(1−λ−1_)λ−2_{ds(x, y).}

Further, h(x, y)/ h(x′_{, y}′_{)≤exp (C(ds(x, y)−ds(x}′_{, y}′_{)). Therefore,}

h(x, y)

h(x′_{, y}′₎≤max{exp(Cdu(x, x

′_{)),exp(Cdu(y, y}′_))},

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 γu _andγu _inΓu _{it holds Vol}

γu(P)≍Vol_γu(P)for

every P ∈℘.

In particular, the volume of the cylinder Vol (P) is comparable with the volume of its unstable sectionsγu _{∩P, i.e., Vol}

γu(P)≍Vol (P), bounded by

uniform constants which do not depend onpneitherγu_∈Γu_.

Proof. It follows from Corollary 3.1 and Lemma 3.2 that for everyn >0

Volγu(℘n(x))

Volγu(℘n(x))≤C

supγu_∈Γu Vol (γu)

infγu_∈Γu Vol (γu),

Corollary 3.3. The holonomy of the stable lamination is absolutely contin-uous with respect to the dynamical class {Dα,γu}γu_∈Γu and it has a bounded

Jacobian.

This follows from Dα,γu(B) ≍ Dα,γu(H(B)), which is bounded by

con-stants C = C(∆, λ,Γ) >1, for every γu _{and γ}u_{, where H = H}

γu_,γu is the

holonomy transformation defined by these unstable admissible manifolds. We shall prove later that h = hγu_,γu below is the Jacobian of H with

respect to the dynamical measure class:

h(x) =

"_+∞ Y

i=1

J(T |γu_)(Ti_(x))

J(T |γu_)(Ti_(H(x)))

#α

.

The following result is a straightforward consequence of Lemma 3.2.

Lemma 3.3. For everyγu _andγu _{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 γu

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_|γu_{)(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(T}n _|γu_{)(x)≤C, for everyx∈P∩γ}u_,

4

Proofs of the main results

Lemma 4.1. The dynamical measure class is non trivial.

Proof. LetT : Ωªbe aC2_{hyperbolic 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 = Wu_(x,

R) and for every P ∈ ℘(Ωn), the family of stable cylinders generating the stable subsets of Ωn, it holds thatµn,W(P)≍ 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−αlnJuT of an or-dinaryC2 _{horseshoe andµ}

W the projection ofµαontoW =Wu(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∈FuVol (W) ≤

µW(P) (VolWP)dimu(Ω)

≤ C sup W∈Fu

Vol (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

Ju_Tm_(x)

Ju_Tm_(y) ∈[e

−d_{, e}d_{] whenever ℘}n

m(x) =℘nm(y),

where wpn

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

µn,W(P) ≍VolW(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. Letµ∗ _{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 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 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) =

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

This follows from the Bounded Distortion Property (7) andkτk_(x)−τk_{(y)k ≍}

d∗_(τk_{(x), τ}k_{(y)) =d}∗_(Tk_{(x), T}k_(y)).

Also, using the Bounded Distortion Property (11) we conclude that

lnh(T(τ k_(x)))

h(T(τk_(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∗_(τk+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}

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-sureµFu_(x) which is equivalent toDα,_Fu_(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 subsetsBs _{simply by setting}

µW(B) = ˜µ

By the absolute continuity of the stable lamination this measure ˜µ does not depend on the unstable leave W = Wu_(x,

R) choosen. Now, W

n≥0TnBs

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.

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)≍(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 thatT}n_{(P) is an unstable cylinder (i.e., such} thatTn_{(P) has full width) diam (P∩W}u_{(x,R))≍ kJ}u_Tn_(x)k−1_{bounded 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 (Ju_Tn_(x))−1_≤r

and (Ju_Tn−1_f(x))−1_{> r. Therefore,}

B(x, C−1r)∩γu_{⊆Sn(x)∩γ}u_∩γu_{⊆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 Ws_(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_(Sn(x)∩γu_{) 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)∩γu _{intersects at mostM atoms in the family℘r(γ}u_{), for every}

x ∈ Λ, 0 < r < 1 and admissible unstable manifold γu_{. M should depend} in principle on γu_{, however, by geometry of admissible manifolds shows that} this dependence can be dropped out.

By Theorem C,µFu_(x)(P)≍diam (P∩Wu(x,R))dimu(Ω). So for everyP ∈

℘, henceµFu_(x)(B(x, r))≍diam (B(x, r) ∩ γu))dimu(Ω) which is comparable

with rdimu(Ω) _{for every x ∈ Ω, 0 < r < 1, using the bounded geometry of}

admissible manifolds. So, dimH(Ω∩Wu(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 µFu_(x) has the strong uniform

distribution property µFu_(x)(B(x, r)) ≍ rdimu(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 laminationFs _{we see that}

where α= dimH(Ω∩W), Hα,γu is the restriction of the Hausdorff measure

to the admissible unstable manifoldγu_andH=H

γu_,γu is the holonomy map

defined by γu _{and γ}u_{. Constants are uniform, by previous remarks and the} bounded geometry of admissible manifolds. Hence,

kH(x)−H(y)k

kx−yk ≍1, ∀x, y∈γ u _∩ [

z∈Ω

Ws(z,R),

meaning thatC−1_{kx−yk ≤ kH(x)−H(y)k ≤Ckx−yk,F}s_{is Lipschitz. We} are done.

Proof of Theorem D

We notice thatD1,γu represents the Lebesgue measure class ofγu. This is a

straightforward consequence of the definitions and the fact that ℘generates the stable lamination. In particular, Volγu(X) > 0 implies dimu(X) = 1.

Thus, dimu(Ω) = 1 implies that µW, the projection of µalong Fs onto the local unstable manifoldsW =Wu_{(x,R), is equivalent to the restriction of the} Lebesgue measure classLW toW ∩ Ω, bounded by constants not depending on x ∈ Ω neither on W ∈ Fu_{. In particular, the volume Vol}

W(Ω) of the unstable Cantor sets Ω ∩Wu_{(x,R) is uniformly bounded away from zero, by} the absolute continuity of the lamination and since its Jacobians are uniformly bounded. As VolW(Ω)>0 for some local unstable manifold W =Wu(x,R) implies dimu(Ω) = 1, we see that all these conditions are equivalent. To finish, we notice that the ergodic basin of µ=µΩ, defined as

Ws_{(µ) ={x∈}

R: lim n→+∞

1

n

n−1

X

k=0

φ(Tk_{(x)) =}

Z

φ(z)dµ(z) ∀φ∈C0(M)},

contains the stable set of Ω, Ws_{(Ω) =} S

x∈ΩFs(x). Now, Ws(Ω) has

posi-tive volume, in view of the absolute continuity of the stable lamination with respect to the Lebesgue measure, using a Fubini’s theorem argument, hence Vol (Ws_{(µ))>0 and we are done.}

Acknowledgments

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