Gibbs–Markov structures and limit laws
for partially hyperbolic attractors with mostly
expanding central direction
✩José F. Alves
a,∗, Vilton Pinheiro
baDepartamento de Matemática Pura, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687,
4169-007 Porto, Portugal
bDepartamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil
Received 23 February 2009; accepted 9 October 2009 Available online 28 October 2009 Communicated by Kenneth Falconer
Abstract
We consider a partially hyperbolic setKon a Riemannian manifoldM whose tangent space splits as
TKM=Ecu⊕Es, for which the center-unstable directionEcu expands non-uniformly on some local unstable disk. We show that under these assumptionsf induces a Gibbs–Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the center-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an almost sure invariance principle and the validity of the central limit theorem.
2009 Elsevier Inc. All rights reserved.
MSC:37A25; 37C05; 37C40; 37D25; 37D30
Keywords:Partial hyperbolicity; Gibbs–Markov structure; Decay of correlations; Large deviations; Almost sure invariance principle; Central limit theorem
✩ Work carried out at the Federal University of Bahia and University of Porto. J.F.A. was partially supported
by FCT through CMUP and by POCI/MAT/61237/2004. V.P. was partially supported by PROCAD/Capes and by POCI/MAT/61237/2004.
* Corresponding author.
E-mail addresses:[email protected] (J.F. Alves), [email protected] (V. Pinheiro). URL:http://www.fc.up.pt/cmup/home/jfalves (J.F. Alves).
Contents
1. Introduction . . . 1707
1.1. Overview . . . 1709
1.2. Gibbs–Markov structures . . . 1709
1.3. Partially hyperbolic attractors . . . 1710
1.4. Limit laws . . . 1712
2. Preliminaries . . . 1713
3. Gibbs–Markov structure . . . 1717
3.1. Returning disks . . . 1717
3.2. Partition on the reference leaf . . . 1719
3.2.1. First step of induction . . . 1720
General step of induction . . . 1720
3.3. Product structure . . . 1722
3.4. Backward contraction and bounded distortion . . . 1722
3.5. Regularity of the foliations . . . 1723
4. Decay estimates . . . 1726
Acknowledgment . . . 1730
References . . . 1730
1. Introduction
Remarkable advances in the study of dynamical systems, specially the statistical properties of those with chaotic behavior, have been achieved through the idea ofinducing. Roughly speaking, this consists of replacing the initial dynamical system by another one whose dynamical features are easier to understand and from which one can recover much information on the initial system. This idea goes back to the 70’s where Markov partitions have been used to study the statistical properties of uniformly hyperbolic dynamical systems via conjugations to shifts. Since then, a main goal in dynamical systems theory is to enlarge that strategy to wider classes of systems.
A main achievement in this direction has been attained by Young in [23,24]. In these works she developed an abstract framework, theYoung towers, that proved usefulness in a systematic treatment of several classes of dynamical systems, including Axiom A attractors, piecewise hy-perbolic maps, billiards with convex scatterers, logistic maps, intermittent maps and Hénon-type attractors. The latter case has actually been treated by Benedicks and Young in [9]. A
prepon-derant role in this context is played byGibbs–Markov structures, which may be understood as
a generalization of the classical Markov partitions and are naturally associated to an inducing scheme that gives rise to a Young tower. Many statistical properties on the dynamics of these
induced structures can be recovered from the Gibbs–Markov map that one obtains
quotient-ing out by stable leaves. Gibbs–Markov maps constitute themselves object of great dynamical interest; see e.g. [2,3] for Aaronson and Denker contributions on the statistical properties of Gibbs–Markov maps, whose main ideas go back to Aaronson’s book [1] on infinite ergodic the-ory.
A Gibbs–Markov structure is characterized by a suitable region of the phase space partitioned
into subsets (possibly infinitely many) each of which with a givenreturn time. Comparing to
flexibility conferred by the chance of arbitrarily large return times is a fundamental step to-wards applications to non-uniformly hyperbolic dynamical systems, where large waiting times are needed to reach good expansion rates in the center-unstable direction of some points. In cer-tain cases we are able to determine the speed at which the return times decay, given in terms of the time that generic points need to achieve the good expansion rates.
Dynamical systems that do not fit the class of uniformly hyperbolic ones but combine non-uniform expanding/contracting central directions with other directions of non-uniformly hyperbolic behavior give rise to a wider class of partially hyperbolic dynamical systems. The case of par-tially hyperbolic diffeomorphisms for which the tangent bundle over some attracting set splits asEcs⊕Eu, the sum of one invariant sub-bundle with non-uniformly contracting behavior and another one with uniformly expanding behavior, has been treated in [12], where some Gibbs– Markov structures were obtained to deduce decay of correlations and the validity of the central limit theorem for the SRB measures which had been obtained in [10]; see also [11,21]. Limit theorems for partially hyperbolic systems of this type were also obtained in [13].
Our main goal in this work is to prove the existence of Gibbs–Markov structures in the case of partially hyperbolic diffeomorphisms with an attracting set over which the tangent bundle splits as Es ⊕Ecu, the sum of a sub-bundle Es having uniformly contracting behavior with
another one Ecuhaving non-uniform expanding behavior. This kind of attracting set has been
previously considered in [5], where the existence ofSRB measureswas established. The method used in [5] is based on the simple idea of iterating forward Lebesgue measure on some center-unstable disk and obtaining the absolute continuity with respect to Lebesgue measure on local unstable disks of weak* accumulation points. Due to the simplicity of the method, it gives not much information on the properties of such SRB measures. As a byproduct of the machinery that we develop here, we are also able to deduce the existence of the SRB measure. Our method uses deeper knowledge on the geometrical structure of the attractor, thus enabling us to prove the existence of a Gibbs–Markov structure inside it, leading to an inducing scheme. As an immediate consequence of our main result, combined with others from [8,18,20], we easily deduce some statistical properties of these dynamical systems, namely decay of correlations, central limit theorem,large deviationsand amultidimensional almost sure invariant principle. Let us point out that the caseEs⊕Ecuthat we consider here is, for our purposes, considerably more difficult to deal with than the dual caseEcs⊕Eu, since the richest part of the dynamics in the neighborhood of an attracting set occurs in the unstable direction, where in our case the expansion is attained just asymptotically.
1.1. Overview
This paper is organized as follows. In the remaining of this introduction we consider three subsections. In the first one we define the Gibbs–Markov structures. In the second subsection we introduce partial hyperbolicity and state our main result on the existence of Gibbs–Markov struc-tures for certain partially hyperbolic attractors. In the final subsection we define some limit laws and state some statistical consequences of our main result. In Section 2 we recall some results on hyperbolic times and bounded distortion from [5]. The construction of Gibbs–Markov structures for partially hyperbolic attractors is performed in Section 3. We begin with some results on the recurrence of disks, and then present an algorithmic construction that gives rise to the product structure. Finally, in Section 3.5 we prove some results on the regularity of the stable and un-stable foliations. This comprises the generalization of classical results for uniformly hyperbolic attractors to our setting, namely the Hölder continuity of the central-unstable direction and the absolute continuity of the stable foliation. Finally, the estimates on the decay of return times is obtained in Section 4.
1.2. Gibbs–Markov structures
Here we present the structures which have been introduced in [23] and constitute the main object of our interest. These structures comprise the dynamical and geometrical essence of the return map to the base of a Young tower.
Letf :M→M be a diffeomorphism of a Riemannian manifoldM. We say thatf isC1+
iff isC1 andDf is Hölder continuous. Let Leb denote the Lebesgue measure on the Borel
sets ofMassociated to the Riemannian structure. Given a submanifoldγ⊂Mwe use Lebγ to
denote the Lebesgue measure onγ induced by the restriction of the Riemannian structure toγ. An embedded diskγ ⊂M is called an unstable manifoldif dist(f−n(x), f−n(y))→0 as
n→ ∞for everyx, y∈γ. Similarly,γ is called astable manifoldif dist(fn(x), fn(y))→0 as
n→ ∞for everyx, y∈γ. Let Emb1(Du, M)be the space ofC1embeddings fromDuintoM. We say thatΓu= {γu}is acontinuous family ofC1 unstable manifoldsif there is a compact setKs, a unit diskDuof someRn, and a mapΦu:Ks×Du→Msuch that
• γu=Φu({x} ×Du)is an unstable manifold;
• ΦumapsKs×Duhomeomorphically onto its image;
• x→Φu|({x} ×Du)is a continuous map fromKs to Emb1(Du, M).
Continuous families ofC1stable manifolds are defined similarly. We say thatΛ⊂Mhas a prod-uct strprod-uctureif there exist a continuous family of unstable manifoldsΓu= {γu}and a continuous family of stable manifoldsΓs= {γs}such that
• Λ=(
γu)∩( γs);
• dimγu+dimγs=dimM;
• eachγs meets eachγuin exactly one point;
• stable and unstable manifolds meet transversally with angles bounded away from 0.
Givenx∈Λ, letγ∗(x)denote the element ofΓ∗containingx, for∗ =s, u. For eachn1 we let(fn)udenote the restriction of the mapfntoγu-disks and detD(fn)udenote the Jacobian ofD(fn)u.
We require that the product structure satisfies several properties that we explicit below in (P1)–(P5). From here on we assume that C >0 and 0< β <1 are constants depending only
onf andΛ.
(P1) Markov: there are pairwise disjoints-subsetsΛ1, Λ2, . . .⊂Λsuch that
(a) Lebγ((Λ\Λi)∩γ )=0 on eachγ∈Γu;
(b) for eachi∈Nthere isRi∈Nsuch thatfRi(Λi)isu-subset, and for allx∈Λi
fRiγs(x)⊂γsfRi(x) and fRiγu(x)⊃γufRi(x).
This allows us to define thereturn timefunctionR:Λ→NasR|Λi=Ri.
(P2) Contraction onΓs: for ally∈γs(x)andn1
dist
fn(y), fn(x)Cβn.
(P3) Backward contraction onΓu: for allx, y∈Λi withy∈γu(x), and 0n < Ri
dist
fn(y), fn(x)CβRi−ndist
fRi(x), fRi(y) .
(P4) Bounded distortion: for allx, y∈Λi withy∈γu(x)
logdetD(f
Ri)u(x)
detD(fRi)u(y)Cdist
fRi(x), fRi(y)η.
(P5) Regularity of the foliations:
(a) for ally∈γs(x)andn0
log
∞
i=n
detDfu(fi(x))
detDfu(fi(y))Cβ n
;
(b) givenγ , γ′∈Γu, defineφ:γ∩Λ→γ′∩Λasφ (x)=γs(x)∩γ. Thenφis absolutely continuous and
d(φ∗−1Lebγ) dLebγ′
(x)= ∞
i=0
detDfu(fi(x))
detDfu(fi(φ (x))).
See Section 3.5 for a precise definition of absolute continuity. A set with a product structure for which properties (P1)–(P5) above hold will be called aGibbs–Markov structure.
1.3. Partially hyperbolic attractors
LetK⊂Mbe a compactinvariantset for aC1diffeomorphismf:M→M, meaning that
splittingTKM=Ecs⊕Ecuand 0< λ <1 such that for some choice of a Riemannian metric
onM
Df Excs ·
Df−1Ecuf (x)
λ, for allx∈K. (1)
We call Ecs the center-stable bundle and Ecu the center-unstable bundle. We say that K is
partially hyperbolicif it has a dominated splittingTKM=Ecs⊕Ecufor whichEcsisuniformly contracting: there is 0< λ <1 such that for some choice of a Riemannian metric onM
Df Excs
λ, for allx∈K. (2)
Fixing some small number c >0, we say that f is non-uniformly expanding in the
central-unstable directionat a pointx∈Kif
lim sup
n→+∞
1
n n
j=1
logDf−1Ecu fj(x)
<−c. (NUE)
We define theexpansion timefunction
E(x)=min N1: 1
n n−1
i=0
logDf−1Ecu fi(x)
<−c, ∀nN
. (3)
Note thatE(x)is finite for the pointsx∈Ksatisfying (NUE).
In this work we consider partially hyperbolic sets of the same type of those considered in [5], for which the center-stable direction is uniformly contracting and the central-unstable direction is non-uniformly expanding. To highlight the uniform contraction in the center-stable direction we shall writeEs instead ofEcs.
Theorem A.Letf :M→Mbe aC1+diffeomorphism and letK⊂Mbe a transitive partially hyperbolic set such thatTKM=Es ⊕Ecu. Assume that for some local unstable diskD⊂K
and τ >0 one hasLebD{E> n} =O(n−τ).Then there existsΛ⊂K with a Gibbs–Markov
structure. Moreover,Lebγ{R > n} =O(n−τ)for anyγ∈Γu.
Remark 1.1.The existence of a diskDfor which (NUE) is satisfied Lebesgue almost everywhere can actually hold under very mild conditions. Indeed, as shown in [7, Theorem A], ifKattracts
a positive Lebesgue measure set of points for which (NUE) holds, thenKcontains some local
unstable diskDfor which (NUE) holds for LebDalmost every point.
Remark 1.2.Under the assumptions of Theorem A we are able to say more about the set Λ
with the product structure. Actually, our construction gives that the setΛitself coincides with the union of the leaves inΓu. This is not always the case, e.g. for Hénon attractorsΛis a Cantor set; see [9].
An open class of diffeomorphisms for whichK=Mis partially hyperbolic and satisfies the
Lebγ{R > n}hassuper-polynomial decay, meaning that it decays faster than any polynomial.
Transitivity of the diffeomorphisms in that class has been proved in [22].
1.4. Limit laws
Anf-invariant Borel probabilityμinMis called anSRB measureif, for a positive Lebesgue measure set of pointsx∈M,
lim
n→+∞
1
n n−1
j=0
ϕfj(x)=
ϕ dμ, for any continuousϕ:M→R. (4)
Defining thecorrelation functionof observablesϕ, ψ:M→Ras
Cn(ϕ, ψ )=
ϕ◦fnψ dμ−
ϕ dμ
ψ dμ ,
it is sometimes possible to obtain specific rates for which Cn(ϕ, ψ )decays to 0 as ntends to
infinity, at least for certain classes of observables with some regularity. Note that taking the observables as characteristic functions of Borel sets we are led to the classical definition of mixing.
The next two corollaries follow from Theorem A together with [8, Theorem B] and [8, Theo-rem C]; see also [8, Remark 2.4].
Corollary B(Decay of correlations). Letf :M→Mbe aC1+diffeomorphism and letK⊂M
be a transitive partially hyperbolic set such thatTKM=Es⊕Ecu. Assume that there is a local
unstable diskD⊂Kandτ >1such thatLebD{E> n} =O(n−τ).Then some power off has
an SRB measureμandCn(ϕ, ψ )=O(n−τ+1)for Hölder continuousϕ, ψ:M→R.
The existence of the SRB measure for f has already been proved in [5, Theorem A]. In
general, we can only assure that the correlation decay holds for some power off. However, if the return times associated to the elements of the Gibbs–Markov structure given by Theorem A are relatively prime, i.e. gcd{Ri} =1, then the same conclusion holds with respect to f. For
simplicity, from here on we assume thatgcd{Ri} =1.Otherwise, the same conclusions hold for
some power off.
In the next result we obtain conditions for the validity of theCentral Limit Theorem(CLT), which states that the deviation of the average values of an observable along an orbit from the
asymptotic average is given by a Normal Distribution: given any Hölder continuousϕ:M→R
which is not acoboundary(ϕ=ψ◦f −ψfor anyψ∈L2) there existsσ >0 such that 1
√ n
n−1
j=0
ϕ◦fj−
ϕ dμ
distr
−−→N (0, σ ), asn→ ∞.
Corollary C(Central limit theorem). Letf :M→Mbe aC1+diffeomorphism and letK⊂M
be a transitive partially hyperbolic set with TKM=Es ⊕Ecu. Assume that there is a local
unstable disk D⊂K and τ >2 such that LebD{E> n} =O(n−τ).Then CLT holds for any
Given a Hölder continuous observableϕ:M→Randǫ >0, we define thelarge deviationof the time average with respect to the mean ofϕas
Dn(ϕ, ǫ)=μ x∈M:
1
n n−1
j=0
ϕfj(x)−
ϕ dμ
> ǫ
.
The next result is a consequence of Theorem A together with [19, Theorem 4.2] and [18, Theo-rem 1.2].
Corollary D(Large deviations). Letf :M→Mbe aC1+diffeomorphism and letK⊂Mbe a transitive partially hyperbolic set withTKM=Es⊕Ecu. Assume that there is a local unstable
disk D⊂K and τ >2 such that LebD{E> n} =O(n−τ).Then, for any Hölder continuous ϕ:M→Rand anyǫ >0, we haveDn(ϕ, ǫ)=O(n−τ+1).
Givend1 and a Hölder continuousϕ:M→Rd, we denoteϕn=nj−=10(ϕ◦fj−
ϕ dμ), for eachn1. We say that the sequence{ϕn}nsatisfies ad-dimensional almost sure invariance
principle(ASIP) if there existsλ >0 and a probability space supporting a sequence of random variables{ϕn∗}nand ad-dimensional Brownian motionW (t )such that
1. {ϕn}nand{ϕn∗}nare equally distributed;
2. ϕn∗=W (n)+O(n1/2−λ), asn→ ∞, almost everywhere.
The ASIP is said to benon-degenerateif the Brownian motionW (t )has non-singular covariance matrixΣ. For the dynamical systems considered in this paper, there is a closed subspaceZof
infinite codimension in the space of all (piecewise) Hölder ϕ:M→Rd such that Σ is
non-singular wheneverϕ /∈Z; see [20, Remark 1.2] and [16, Section 4.3].
The next result follows from Theorem A and [20, Theorem 1.6], and it strengthens Corollary C above.
Corollary E(Almost sure invariance principle). Letf :M→Mbe aC1+diffeomorphism and letK⊂Mbe a transitive partially hyperbolic set withTKM=Es⊕Ecu. Assume that there is
a local unstable diskD⊂Kandτ >2such thatLebD{E> n} =O(n−τ).Then{ϕn}nsatisfies
an ASIP for any Hölder continuousϕ /∈Z.
2. Preliminaries
With the only exception of Lemma 2.7, the material contained in this section comes from [5]. Our aim is to recall some results on the Hölder continuity of the tangent direction of center-unstable submanifolds, to introduce hyperbolic times and recall their main properties.
We fix continuous extensions of the two subbundlesEs andEcuto some neighborhoodUof
Kthat we denote byE˜s andE˜cu. We do not require these extensions to be invariant underDf. Given 0< a <1, we define thecenter-unstable cone fieldCacu=(Cacu(x))x∈U of widthaby
Cacu(x)=v1+v2∈ ˜Exs⊕ ˜Excusuch thatv1av2
. (5)
We define the center-stable cone fieldCas =(Cas(x))x∈U of width a in a similar way, simply
slightly increasingλ <1, the domination condition (1) remains valid for any pair of vectors in the two cone fields:
Df (x)vs·
Df−1f (x)vcuλ vs
vcu
for every vs∈Cas(x),vcu∈Cacu(f (x)), and any pointx∈U∩f−1(U ). Note that the center-unstable cone field is positively invariant:
Df (x)Ccua (x)⊂Cacuf (x), wheneverx, f (x)∈U.
Actually, the domination property together with the invariance ofEcu= ˜Ecu|Kimply that
Df (x)Cacu(x)⊂Ccuλa f (x)
⊂Cacu f (x)
,
for everyx∈K, and this extends to anyx∈U∩f−1(U )just by continuity.
We say that an embedded C1 submanifold N ⊂U is tangent to the center-unstable cone
field, if the tangent subspace toN at each pointx ∈N is contained in the corresponding cone
Cacu(x). Thenf (N )is also tangent to the center-unstable cone field, if it is contained inU, by the domination property. The tangent bundle ofN is said to beHölder continuousifx→TxN
defines a Hölder continuous section fromNto the corresponding Grassman bundle ofM. Given
aC1submanifoldN⊂U, we define
κ(N )=inf
C >0: the tangent bundle ofN is(C, ζ )-Hölder
. (6)
The next result contains all the information we need on the Hölder control of the tangent direction and its proof can be found in [5, Corollary 2.4].
Proposition 2.1.There existsC1>0such that, given anyC1submanifoldN⊂Utangent to the
center-unstable cone field, then
1. there existsn01such thatκ(fn(N ))C1for everynn0such thatfk(N )⊂Ufor all
0kn;
2. ifκ(N )C1, thenκ(fn(N ))C1for everyn1such thatfk(N )⊂Ufor all0kn; 3. ifN andnare as in the previous item, then the functions
Jk:fk(N )∋x→logdet
DfTxfk(N )
, 0kn,
are(L1, ζ )-Hölder continuous withL1>0depending only onC1andf.
The notion we introduce next allows us to derive uniform expansion and bounded distortion estimates from the non-uniform expansion assumption in the center-unstable direction.
Definition 2.2.Given 0< σ <1, we say thatnis aσ-hyperbolic timeforx∈Kif
n
j=n−k+1
Df−1Efcuj(x)
In particular, ifnis aσ-hyperbolic time forx, thenDf−k|Efcun(x)is a contraction for every
1kn:
Df−kEfcun(x)
n
j=n−k+1
Df−1Ecu fj(x)
σk. (7)
Ifa >0 is sufficiently small and we chooseδ1>0 also small so that theδ1-neighborhood of Kis contained inU, then, by continuity,
Df−1
f (y) v
1
√ σ
Df−1Ecuf (x)
v, (8)
wheneverx∈K, dist(x, y)δ1, andv∈Cacu(y).
Given any diskΔ⊂M, we use distΔ(x, y)to denote the distance betweenx, y∈Δ,
mea-sured along Δ. The distance from a point x ∈Δ to the boundary of Δ is distΔ(x, ∂Δ)=
infy∈∂ΔdistΔ(x, y). The next result has essentially been proved in [5, Lemma 2.7]; see [7,
Lemma 4.2] for a detailed proof.
Lemma 2.3.Let0< δ < δ1andΔ⊂Ube aC1disk of radiusδtangent to the center-unstable cone field. Then, there isn01such that forx∈Δ∩KwithdistΔ(x, ∂Δ)δ/2andnn0a σ-hyperbolic time forx there is a neighborhoodVnofxinΔsuch that:
1. fnmapsVndiffeomorphically onto a center-unstable disk of radiusδ1aroundfn(x);
2. for every1knandy, z∈Vn,
distfn−k(V n)
fn−k(y), fn−k(z)
σk/2distfn(V n)
fn(y), fn(z) .
We call the setsVn hyperbolic pre-ballsand their imagesfn(Vn)hyperbolic balls. Notice
that the latter are indeed center-unstable balls of radiusδ1. The next result follows from
Proposi-tion 2.1 and Lemma 2.3 above exactly as in the proof of [5, ProposiProposi-tion 2.8].
Corollary 2.4.There existsC2>1such that given a diskΔas in Lemma2.3withκ(Δ)C1,
and given any hyperbolic pre-ballVn⊂Δwithnn0, then for ally, z∈Vn
log|detDf
n |TyΔ| |detDfn|T
zΔ|
C2distfn(D)fn(y), fn(y)ζ.
The next result states the existence ofσ-hyperbolic times for points satisfying (NUE) and its proof can be found in [5, Lemma 3.1, Corollary 3.2].
Proposition 2.5.There areθ >0andσ >0such that for everyx∈KwithE(x)nthere exist
σ-hyperbolic times1n1<· · ·< nlnforxwithlθ n.
Givenn1, we define
The next result follows from the proposition above as in [5, Proposition 3.5] or [6, Corollary 2.3]. It plays important role in the metric estimates of Section 4.
Corollary 2.6.LetDbe a local unstable disk for which(NUE)holdsLebD almost everywhere.
Givenn1andA⊂D\ {E> n}withLebD(A) >0we have
1
n n
j=1
LebD(A∩Hj)
LebD(A)
θ.
We finish this section with a technical lemma which will be useful in the proof of Proposi-tion 4.3. We takeδs>0 sufficiently small so that local stable manifoldsWδss(x)are defined for
all pointsx∈Kand
log
Df−1Ecux −log
Df−1E˜cuy <
c
4, (9)
for allx∈Kandy∈Wδs
s(x), wherec >0 is given by (NUE).
Lemma 2.7. Given0< ε < δ1, there existsNε>0 such that for everynn0 and everyx∈ D∩Hn, the ball of radiusεcentered atfn(z)inside the hyperbolic ballfn(Vn(x))contains a hyperbolic pre-ballVk(z)withkNε.
Proof. Givennn0andx∈D∩Hn, letBxn=fn(Vn(x))be the hyperbolic ball associated to x with hyperbolic timen. Recall thatBxnis a center-unstable ball of radiusδ1aroundfn(x). We
define the cylinder
Cxn= y∈Bn
x
Wδs
s(y).
Since (NUE) holds for LebD almost every point and it remains valid by forward iteration, it
follows that LebBn
x almost every point inB
n
x also satisfies (NUE). Givenz∈Cnx, we define
˜
E(z)=min N1: 1
n n−1
i=0
logDf−1E˜cu fi(w)
<−
3c
4 ,∀nN
.
The fact that (NUE) holds for LebBn
x almost every point inB
n
x together with (9) imply thatE˜(z)
is well defined for Leb almost every pointz∈Cxn. HenceE˜(z)is well defined for Leb almost every pointzbelonging to the set
C=
nn0
x∈D∩Hn
Using Lemma 2.3 we may choosenεlarge enough so that any hyperbolic pre-ball ofσ
-hyper-bolic timennε will have diameter not exceeding ε/2. Let nowBxn(ε/2)denote the ball of
radiusε/2 aroundfn(x)insideBxn, and take
vε= min
nn0
x∈D∩Hn
Leb
y∈Bn x(ε/2)
Wδs
s(y)
.
Since the sizes and tangent directions of hyperbolic balls and stable disks are uniformly con-trolled, this minimumvεmust be strictly positive. Hence, as
Leb
z∈C: E˜(z) > n
→0, whenn→ ∞,
it is possible to chooseNε∈Nlarge enough so that
Leb
z∈C: E˜(z) > Nε
vε. (10)
We takeNε also satisfyingθ Nε> nε. By the choice ofvε, givennn0andx∈D∩Hn there
must be somez∈Wδs
s(y)withy∈B
n
x(ε/2)such thatE˜(z)Nε. Using (9) we easily deduce
thatE(z)Nε; recall (3). Hence, by Proposition 2.5 there exists someσ-hyperbolic timehfor ywithθ Nε< hNε. Since we have takenθ Nε> nε, then by the choice ofnεwe are done. ✷
3. Gibbs–Markov structure
In this section we describe the geometric construction of the product structure. This will be made in three steps. In the first one we prove the existence of a center-unstable disk (a reference disk) for which forward iterates come back to a neighborhood of itself, and whose projection along stable leaves cover the disk completely. In the next step we use these returns to define a partition on the reference disk. This part of the construction follows ideas from [6, Section 3]. Finally we use the partition on the reference leaf and the returns to define the product structure.
3.1. Returning disks
LetDbe a local unstable disk as in Theorem A. Diminishingδ1>0, if necessary, we may
assume thatDhas radiusδ1. Take 0< δs< δ1/2 such that points inKhave local stable manifolds
of radiusδs. In particular, these local stable manifolds are contained inU; recall (8).
Lemma 3.1.There areN01andq∈Ksuch that:
1. Wδs
s/2(q)intersectsDin a pointpwithdistD(p, ∂D) > δ1/2;
2. for each center-unstable diskγ1uof radiusδ1centered at a point inKthere are0jN0
and a diskγ2u⊂γ1uof radiusδ1/2centered at a pointz∈Wδss/4(f −j(q)).
Proof. We start by observing that there is a constant α=α(ρ) >0 withα→0 asρ→0 for which the following holds:givenx∈K,ρ >0and y∈K withdist(x, y) < ρ having a local unstable disk of radiusδ1centered aty, thenWδss(x)intersectsW
u
δ1(y)in a pointzwith
distWδss(x)(z, x) < α and distWδu
In particular, such a pointzhas a neighborhood of radiusδ1/2 insideWδu
1(y).
Takeρ >0 small so that 4α < δs. Since we are assumingf|Ktransitive, we may fixq∈K
andN0∈Nsuch that both (i)Wδss/2(q)intersectsDin a pointpwith distD(p, ∂D) > δ1/2, and (ii){f−N0(q), . . . , f−1(q), q}isρ-dense inK. Byρ-dense we mean that any other point inK has one of the points in the set above at a distance less thanρ.
Hence, given any center-unstable disk γ1u of radius δ1 centered at a point y ∈K, there
is 0j N0 such that dist(f−j(q), y) < ρ. Then, by the choice of α and ρ, we have
that Wδs
s(f
−j(q)) intersects γu
1 in a point z with distWs
δs(f−j(q))(z, f
−j(q)) < α < δ s/4 and
distγu
1(z, y) < δ1/2. ✷
Lemma 3.2.There isδ2>0such that ifγuis a center-unstable disk of radiusδ1/2centered at a
pointz∈Wδs
s(w)withw∈K, thenf
j(γu)contains a center-unstable disk of radiusδ
2centered
atfj(z), for each1j N0.
Proof. Let us first prove the result for j =1. Letf (y) be a point in∂f (γu)minimizing the distance fromf (z)to∂f (γu), and letη1be a curve of minimal length inf (γu)connectingf (z)
tof (y). Defineη0=f−1(η1). Denote byη˙1(x)the tangent vector to the curveη1at the pointx.
Then,
Df−1(w)η˙1(x) C
η˙1(x)
,
where
C=max
x∈M
Df−1(x)
1.
Hence,
length(η0)Clength(η1).
Noting thatη0is a curve connectingztoy∈∂γu, this implies that length(η0)δ1/2. Hence
length(η1)C−1length(η0)C−1δ1/2.
Thusf (γu)contains the diskγ1uof radiusC−1δ1/2 aroundf (z). Moreover, dist
f (z), f (w)
λδs< δs,
and so, by the choice ofδs, we have thatγ1uis also a center-unstable disk. Making nowγ1uplay
the role ofγuandf2(z)play the role off (z)we prove that:
(a) f (γ1u)contains a center-unstable disk of radiusC−2δ1/22centered atf2(z);
(b) dist(f2(z), f2(w))λ2δs< δs.
Item (a) gives in particular thatf2(γu)contains a center-unstable disk of radiusC−2δ1/22
cen-tered at f2(z). Arguing inductively we are able to prove thatfj(γu)contains a disk of radius
C−jδ1/2j C−N0δ
3.2. Partition on the reference leaf
The construction that we are going to explain below requires the use several constants. First we takeδ1>0 as in (8), and 0< δ2< δ1as in Lemma 3.2. Then we takeδ0>0 andε >0 so that
δ0≪δ2 and ε≪δ0.
Next we describe the construction of the (mD mod 0) partitionP of the unstable disk of radius δ0centered atpcontained inD. For that we consider the following neighborhoods ofpinD
Δ00=Bu(p, δ0), Δ10=Bu(p,2δ0), Δ02=Bu(p,√δ0) and Δ30=Bu(p,2√δ0),
and the cylinders over these sets,
Ci=
x∈Δi0 Wδs
s(x), fori=0,1,2,3.
Lettingπdenote the projection fromC3ontoΔ3
0along local stable leaves, we have πCi
=Δi0, fori=0,1,2,3. We say that a center-unstable diskγuu-crossesCi ifπ(γu)=Δi0.
Remark 3.3.To simplify the exposition we shall pretend that each center-unstable diskγu u -crossingCi is still a disk centered at a point inWs
δs(p)with the same radius ofΔ
i
0. Actually, the
radius of such a diskγuis proportional to the radius ofΔi0, with the proportionality depending only on the height of the cylinder and the angles of the two fiber bundles in the dominated splitting.
Let∂uC3
1denote thetopandbottomcomponents of∂C3, i.e. the set of pointsz∈∂C3such that z∈∂Wδs
s(x)for somex∈Δ
3
0. By the domination property, we may takeδ0>0 small so that no
center-unstable disk contained inC3and intersectingWδs
s/2(p)can reach∂
uC3. For 0< σ <1
given by Lemma 2.5, let
Ik=
x∈Δ10: δ01+σk/2<distD(x, p) < δ01+σ(k−1)/2, k1,
be a partition (LebDmod 0) into countably many rings ofΔ10\Δ00.
We are now able to start with the construction of the partitionP of Δ0. The construction
requires that we introduce inductively several objects. In particular, we will consider sequences of sets(Δn),(An), and(Bn). For eachn0, the setΔnis that part ofΔ0that has not yet been
partitioned up to timen. The setΔnis the disjoint union ofAnandBn, whereAn is essentially
the part ofΔnwhere new elements of partition may be appear in the next step of the construction,
andBn is some protection that we put around the sets previously constructed in order to avoid
3.2.1. First step of induction
We fix R0 some large integer, and we ignore any dynamics occurring up to time R0. Let
kR0+1 be the first time thatΔ0∩Hk= ∅. Forj < kwe define formally the objects Aj=Aεj=Δj=Δ0 and Bj= ∅.
Let(ωk,j3 )j be all the center-unstable disks inAεk−1contained in hyperbolic pre-ballsVm, with k−N0mk, which are mapped byfk onto a center-unstable disku-crossingC3and
inter-sectingWδs
s/4(p). Then we let
ωik,j=ω3k,j∩f−kCi
, i=0,1,2
and setR(x)=kforx∈ωk,j0 . We take
Δk=Δk−1\ {R=k},
and define a functiontk:Δk→Nby
tk(x)=
s, ifx∈ω1k,j andπ(fk(x))∈Isfor somej;
0, otherwise.
Finally let
Ak=x∈Δk: tk(x)=0, Bk=x∈Δk: tk(x) >0
and
Aεk=x∈Δk: distfk+1(D)fk+1(x), fk+1(Ak)< ε. General step of induction
The general step of the construction follows the ideas above with minor modifications. As-sume that the sets Δi, Ai, Aεi Bi, {R =i} and functions ti :Δi →N are defined for each in−1. Let(ω3n,j)j be all the center-unstable disks in Aεn−1 contained in hyperbolic
pre-ballsVm, withn−N0mn, which are mapped byfnonto a center-unstable disku-crossing C3and intersectingWδs
s/4(p). Take
ωin,j=ω3n,j∩f−nC1i
, i=0,1,2 (11)
setR(x)=nforx∈ω0n,j, and let
Δn=Δn−1\ {R=n}.
The definition of the functiontn:Δn→Nis slightly different in the general case:
tn(x)= ⎧ ⎪ ⎪ ⎨
⎪ ⎪ ⎩
x, ifx∈ω1n,j\ωn,j0 andfn(x)∈Is for somej,
0, ifx∈An−1\jωn,j1 ,
Finally, we let
An=x∈Δn: tn(x)=0, Bn=x∈Δn: tn(x) >0
and
Aεn=
x∈Δn: distfn+1(D)fn+1(x), fn+1(An)< ε.
At this point we have described the construction of the setsAn,Aεn,Bnand{R=n}.
Since the components of{R=n}are taken inAεn−1, it could happen that these new compo-nents intersectBn−1. The next lemma shows that this is not the case as long asǫ >0 is taken
small enough. For notational simplicity we will drop the indexj in the elements defined at (11).
Lemma 3.4.Ifǫ >0is small, thenωn1∩ {tn−11} = ∅for alln1.
Proof. Takek1 and letω0n−k be a component of{R=n−k}. LetQk be the part ofω1n−k
that is mapped byπ◦fn−k ontoIk, and assume thatQk intersects someω3n. Recall that, by construction,Qk is precisely that part of ω1n−k on which tn−1=1, andω3n is contained in a
hyperbolic pre-ballVmwithn−N0mn.
Letq1andq2be any two points in distinct components (inner and outer, respectively) of the
boundary ofQk. If we assume thatq1, q2∈ω3n, thenq1, q2∈Vm, and so by Lemma 2.3 we have
distfn−k(D)fn−k(q1), fn−k(q2)C0σk/2distfn(D)fn(q1), fn(q2) (12) for someC0 depending on N0. We also have for some C1>0 depending on the angle of the
stable and center-unstable spaces overK∞
distfn−k(D)fn−k(q1), fn−k(q2)C1δ01+σ(k−1)/2−δ01+σk/2
=C1δ0σk/2σ−1/2−1,
which combined with (12) gives
distfn(D)fn(q1), fn(q2)C1 C0
δ0σ−1/2−1.
On the other hand, sinceω3n⊂Aεn−1by construction ofω3n, taking
ε <C1 C0δ0
σ−1/2−1
(13)
3.3. Product structure
Consider the center-unstable diskΔ0⊂Dand the partitionPofΔ0(LebDmod 0) defined in
Section 3.2. We shall use the elements ofPto define thes-subsets that give rise to the hyperbolic structure. Given an arbitrary elementω∈P, we have by construction someR(ω)∈Nsuch that fR(ω)(ω)is a center-unstable disku-crossingC0. We defineCωas the cylinder made by the stable
leaves passing through the points inω, i.e.
Cω=
x∈ω Wδs
s(x).
The sets Cω, with ω∈P, are by definition the pairwise disjoint s-subsets Λ1, Λ2, . . . which
define the Markovian structure.
Now we define inductively some sets of center-unstable manifolds u-crossing C0 that will
give rise to the familyΓu. The first one is
Γ0= {Δ0}.
Having definedΓj, for somej0, we define
Γj+1=fR(ω)(Cω∩γ ): ω∈Pandγ∈Γj.
Observe that each element ofΓj is equal to an iterate of a subset ofΔ0. In particular, the
ele-ments of eachΓj are unstable manifolds. Moreover, since by constructionfR(ω)(ω)intersects Wδs
s/4(p), then according to the choice ofδ0and the invariance of the stable foliation, we have that each element ofΓj mustu-crossC0.
Since the union of the leaves of the setsΓj, withj 0, is not necessarily compact, we still
need to take accumulation points of that union. Let
Δ∞= j0
γj∈Γj
γj.
Given x∈Δ∞, there are(jk)k→ ∞, disksγjk ∈Γjk and points xk ∈γjk converging tox as
k→ ∞. Using the domination property and Ascoli–Arzela theorem we conclude that the disks
γjk converge to a diskγ∞containingx. Since the diskγ∞is accumulated by disksu-crossing
C0then it also mustu-crossC0. We defineΓ∞as the set of all these accumulation disks. Finally, we take
Γu= j0
Γj∪Γ∞.
3.4. Backward contraction and bounded distortion
The backward contraction property (P3) follows from Lemma 3.5 below. Bounded
distor-tion is typically a consequence of backward contracdistor-tion together with some Hölder control of log|detDfu|. Property (P4) follows naturally from Proposition 2.1 together with Lemma 3.5
Lemma 3.5.There isC >0such that, givenω∈P andγ∈Γu, we have for all1kR(ω)
and allx, y∈Cω∩γ
distfR(ω)−k(Cω∩γ )
fR(ω)−k(x), fR(ω)−k(y)
Cσk/2distfR(ω)(Cω∩γ )
fR(ω)(x), fR(ω)(y) .
Proof. Recall that, by construction, for eachω∈P there is a hyperbolic pre-ballVn(ω)(x)
con-tainingωassociated to some pointx∈Dwithσ-hyperbolic timen(ω)satisfyingR(ω)−N0
n(ω)R(ω). Takingδs, δ0< δ1/2, it follows from (8) thatn(ω)is a √σ-hyperbolic time for
every point inCω∩γ. Then, recall (7), this implies that for all 1kn(ω)and allx, y∈Cω∩γ
we have
distfn(ω)−k(Cω∩γ )
fn(ω)−k(x), fn(ω)−k(y)σk/2distfn(ω)(Cω∩γ )
fn(ω)(x), fn(ω)(y).
Since the difference betweenR(ω)andn(ω)is at mostN0, the result follows withCdepending
only onN0and the derivative off. ✷
3.5. Regularity of the foliations
Here we prove property (P5). This is standard for uniformly hyperbolic attractors, and we
shall adapt classical ideas to our setting. We begin with the statement of a useful result on vector bundles whose proof can be found in [15, Theorem 6.1].
Lemma 3.6.Letp:Y →Xbe a vector bundle over a metric spaceXendowed with an admissi-ble metric. LetD⊂Y be the unit ball bundle, andF:D→Da map covering a homeomorphism
f:X→X. Suppose0κ <1and that for eachx∈X, the restrictionFx:Dx→Dxhas Lips-chitz constantLip(Fx)κ. Then
1. there is a unique sectionσ0:X→Dwhose image is invariant underF;
2. letLip(f )=c <∞and0< α1be such thatκcα<1. Thenσ0satisfies a Hölder
condi-tion of exponentα.
For the sake of completeness, let us mention that a metricd onEisadmissibleif there is a complementary bundleE′overX, and an isomorphismh:E⊕E′→X×Ato a product bundle, whereAis a Banach space, such thatdis induced from the product metric onX×A.
Theorem 3.7.Letf:M→Mbe aC1diffeomorphism and letK⊂Mbe a compact invariant set with a dominated splittingTKM=Ecs⊕Ecu. Then the fiber bundlesEcsandEcuare Hölder
continuous onK.
Proof. We consider the center-unstable bundle, the other one is similar. For eachx∈K letLx
be the space of bounded linear mapsL(Excu, Excs). For eachx∈K, letLx(1)denote the unit ball
around 0∈Lx, and defineΓx:Lx(1)→Lf (x)(1)as the graph transform induced byDf (x):
Γx(μx)=Df Ecsx
·μx·Df−1 Ef (x)cu
ConsiderL(Ecu, Ecs)the vector bundle overKwhose fiber overx∈K isLx, and letDbe its
unit ball bundle. ThenΓ :D→Dis a bundle map coveringf|Kwith
Lip(Γx)
DfExcs ·
Df−1Ef (x)cu
λ <1.
Letcbe a Lipschitz constant for|K, and choose 0< α1 small so thatλcα<1. By Lemma 3.6 there exists a unique section σ0:X→D whose image is invariant underF and it satisfies a Hölder condition of exponentα. This unique section is necessarily the null section. ✷
The next result gives precisely (P5)(a).
Corollary 3.8.There areC >0and0< β <1such that for ally∈γs(x)andn0
log
∞
i=n
detDfu(fi(x))
detDfu(fi(y))Cβ n.
Proof. As we are assuming that Df is Hölder continuous, it follows from Theorem 3.7 that log|detDfu| is Hölder continuous. The conclusion is then an immediate consequence of the
uniform contraction on stable leaves. ✷
Now we are going to prove (P5)(b). We start by introducing some useful notions. We say that φ:N→P, whereN andP are submanifolds ofM, isabsolutely continuousif it is an injective map for which there existsJ:N→R, called theJacobianofφ, such that
LebP
φ (A)=
A
J dLebN.
Property (P5)(b) can be restated in the following terms:
Proposition 3.9.Givenγ , γ′∈Γu, defineφ:γ′→γbyφ (x)=γs(x)∩γ. Thenφis absolutely continuous and the Jacobian ofφis given by
J (x)=
∞
i=0
detDfu(fi(x))
detDfu(fi(φ (x))).
One can easily deduce from Corollary 3.8 that this infinite product converges uniformly. The remaining of this section is devoted to the proof of Proposition 3.9. We start with a general result about the convergence of Jacobians whose proof is given in [17, Theorem 3.3].
Lemma 3.10.LetNandP be manifolds,P with finite volume, and for eachn1,φn:N→P
an absolutely continuous map with JacobianJn. Assume that
1. φnconverges uniformly to an injective continuous mapφ:N→P;
2. Jnconverges uniformly to an integrable functionJ:N→R.
For the sake of completeness, let us mention that there is a slight difference in our definition of absolute continuity of maps. Contrarily to [17], and for reasons that will become clear in later, we do not impose continuity of the mapsφn. However, the proof of [17, Theorem 3.3] uses only
the continuity of the limit functionφ, and so it still works in our case.
Consider nowγ , γ′∈Γu andφ:γ′→γ as in Proposition 3.9. The proof of the next lemma is given in [17, Lemma 3.4] for uniformly hyperbolic diffeomorphisms. Nevertheless, one can easily see that it is obtained as a consequence of [17, Lemma 3.8] whose proof uses only the existence of a dominated splitting.
Lemma 3.11.For each n1, there is an absolutely continuous πn:fn(γ )→fn(γ′) with
JacobianGnsatisfying
1. limn→∞supx∈γ{distfn(γ′)(πn(fn(x)), fn(φ (x)))} =0;
2. limn→∞supx∈fn(γ ){|1−Gn(x)|} =0.
We consider the sequence of consecutive return times for points inΛ,
s1=R and sn+1=sn+R◦fsn, forn1.
Notice that these return time functions are defined Lebγ almost everywhere on eachγ∈Γuand
are piecewise constant.
Remark 3.12.Using the sequence of return times one can easily construct a sequence of (Lebγ
mod 0 on eachγ ∈Γu) partitions(Pn)n bys-subsets ofΛwithsn constant on each element
ofPn, for which (P1)–(P5) hold when we takesnplaying the role ofR and the elements ofPn
playing the role of thes-subsets. Moreover, the constantsC >0 and 0< β <1 can be chosen not depending onn.
We define, for eachn1, the mapφn:γ→γ′as
φn=f−snπsnf
sn. (14)
It is straightforward to check thatφnis absolutely continuous with Jacobian
Jn(x)= |
det(Dfsn)u(x)|
|det(Dfsn)u(φn(x))|·Gsn
fsn(x). (15)
Observe that these functions are defined Lebγ almost everywhere. So, we may find a Borel set
F ⊂γ with full Lebγ measure on which they are all defined. We extend φn toγ simply by
consideringφn(x)=φ (x)andJn(x)=J (x)for alln1 andx∈γ\F. SinceF has zero Lebγ
measure one still has thatJnis the Jacobian ofφn.
Proposition 3.9 is and immediate consequence of Lemma 3.10 together with the next one.
Lemma 3.13.The mapsφnconverge uniformly toφ, and the JacobiansJn converge uniformly
Proof. It is enough to prove the convergence of each sequence restricted toF, i.e. restricted to the set of points where the expressions ofφnandJnare given by (14) and (15) respectively.
Let us prove first thatφnconverges uniformly toφ. Using the backward contraction on
unsta-ble leaves given by (P3), recall Remark 3.12, we may write forx∈γ
distγ′φn(x), φ (x)=distγ′f−snπsnf
sn(x), f−snfsnφ (x)
Cβsndist
fsn(γ′)
πsnfsn(x), fsnφ (x) .
Sincesn→ ∞asn→ ∞and distfsn(γ′)(πsnf
sn(x), fsnφ (x))is bounded by Lemma 3.11, we
have the uniform convergence ofφntoφ.
By (15) we have
Jn(x)= |
det(Dfsn)u(x)|
|det(Dfsn)u(φ (x))|·
|det(Dfsn)u(φ (x))|
|det(Dfsn)u(φn(x))|·Gsn
fsn(x)
.
Using the chain rule and Corollary 3.8 it easily follows that the first term in the product
above converges uniformly to J (x). Moreover, the third term converges uniformly to one, by
Lemma 3.11. It remains to see that the middle term also converges uniformly to one. Recalling Remark 3.12, by bounded distortion we have
|det(Dfsn)u(φ (x))|
|det(Dfsn)u(φn(x))|exp
Cdistfsn(γ′)fsnφ (x), fsnφn(x) η
=exp
Cdistfsn(γ′)fsnφ (x), πsn
fsn(x)η.
Similarly we obtain
|det(Dfsn)u(φ (x))|
|det(Dfsn)u(φn(x))|exp
−Cdistfsn(γ′)
fsnφ (x), π
sn
fsn(x)η.
The conclusion then follows from Lemma 3.11. ✷
4. Decay estimates
In this section we obtain the metric estimates on the decay of LebD{R > n}of Theorem A.
These estimates are an adaptation of similar ones from [6, Section 5] to our setting. We start by proving estimates arising directly from the construction preformed in Section 3.2. In the final part of the argument we use some results that have been put into an abstract setting in [4, Section 4.5.2] and get the desired conclusion.
Lemma 4.1.There existsa0>0such that
LebD(Bn−1∩An)a0LebD(Bn−1),
Proof. It is enough to see this for each component ofBn−1. Let C be a component of Bn−1
and letQ be its outer ring, corresponding totn−1=1. Observe that by Lemma 3.4 we have
Q=C∩An. Moreover, there must be somek < nand a componentω0k of{R=k}such that
π◦fk mapsC diffeomorphically onto ∞
i=kIi and Qonto Ik, both with uniform bounded
distortion as in Corollary 2.4. Thus, it is sufficient to compare the Lebesgue measures of∞ i=kIi
andIk. We have
LebD(Ik)
LebD(∞i=kIi)
≈[δ0(1+σ
(k−1)/2)]u
− [δ0(1+σk/2)]u [δ0(1+σ(k−1)/2)]u−δu
0
≈1−σ1/2,
whereuis the dimension ofEcu. ✷
Lemma 4.2.There existb0, c0>0withb0, c0→0asδ0→0, such that
1. LebD(An−1∩Bn)b0LebD(An−1),
2. LebD(An−1∩ {R=n})c0LebD(An−1),
for alln1.
Proof. It is enough to prove this for each neighborhood of a componentω0nof{R=n}. Observe that by construction we haveω3n⊂Aεn−1, which means thatω2n⊂An−1, because we are taking ε < δ0. Using the uniform bounded distortion of Corollary 2.4 we obtain
LebD(ω1n\ω0n)
LebD(ω2n\ω1n)
≈LebD(Δ 1 0\Δ00)
LebD(Δ20\Δ10) ≈ δ
d 0 δ0d/2 ≪
1,
which gives the first estimate. Moreover,
LebD(ω0n)
LebD(ω2n\ω1n)
≈ LebD(Δ 0 0)
LebD(Δ20\Δ10) ≈ δ
d 0 δ0d/2 ≪
1,
and this gives the second one. ✷
Proposition 4.3.There existc1>0and a positive integerN=N (ε)such that
LebD
N
i=0
{R=n+i}
c1LebD(An−1∩Hn),
for alln1.
Proof. Let K0=maxx∈ΛDf−1 and take r=5δ0K0N0, whereN0 is given by Lemma 3.1.
Recall that by Lemma 2.3, for each z∈fn(An−1∩Hn)there is x ∈Hn and a σ-hyperbolic
pre-ball Vn(x)⊂Dwhich is sent diffeomorphically onto the center-unstable ball of radius δ1
pairwise disjoint, where eachBr(zj)is the ball of radiusr centered atzj inside the hyperbolic
ball aroundzj. By maximality we have
j
B2r(zj)⊃fn(An−1∩Hn). (16)
For eachj letxj∈Hnbe the point such thatfn(xj)=zj.
Claim 1.There is0kNǫ+N0such thattn+kis not identically zero inf−n(Bǫ(z)).
Assume, by contradiction, thattn+k|f−n(Bǫ(z))=0 for all 0kNǫ+N0. This implies
thatf−n(Bǫ(z))⊂An+kfor all 0kNǫ+N0. Using Lemma 2.7 we may find a hyperbolic
pre-ball Vm⊂Bǫ(z)withσ-hyperbolic timemNǫ. Now, sincefm(Vm)is a center-unstable
disk of radiusδ1, it follows from Lemma 3.1 and Lemma 3.2 that there areV ⊂fm(Vm)and m′N0such thatu-crossingC3and intersectingWδss/4(p). Thus, takingk=m+m
′ we have
that 0kNǫ+N0andf−n(Vm)contains an element of{R=n+k}insidef−n(Bǫ(z)). This
contradicts the fact thattn+k|f−n(Bǫ(z))=0 for all 0kNǫ+N0.
Claim 2.f−n(Bδ1/4(z))contains a component of{R=n+k}with0kNǫ+N0.
Let k be the smallest integer 0kNǫ +N0 for which tn+k is not identically zero in
f−n(Bǫ(z)). Since f−n(Bǫ(z))⊂Aǫn−1⊂ {tn−11}, there must be some component ω0n+k
of {R =n+k} for which f−n(Bǫ(z))∩ω1n+k = ∅. Recall that, by definition, fn+k sends
ωn1+k diffeomorphically onto a center-unstable disk (of radius 2δ0) u-crossing C1 and
inter-secting Wδs
s/4(p). Thus, the diameter off
n(ω1
n+k)is at most 4δ0K N0
0 . Since Bǫ(z) intersects fn(ωn1+k)andǫ < δ0< δ0K0N0, we havef−n(Bδ1/4(z))⊃ωn0+k, as long as we takeδ0>0 small
so that 5δ0K0N0< δ1/4.Hence, we have shown thatf−n(Bδ1/4(z))contains some component of
{R=n+k}with 0kNǫ+N0, and so we have proved the claim.
Sincenis a hyperbolic time forxj, we have by the distortion control given by Corollary 2.4 that there is some constantConly depending onC2andδ1for which
LebD(f−n(B2r(zj)))
LebD(f−n(Br(zj)))
CLebfn(D)(B2r(zj))
Lebfn(D)(Br(zj)) (17)
and
LebD(f−n(Br(zj)))
LebD(ω0n+k)
C Lebfn(D)(Br(zj))
Lebfn(D)(fn(ω0
n+k))
. (18)
Recalling that from timenup ton+kwe have at mostN0iterates, from (17) and (17) we easily
deduce that there is some positive constant, that we still denote byC, for which
LebDf−nB2r(zj)CLebDf−nBr(zj)
and
LebDf−nBr(zj)CLebDω0n+kj
Finally, let us compare the Lebesgue measure of the setsNi=0{R=n+i}andAn−1∩Hn. By
(16) we have
LebD(An−1∩Hn)
j
LebDf−nB2r(zj)C
j
LebDf−nBr(zj).
On the other hand, by the disjointness of the ballsBr(zj)we have
j
LebDf−nBr(zj)C
j
LebDω0n+k
CLebD N
i=0
{R=n+i}
.
We just have to takec1=C−2. ✷
For completing the proof of Theorem A, it is enough to show that
LebD{E> n} =On−τ ⇒ LebD{R > n} =On−τ.
Recall that we have definedHn, forn1, as the set of points for whichnis aσ-hyperbolic time.
In Corollary 2.6 we obtained the following estimate:
(m1) There isθ >0such that for alln1andA⊂M\ {E> n}withLebD(A) >0
1
n n
j=1
LebD(A∩Hj)
LebD(A)
θ.
In the construction of the Markov structure have taken a diskΔof radiusδ0>0 and defined
inductively the subsetsAn,Bn,{R=n}andΔnrelated in the following way:
Δn=Δ\ {Rn} =An ˙∪Bn.
Moreover, we have proved in Lemma 4.1, Lemma 4.2 and Proposition 4.3 that the following metric relations hold:
(m2) There isa0>0(bounded away from 0 for allδ0) such that for alln1
LebD(Bn−1∩An)a0LebD(Bn−1).
(m3) There areb0, c0>0withb0, c0→0asδ0→0, such that for alln1
LebD(An−1∩Bn)
LebD(An−1)
b0 and
LebD(An−1∩ {R=n})
LebD(An−1)
c0.
(m4) There isr0>0and an integerN0such that for alln1
LebD
N
i=0
{R=n+i}
Estimates (m1)–(m4) are enough to use the results of [4, Section 4.5.2] and obtain the decay of
Lebγ{R > n}as in the conclusion of Theorem A.
Acknowledgment
The authors acknowledge the referee for valuable comments and references.
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