Statistical properties of diffeomorfisms with weak invariant manifolds

We consider diffeomorphisms of compact Riemmanian manifolds which have a Gibbs-Markov-Young structures, consisting of a reference set $\Lambda$ with a hyperbolic product structure and a countable Markov partition. We assume polynomial contraction on stable leaves, polynomial backward contraction on unstable leaves, a bounded distortion property and a certain regularity of the stable foliation. We establish a control on the decay of correlations and large deviations of the SRB measure of the dynamical system, based on a polynomial control on the Lebesgue measure of the tail of return times. Finally, we present an example of a dynamical system defined on the torus and prove that it verifies the properties of the Gibbs-Markov-Young structure that we considered.


Introduction
In this work we consider discrete-time dynamical systems f : M → M , where f is a diffeomorphism of a compact Riemannian manifold M . We are interested in the study of the statistical behavior of typical orbits when f has non-uniformly hyperbolic behavior; more specifically, to study the large deviations and decay of correlations with respect to the SRB measure for diffeomorphisms having some Gibbs-Markov-Young structure.
Bowen, Ruelle and Sinai [16,5,14] obtained exponential decay of correlations for uniformly hyperbolic diffeomorphisms with respect to some measures which describe the statistics of a large set of initial states in the phase space, the so-called Sinai-Ruelle-Bowen (SRB) measures. Later, some classes of non-uniformly hyperbolic diffeomorphisms were considered. First, Young [19] proved an exponential rate for the decay of correlations assuming there exists a reference set Λ ⊆ M with a hyperbolic product structure and, among other properties, exponential contraction along stable leaves and exponential backward contraction on unstable leaves. Alves and Pinheiro in [1] weakened these assumptions, removing the backward contraction but still imposing an exponential contraction along stable leaves. In that paper, they proved exponential or polynomial decay of correlations, depending on different hypothesis that we will explain later. In addition, Young also obtained, in [20], a control on the rate of decay of correlations for non-invertible dynamical systems and, together with Benedicks in [6], for Hénon maps.
Many authors have proved results on large deviations for uniformly hyperbolic dynamical systems, some of which can be found in [13,7,8,18,17]. Later, Araújo and Pacifico, in [4], studied large deviations for certain classes of non-uniformly expanding maps and partially hiperbolic non-uniformly expanding diffeomorphisms. In [3], Araújo extended these results to a more general case. Melbourne and Nicol, in [12], obtained a control on large deviations for non-uniformly hyperbolic systems that verify certain properties, including exponential contraction on stable leaves and exponential backward contraction on unstable leaves. In [10], Melbourne obtained a slightly better result for large deviations.
Using a tower structure, introduced by Young in [19], it is possible, under certain conditions, to obtain a relation between the measure of the tail of the return time function and both the decay of correlations and large deviations. Young, in [19], for systems with exponential behaviour in stable and unstable leaves, proved exponential decay of correlations when the measure of the tail of the return time decreases exponentially. In [20], Young also proved, for non-invertible systems, both polynomial and exponential decay of correlations based, respectively, on polynomial and exponential control on the tail of the return time. Alves and Pinheiro, in [1], extended the result of [19] to a more general case, obtaining, in addition to the exponential decay of correlations, a polynomial decay of correlations assuming a polynomial return time. As for the large deviations, Melbourne and Nicol, in [12], also obtained exponential and polynomial control of large deviations, with the corresponding hypothesis on the tail of the return time.
1.1. Overview. The main goal of Section 2 is to define Gibbs-Markov-Young structures and present the two main results of this paper, which give a polynomial control on both the decay of correlations and large deviations, from a polynomial control of the tail of the return time associated to such a structure. In Subsection 2.3 we present an example on the torus having a Gibbs-Markov-Young structure and the return time function with polynomial tail. In Section 3 we introduce the Young tower and quotient tower, and present some auxiliary results. Sections 4 and 5 are devoted to the proof of the two main theorems. In the Appendix we prove a result in the quotient tower that is important for the control of large deviations.

Statement of results
Let M be a finite dimensional Riemannian compact manifold, d be the distance in M and Leb be the Lebesgue measure on the Borel sets of M . Given a submanifold γ of M , let Leb γ denote the measure on γ induced by the restriction of the Riemannian structure to γ and d γ the distance induced in γ. Consider a diffeomorphism f : M → M .
Analogously, an embedded disk γ ⊆ M is called a stable manifold if, for every x, y ∈ γ, d(f n (x), f n (y)) → n 0.
We say that Γ u = {γ u } is a continuous family of C 1 unstable manifolds if there is a compact set K s , a unit disk D u of some R n and a map φ u : K s × D u → M such that: (a) γ u = φ u ({x} × D u ) is an unstable manifold; (b) φ u maps K s × D u homeomorphically onto its image; (c) x → φ u | {x}×D u defines a continuous map from K s to Emb 1 (D u , M ), where Emb 1 (D u , M ) denotes the space of C 1 embeddings from D u into M . Continuous families of C 1 stable manifolds are defined similarly. We say that Λ ⊆ M has a hyperbolic product structure if there exists a continuous family of stable manifolds Γ s = {γ s } and a continuous family of unstable manifolds Γ u = {γ u } such that: (a) Λ = ( γ s ) ( γ u ); (b) dim γ s + dim γ u = dim M ; (c) each γ s intersects each γ u in exactly one point; (d) stable and unstable manifolds are transversal with angles bounded away from 0. A subset Λ 1 ⊆ Λ is called an s-subset if Λ 1 also has a hyperbolic product structure and its defining families Γ s 1 and Γ u 1 can be chosen with Γ s 1 ⊆ Γ s and Γ u 1 = Γ u . A subset Λ 2 ⊆ Λ is called a u-subset if Λ 2 also has a hyperbolic product structure and its defining families Γ s 2 and Γ u 2 can be chosen with Γ s 2 = Γ s and Γ u 2 ⊆ Γ u . Given x ∈ Λ, denote by γ * (x) the element of Γ * containing x, for * ∈ {s, u}. For each n ≥ 1 denote by (f n ) u the restriction of the map f n to γ u -disks, and by det D(f n ) u the Jacobian of (f n ) u .
From now on we consider Λ ⊆ M having a hyperbolic product structure, with Γ s and Γ u as their defining families. We say that Λ has a Gibbs-Markov-Young (GMY) structure if the properties (P 0 )-(P 5 ) listed bellow hold.
We introduce a return time function R : Λ → N and a return function f R : Λ → Λ defined for each i ∈ N as For x, y ∈ Λ, let the separation time s(x, y) be defined as For the remaining properties we assume that C > 0, α > 1 and 0 < β < 1 are constants depending only on f and Λ.
(P 2 ) Polynomial contraction on stable leaves: Backward polynomial contraction on stable leaves: (P 5 ) Regularity of the stable foliation: then (a) Θ is absolutely continuous and The properties of f that we present here are related to similar properties defined in [19] and [1]. The main difference here is that we only assume polynomial contraction on stable leaves, in opposition to the exponential contraction in [19] and [1]. Our Markovian property (P 1 ) is the same as in [1] and weaker than the Markov property in [19]; see [19,Remark 2.3]. Properties (P 2 ) and (P 3 ), about polynomial contraction on stable leaves and backward polynomial contraction on unstable leaves, are an improvement over [19], where exponential contraction is assumed. In [1], there is no backward contraction assumed on unstable leaves, however, exponential contraction is assumed on the stable ones. Properties (P 4 ) and (P 5 ) coincide with properties (P 4 ) and (P 3 ) in [1], respectively. Our properties (P 4 ) and (P 5 ) are weaker than the corresponding ones in [19]; see [19,Remark 2.4].

2.2.
Main results. An f -invariant probability measure µ is called a Sinai-Ruelle-Bowen (SRB) measure if all the Lyapunov exponents of f are nonzero µ almost everywhere and the conditional measures on local unstable manifolds are absolutely continuous with respect to the Lebesgue measures on these manifolds.
It was proved in [19,Theorem 1] that if f has a hyperbolic structure Λ such that the return time R is integrable with respect to Leb γ , for some γ ∈ Γ u , then f has some SRB measure µ. Given 0 < η ≤ 1, we define the space of η-Hölder continuous functions We denote the correlation of observables ϕ, ψ ∈ H η by Theorem A. Suppose that f admits a GMY structure Λ with gcd{R i } = 1 for which there are γ ∈ Γ u , ζ > 1 and C 1 > 0 such that Then, given ϕ, ψ ∈ H η , there exists C 2 > 0 such that for every n ≥ 1 where α > 0 is the constant in (P 2 ) and (P 3 ).
The proof of this theorem will be given in Section 4. If µ is an ergodic probability measure and ε > 0, the large deviation at time n of the time average of the observable φ from its spatial average is given by Theorem B. Suppose that f admits a GMY structure Λ with gcd{R i } = 1 for which there are γ ∈ Γ u , ζ > 1 and C 1 > 0 such that Then there are η 0 > 0 and ζ 0 = ζ 0 (η 0 ) > 1 such that for all η > η 0 , 1 < ζ < ζ 0 , ε > 0, p > max{1, ζ − 1} and φ ∈ H η , there exists C 2 > 0 such that for every n ≥ 1 This theorem will be proved in Section 5. We note that, in both theorems, the conclusions remain valid for a power of f when the assumption gcd{R i } = 1 is removed.

2.
3. An example. Here we give an example of a diffeomorphism f of the two-dimensional torus T 2 = R 2 /Z 2 with a GMY structure Λ having polynomial decay for the Lebesgue measure of the tail of the return time. As a consequence, we deduce that f satisfies the results on polynomial decay of correlations and large deviations from Section 2.2.
We start with an orientation preserving C 2 Anosov diffeomorphism f 0 of T 2 and we consider a finite Markov partition W 0 , . . . , W d for f 0 such that the fixed point (0, 0) belongs to the interior of W 0 . Considering the hyperbolic decomposition into stable and unstable sub-bundles T M = E s ⊕ E u , we assume that there is 0 < λ < 1 such that We assume moreover that the transition matrix A of f 0 is aperiodic, i.e. some power of A having all entries strictly positive. By a suitable change of coordinates we can suppose that f 0 (a, b) = φ 0 (a), ψ 0 (b) for all (a, b) ∈ W 0 , the local stable manifold of (0, 0) is {a = 0}, the local unstable manifold of (0, 0) is {b = 0} and both φ 0 and ψ 0 are orientation preserving.
Now we consider f : Observe that V 0 is a neighborhood of (0, 0) strictly contained in W 0 . We assume that for some 0 < θ < 1 we have and assume that φ and ψ coincide respectively with φ 0 and ψ 0 in W 0 \ V 0 . Note that φ is the so-called intermittent map of the type considered in [20, Section 6.2] and (0, 0) is a fixed point of f with φ ′ (0) = 1 = ψ ′ (0).
The following theorems, on the decay of correlations and large deviations for the diffeomorphism f , will be proved in Section 6.

Tower structures
In this section we are going to define a tower structure and a quotient tower, originally introduced by Young in [19]. We assume that f : M → M has a GMY structure Λ with return time function R : Λ → N.
3.1. Tower maps. We define a tower ∆ = {(x, l) : x ∈ Λ and 0 ≤ l < R(x)} and a tower map F : ∆ → ∆ as The l-th level of the tower is defined as There is a natural identification between ∆ 0 , the 0-th level of the tower, and Λ. So, we will make no distinction between them. Under this identification we easily conclude from the definitions that F R = f R for each x ∈ ∆ 0 . The l-th level of the tower is a copy of the set {R > l} ⊆ ∆ 0 . We define a projection map and observe that f • π = π •F . Let P be a partition of ∆ 0 into subsets ∆ 0,i with ∆ 0,i = Λ i for i ∈ N. We can now define a partition on each level of the tower, ∆ l , by defining its elements as So, the set Q = {∆ l,i } l,i is a partition of ∆. We introduce a sequence of partitions (Q n ) n≥0 of ∆ defined for n ∈ N as follows For each point x ∈ ∆, let Q n (x) be the element of Q n that contains that point. Next, we establish a polynomial upper bound on the diameter of the elements of the tower partition.
Lemma 3.1. There exists C > 0 such that, for all k ∈ N and Q ∈ Q 2k , diam(πF k (Q)) ≤ C k α . Proof. Take k > 0 and Q ∈ Q 2k . Fixing x, y ∈ Q, there exists z = γ u (x) ∩ γ s (y). Choosing l such that Q ⊆ ∆ l , then y 0 = F −l (y) and z 0 = F −l (z) are both in ∆ 0 and are in the same stable leaf. So, using (P 2 ), because M is compact.
3.2. Quotient towers. We now introduce a quotient tower, obtained from the tower by identifying points in the same stable leaf. Let ∼ be the equivalence relation defined on Λ by x ∼ y if y ∈ γ s (x). ConsiderΛ = Λ/∼ and the quotient tower∆, whose levels are∆ l = ∆ l /∼ and set∆ l,i = ∆ l,i /∼. Since the tower map F takes γ s -leaves to γ s -leaves, we can definē F :∆ →∆ as the function obtained from F by this identification. We introduce a partition of∆,Q = {∆ l,i } l,i and a sequence of partitions (Q n ) n∈N 0 of∆, defined analogously to (2), as followsQ Since R is constant on each stable leaf and f R takes γ s -leaves to γ s -leaves, then the definitions of the return timeR :∆ 0 → N and the separation times :∆ 0 ×∆ 0 → N are naturally induced by the corresponding definitions in ∆ 0 . We extend the separation times to∆ ×∆ in the following way: if x and y belong to the same∆ l,i , takes(x, y) =s(x 0 , y 0 ), where x 0 , y 0 are the corresponding elements of∆ 0,i ; otherwise, takes(x, y) = 0. We now present an auxiliary result whose proof can be found in [1,Lemma 3.4].
Lemma 3.2. There exists a constant C F > 0 such that, given k ∈ N and x, y ∈∆ belonging to the same element ofQ k−1 , we have Using property (P 5 ) we will be able to define a reference measurem on the quotient tower∆. We start by defining measures m γ on each γ ∩ Λ, γ ∈ Γ u . Fix γ ∈ Γ u and, for any given γ ∈ Γ u and x ∈ γ ∩ Λ, let x be the point in γ s (x) ∩ γ. Define and note that u satisfies (P 5 )-(b). For each γ ∈ Γ u , define m γ as the measure in γ such that We are going to see that, if Θ = Θ γ,γ ′ is as defined in (P 5 ), then To show this it is enough to verify that the density of both measures with respect to Leb γ ′ coincide. Indeed, from (P 5 )-(a) we have and so, Now define m as the measure on Λ whose conditional measures on γ ∩ Λ for γ ∈ Γ u are the measures m γ . Take a measure in ∆, also denoted by m, by letting m |∆ l be induced by the natural identification of ∆ l and a subset of Λ. Finally, since (5) holds, we can define a measurem on∆ whose representative on each γ ∈ Γ u is the measure m γ .
Given 0 < β < 1, we define the functional spaces From now on, we denote by C ϕ both the infimum of the constant in the definition of F β and of F + β with respect to ϕ. We also denote by F β and F + β the analogous sets defined for functions with domain M or ∆. The proof of the following result can be found in [20, Lemma 2] and [19, Theorem 1].

Theorem 3.3. Assume thatR is integrable with respect tom. Then
(1)F has a unique invariant probability measureν equivalent tom; (2) dν/dm ∈ F + β and is bounded from below by a positive constant; The next theorem plays a key role in the proof of Theorem A and is an improved version of [20,Theorem 2] given in [1, Theorem 3.6] whose proof can be found in [1, Appendix A].
Theorem 3.4. For ϕ ∈ F + β letλ be the measure whose density with respect tom is ϕ.
To prove Thoerem B we need to consider more general functional spaces, due to the fact that the polynomial estimates in the stable manifolds interfere with the regularity of the function ψ that will appear in Proposition 5.2. Given θ > 0, we define As above, we denote by D ϕ both the infimum of the constant in the definition of G θ and of G + θ with respect to ϕ. The sets G θ and G + θ also represent the analogous sets defined for functions with domain M or ∆.
This theorem is similar to [20, Theorem 3] and will be proved in Appendix A. Note that we are only assuming ϕ ∈ G + θ , instead of F + β , which forces us to impose some extra assumptions. In practice, we only need the following consequence of Theorem 3.5.

Decay of correlations
This section is dedicated to the proof of Theorem A, adapting the approach of [19] and [1] to our more general conditions on the definition of GMY structure. First of all we note that it was shown in [20, Sections 2 and 4] that there exists a measure ν on the tower ∆ such that µ = π * ν andν =π * ν. Fixing ϕ, ψ ∈ H η , let Observe that and, arguing as above, Hence we have C n (ϕ, ψ, µ) = C n ( ϕ, ψ, ν). Given n ∈ N, we fix a positive integer k < n/2 and define the discretizationφ k of ϕ on the tower ∆ as Lemma 4.1. There exists C 2 > 0 depending only on |ϕ| η and on ψ ∞ such that Using the fact that ϕ is Hölder continuous and Lemma 3.1, we observe that for Q ∈ Q 2k and all x, y ∈ Q, which implies that, for any x ∈ Q, Applying (7), (8) and the F -invariance of ν we obtain We only need to take C 2 = 2 ψ ∞ |ϕ| η C η . Now we defineψ k in a similar way toφ k . Denote byψ k ν the signed measure whose density with respect to ν isψ k and by ψ k the density of F k * ψk ν with respect to ν. Let |ν| denote the total variation of a signed measure ν.
Proof. By definition of ψ k we have Sinceφ k is constant on γ s leaves and F andF are semi-conjugated byπ, then So, we have proved that Additionally, asφ k is constant on γ s leaves, and using the definition of ψ k and theF -invariance ofν, we may write Gathering (11) and (12), we obtain the conclusion.
Without loss of generality we may assume thatψ k is not the null function. Defining Sinceψ k is constant on elements of Q 2k , the same holds for ψ k . Let λ k be the probability measure on∆ whose density with respect tom is ψ kρ .
Lemma 4.4. There exists C 4 > 0 depending only on ϕ ∞ and on ψ ∞ such that Proof. Notice that, by the definition of ψ k and theF -invariance ofν, and, similarly, Then, using the last two equalities and the definitions ofρ and λ k , we obtain Settingλ k =F 2k * λ k and since k < n/2, we have d dmF and so, using (13) and Gathering everything that was proved in the previous lemmas, we get Let φ k be the density of the measureλ k with respect tom. The next lemma, whose proof is given in [1, Lemma 4.1], gives that φ k ∈ F + β . Lemma 4.5. There is C > 0, not depending on φ k , such that ∀x,ȳ ∈∆. Now Lemma 4.5 together with (14) allow us to use Theorem 3.4 and obtain This concludes the proof of Theorem A.

Large deviations
In this section we prove Theorem B. Though our assumptions are different from [12] and [10], we will follow the approach in these papers. The proof of Theorem B uses the construction of a function ψ ∈ G θ (∆), which will be done in Proposition 5.2, for θ = αη − 1.
Proof. Let n ∈ N be such that s(x, y) = n. Using (P 3 ), we get We say that ψ : ∆ → R depends only on future coordinates if, given x, y ∈ ∆ with y ∈ γ s (x), then ψ(x) = ψ(y). In particular, a function ψ : ∆ → R depending only on future coordinates can be interpreted as defined in the quotient∆. The following result is an adaptation of [11,Lemma 3.2] to the polynomial case.
Proposition 5.2. Let f have a GMY structure Λ and φ : M → R be a function in H η with η > 1/α. Then there exist functions χ, ψ : ∆ → R such that: (1) χ ∈ L ∞ (∆) and χ ∞ depends only on |φ| η ; since αη > 1. So, the first item holds. Defining ψ = φ • π − χ + χ • F , the second item is verified and, as ψ depends only on future coordinates. So, the third item is proved. We are left to prove the last item. Let n ∈ N and p, q ∈ ∆. Then Since the choice of n is arbitrary we can assume that s(p, q) ≈ 2n. This means that there will be no separation during the calculations of the first two terms. We will consider separately each term of the right-hand side of (15). We start with the third term. When p = (x, R(x)−1) But f R x and f R x belong to the same stable leaf, and then, using (P 2 ), and then, recalling that s(p, q) ≈ 2n, The calculations for the fourth term of the right-hand side of (15) are similar.
Consider now the first term and take p = (x, l) and q = (y, l). Then and analogously for πF j ( q). Then, since φ ∈ H η and using the calculations above and Lemma 5.1, The calculations for the second term of the right-hand side of (15) are analogous. From (16) and (17), we obtain We now present an auxiliary result presented in [15,Theorem 2.5]. Then Given ψ :∆ → R, we define In the next proposition we prove that, in the quotient tower, a control on the decay of correlations implies a control on large deviations. This proof is based on [10, Theorem 1.2].
Proof. We may assume, without loss of generality, that ψ dν = 0. By Markov's Inequality, and so we only need to prove that where C 5 > 0 depends only on p, D ψ and ψ ∞ . By the definition of the Perron-Frobenius operator and the hypothesis, we have for all w ∈ L ∞ (∆) Choosing w = sgn P n (ψ) in (19) we get P n (ψ) 1 = ∆ P n (ψ) sgn(P n ψ) dν ≤ C 4 n ζ .
Note that C 4 depends only on D ψ as sgn P n (ψ) ∞ = 1. Since P n (ψ) ∞ ≤ ψ ∞ we have where C ′ depends only on p, D ψ and ψ ∞ . Define Observe that χ k , ϕ k ∈ L p (∆), and for k sufficiently large. Now we are going to prove that P (ϕ k ) = 0. In fact, given w ∈ L 2 (∆), we have, sinceν is F -invariant, On the other hand, The operator P is the adjoint operator of U : is a sequence of reverse martingale differences. Passing to the natural extension (see [15]), {ϕ k •F n : n ∈ N 0 } is a sequence of martingale differences with respect to a filtration {G n : n ∈ N 0 }. Defining X j = ψ •F j in Theorem 5.3, we have and, by that theorem, we obtain Recalling the definition of ϕ k in (5), we have and so, using (20), (20) and (21), we obtain where C ′′ depends only on p, D ψ and ψ ∞ . Then, recalling (22) and choosing k = n, we conclude that where C 5 depends only on p, D ψ and ψ ∞ .

The example
Let f be the diffeomorphism of the two-dimensional torus T 2 = R 2 /Z 2 introduced in Subsection 2.3. As described in Subsection 2.3, f coincides with an Anosov diffeomorphism f 0 in all rectangles W 1 , . . . , W d of a Markov partition of f 0 but one, W 0 . Recall that we have Observe that as we have not modified the geometric structure of f 0 , then the set W 1 is completely foliated by a set Γ s of stable leaves and a set Γ u of unstable leaves. To obtain the conclusions of Theorems C and D we shall prove that f satisfies the properties (P 1 )-(P 5 ) on the set Λ = W 1 (any other W i = W 0 would be fine) and that we have recurrence times with polynomial decay to some unstable leaf on W 1 , thus being in the conditions of Theorems A and B.
We consider the sequences (a n ) n and (a ′ n ) n defined recursively for n ≥ 1 as a n = φ −1 (a n−1 ) and a ′ n = φ −1 (a ′ n−1 ). For all n≥ 0, set . Observe that these sets form a (lebesgue mod 0) partition of W 0 . Setting for i = 1, . . . , k and n ≥ 0 R| W i = 1, R| Jn = n + 1 and R| J ′ n = n + 1, define and, for x ∈ W 1 , let R(x) be equal to the smallest R i such that f R i (x) ∈ W 1 . Note that as we are assuming the transition matrix of f 0 (and thus of f ) with respect to the partition W 0 , . . . , W k to be aperiodic, then R is well defined.

Invariant manifolds.
Here we prove that the manifolds in Γ s and Γ u satisfy (P 2 ) and (P 3 ). We start by proving some useful estimates about the map φ. It follows from the results in the beginning of [20, Section 6.2] that (a n ) n and (a ′ n ) n have the same asymptotics of the sequence 1/n 1/θ . In particular, there is C > 0 such that for all n ≥ 1 we have ∆a n := a n − a n+1 ≤ C n 1+1/θ , and a similar estimate holds for (a ′ n ) n . For the sake of notational simplicity we shall consider τ = 1/θ. Lemma 6.1. There exists C > 0, such that for all n ≥ 0 and all x ∈ [a n+1 , a n ], we have Proof. By the definition of a n , we have |φ n (a n ) − φ n (a n+1 )| = |a 0 − a 1 | and so, using the Mean Value Theorem and (26), we get, for some ξ ∈ [a n+1 , a n ], Using the previous lemma for a = ξ and any b ∈ [a n+1 , a n ], we obtain the same conclusion for any point in [a n+1 , a n ], concluding the proof.
To simplify notation, we write f ′ u to mean the derivative of f in the unstable direction. Proposition 6.2. There exists C > 0 such that (a) for all n ∈ N and x, y ∈ γ u ∈ Γ u we have Proof. We shall prove (a). The proof of (b) follows similar arguments.
Consider x, y ∈ γ u and let n ∈ N. We first assume that the orbits of x and y visit W 0 exactly at the same moments. Then, it is enough to prove that there is C > 0 such that for a given point z ∈ T 2 we have Considering P = k i=1 p i and Q = k i=1 q i , we have P + Q = n. Note that, since k i + p i ∈ K and k i + p i + 1 ∈ K , then f k i +p i (x) ∈ J 0 . Since we assumed that the orbits of x and y visit W 0 exactly at the same moments, then f k i +p i (x) ∈ J 0 . Observe that f coincides with φ in K ∩ γ u . Using the Mean Value Theorem and Lemma 6.1 we get, for some ξ ∈ J 0 , (27) For the iterates m ∈ M we have f −m (x) / ∈ W 0 , and so the behavior of (f u ) ′ is the same of the unperturbed Anosov case. In particular, there is exponential backward contraction: there is λ > 1 such that Gathering (27) and (28), we obtain, for any n ∈ N, We have for each i C p τ +1 With no loss of generality, we may assume that each p i /C 1 τ +1 ≥ 2. Actually, if this were not the case we would have the p i 's uniformly bounded, meaning that the corresponding p i iterates would be uniformly bounded away from the stable leaf of (0, 0). In particular, there would be some 0 < λ 0 < 1 such that |(f −1 u ) ′ | ≤ λ 0 and this case could be treated as the case of the previous case with λ 0 playing the role of λ.
Let us now prove (29) under the assumption that

Using this we get
thus proving (29).
Let us finally consider the case where the orbits of x and y do not visit W 0 at the same moments. Assume that there is j ≤ n such that f j (x) ∈ J ∪ J ′ and f j (y) / ∈ J ∪ J ′ . Choosing the size of the rectangle W 1 sufficiently small (and thus the length of γ u (x)), we may assure that we necessarily have f j (x) (uniformly) bounded away from γ s (0, 0). In particular, there is some λ 0 such that |f ′ u | ≥ λ 0 , and so we may repeat the calculations above we λ 0 playing the role of λ.

Bounded distortion.
Here we prove the bounded distortion property (P 4 ).
The following lemma is proved in [20,Lemma 5].
Lemma 6.3. There exists C > 0 such that, for all i, n ∈ N with i ≤ n, and for all a, b ∈ [a n+1 , a n ], Lemma 6.4. There exists C > 0 and 0 < β < 1 such that for all x, y ∈ γ u ∈ Γ u we have d(x, y) ≤ Cβ s(x,y) .
Proof. We will start by showing that there exists 0 < β < 1 such that, for x, y ∈ Λ ∩ γ u with s(x, y) = 0, we have d(x, y) ≤ β d(f R (x), f R (y)). In fact, since f R (x), f R (y) ∈ W 0 and f behaves like an Anosov diffeomorphism outside W 0 , then Applying this inequality successively, we obtain where C is the diameter of M .
Proposition 6.5. For γ ∈ Γ u and x, y ∈ Λ ∩ γ, Proof. Let γ ∈ Γ u and x, y ∈ Λ ∩ γ. We have As in Proposition 6.2, without loss of generality we may assume that the orbits of x and y visit W 0 exactly at the same moments.
We will consider now the terms of the right hand side of (30) which belong to some K i . Note that, since k i + p i ∈ K and k i + p i + 1 ∈ K, then f k i +p i ∈ J 0 . From Lemma 6.3, So, adding the term j = p i , we obtain Let us now consider the terms belonging to some M i . Since f is of class C 2 outside W 0 , using the Mean Value Theorem and the fact that f is uniformly expanding on unstable leaves, Gathering the conclusions we obtained for K and M , using Proposition 6.2-(a), and so, ) for some C 2 > 0, thus concluding the proof. Lemma 6.6. Let N and P be manifolds, where P has finite volume, and, for every n ∈ N, let Θ n : N → P be an absolutely continuous map with Jacobian J n . If we assume that (a) Θ n converges uniformly to an injective continous map Θ : N → P , (b) J n converges uniformly to an integrable continous map J : N → R, then Θ is absolutely continuous with Jacobian J.
Until the end of this section we denote Θ = Θ γ ′ ,γ (x) to simplify the notation. The next lemma can be found in [2,Lemma 3.11] and it is a consequence of [9, Lemma 3.8].
Lemma 6.7. Given γ, γ ′ ∈ Γ u and Θ : γ ′ → γ, then, for every n ∈ N, there exists an absolutely continous function π n : f n (γ ′ ) → f n (γ) with Jacobian G n such that Lemma 6.8. There exists C > 0 such that for all x, y ∈ γ s ∈ Γ s and n ∈ N we have We now need to control each term of the above sum. We divide this in three cases.
Assume first that f i (x), f i (y) ∈ W 0 . Since f i (y) ∈ γ s (f i (x)) and f has a product form in W 0 , then log det Df (f i (x)) − log det Df (f i (y)) = 0.
Assume now that f i (x), f i (y) ∈ W 0 . As f behaves like an Anosov diffeomorphism outside W 0 , then log det Df is Lipschitz. So, using the polynomial contraction on stable leaves, we get Finally, for f i (x) ∈ W 0 and f i (y) ∈ W 0 , choose the point z in the same stable leaf as f i (x) such that z is in the boundary of W 0 and between f i (x) and f i (y). Then, applying the first case to f i (x) and z, and the second case to z and f i (y), we obtain the conclusion.
Adding all the terms, we conclude that We define, for n ∈ N, the map Θ n : γ ′ → γ as Θ n = f −Rn •π Rn •f Rn . Note that Θ n is absolutely continuous, its Jacobian is and the Jacobian of Θ is given by Proposition 6.9. For γ ′ , γ ∈ Γ u , the function Θ is absolutely continuous and its Jacobian is given by .
Note that Lemma 6.8 implies that the product in the above proposition is finite. The proof of this proposition is a direct consequence of the following lemma together with Lemma 6.6. Lemma 6.10. The functions Θ n converge uniformly to Θ and their Jacobians J n converge uniformly to J.
Proof. Using (P 3 ), we have, for x ∈ γ, and, since R n → n ∞ and d f Rn (γ) (π Rn f Rn (x), f Rn Θ(x)) is bounded, then the uniform convergence follows.
We write By Lemma 6.7, G Rn (f Rn (x)) converges uniformly to one. To control the second factor note that, by (31) applied to the point Θ(x) and Θ n (x), we have We are left to prove that the first factor converges uniformly to J. Notice that which converges uniformly to zero, by Lemma 6.8.
Proposition 6.11. For each γ, γ ′ ∈ Γ u , the map Θ is absolutely continuous and denoting Proof. It is known that (P 5 )-(b) is satisfied by Anosov diffeomorphisms. But f is topologically conjugate to the Anosov diffeomorphism f 0 . Since the separation time is invariant by topological conjugacy, then so is (P 5 )-(b).
6.4. Recurrence times. Our goal in this subsection is to prove that there exists C > 0 such that for all γ ∈ Γ u and n ∈ N we have Since we have assumed the transition matrix of the initial Markov partition aperiodic, then there is n 0 ∈ N such that f n (W j ) intersects W k , for all j, k and all n ≥ n 0 . Lemma 6.12. For L ∈ {W 1 , . . . , W d , J 0 , J ′ 0 }, there exists n 0 ∈ N and δ 0 > 0 such that, for all n ≥ n 0 and j ∈ {1, . . . , d}, we have Proof. Choosing n 0 as in above, we know that, for all n ≥ n 0 , we have f n (L) intersects W k , for all k. Since, in addition, f n (L) must cross the entire length of the unstable direction of any W k it intersects, then f n (L) crosses the entire length of the unstable direction of every W k . Then Analogously, since (f Consequently, 1 . (33) Applying (P 4 ), there exists C > 0 such that, for all m ∈ N, Fixing w 0 we have Then, from (32), (33) and the previous inequalities, we obtain using (P 4 ) in the last step. Finaly, Define the σ-algebra Lemma 6.13. There exists ε 0 > 0 such that, for all i ∈ N and all ω ∈ B i with R |ω > R i−1 , Proof. Fix i ∈ N and let ω ∈ B i be such that R |ω > R i−1 . It follows from the definition of From Lemma 6.12, we know that Leb γ (A) ≥ δ 0 > 0. We are left to prove that Leb To prove that f −n u ′ | A ≥ δ 1 > 0, we only need to find an upper bound for (f n u ) ′ in B. If z ∈ A then z = f n (x), for some x ∈ B and R(x) = R i (x) = n + n 0 . So, Fixing concluding the proof.
Lemma 6.14. For all i, n ∈ N and all ω ∈ B i , In the proof of the next result we use ideas from [20, Section 4.1] and [1, Section A.2.1].
Proposition 6.15. There exists C > 0 such that, for sufficiently large n, Proof. We start by noting that Defining R 0 = 0, observe that Leb γ {R > n} = (I) + (II), where First we will see that there exists ε 0 > 0 and C > 0, a constant depending on f , but not on n, such that In fact, taking n ≥ 4n 0 , and so 1 2 n n 0 ≥ 2, we have (II) = Leb γ R > n; n ≥ R 1 2 n n 0 n n 0 , applying Lemma 6.13 to each factor.
We will now focus on (I). Let k ≥ 2n 0 . By (34), Fixing i, we have The last inequality is true because there exists j ≤ i such that R j − R j−1 > n i . In fact, if we assume the opposite, then n i i ≥ i j=1 R j − R j−1 = R i , which contradicts the assumption.
We will now prove that each term of the sum (37) is less then or equal to C(1 − ε 0 ) i i τ +1 n τ +1 . Considering first the case i, j ≥ 2, define where if j = 2 or j = 3 we take a = 1 and if j = i we take c = 1. Note that Applying Lemma 6.13 to each factor in a, we get a ≤ (1 − ε 0 ) j−1 . Each factor in c is of the Using again Lemma 6.13, we conclude that c ≤ (1 − ε 0 ) i−j . Using Lemma 6.14 and (36), we get .
Gathering all the estimates above we get For the term i = 1 of (I), we have, by the definition of R 1 , for any n ≥ n 1 , with n 1 sufficiently large. For i ≥ 2 and j = 1, considering each term of the sum in (37), arguing as we did to estimate c in the general case. Finally, from (35), (38) and the calculations for the small terms, we have, for sufficiently large n,

Appendix A. Coupling measures
In this appendix we prove Theorem 3.5. To simplify notation, we shall remove all bars. Though the proof follows the same steps of [20,Theorem 3], we have decided to include it here, as our polynomial assumptions imply some changes in the estimates.
Assume that there is C > 0 such that m{R > n} ≤ C n ζ . Let λ and λ ′ be probability measures in ∆ whose densities with respect to m are in the space G + θ and denote ϕ = dλ dm and ϕ ′ = dλ ′ dm .
Consider the partition Q = {∆ l,i } of ∆ introduced in Section 3.2 and the partition Q × Q of ∆ × ∆. Observe that each element of Q × Q is sent bijectively by F × F onto a union of elements of Q × Q. For n ∈ N, we define and denote by (Q × Q) n (x, x ′ ) the element of (Q × Q) n that contains the pair (x, x ′ ) of ∆ × ∆. Define R : ∆ → N as R(x) = min{n ∈ N 0 : F n (x) ∈ ∆ 0 }.
Observe that τ i+1 − τ i ≥ n 0 for all i ∈ N. We introduce now the simultaneous return time Note that, as (F, ν) is mixing, then (F × F, ν × ν) is ergodic. So T is well defined m × m a.e.. We define a sequence of partitions of ∆ × ∆, ξ 1 < ξ 2 < · · · as follows: The elements of ξ 1 are of the form Γ = A × ∆, where τ 1|A×∆ is constant and A is sent bijectively to ∆ 0 by F τ 1 ; • for i even, ξ i is the refinement of ξ i−1 obtained by partitioning Γ ∈ ξ i−1 in the x ′ direction into sets Γ such that τ i | Γ is constant and π ′ ( Γ) is sent bijectively to ∆ 0 by F τ i ; • for i odd, i > 1, we do the same as in the previous point replacing the x ′ direction by the x direction and π ′ by π.
For convenience we define ξ 0 = {∆ × ∆}. Note that, for all i ∈ N, {T = τ i } and {T > τ i } are ξ i+1 -measurable and, for all n ≤ i, τ n is ξ i -measurable. Define a sequence of stopping times in ∆ × ∆, 0 ≡ T 0 < T 1 < · · · , as We consider the dynamical system F = (F × F ) T : ∆ × ∆ → ∆ × ∆. Observe that, for all n ∈ N, F n = (F × F ) Tn . Let ξ 1 be a partition of ∆ × ∆ composed by rectangles Γ such that T | Γ is constant and F : Γ → ∆ 0 × ∆ 0 is bijective. Define a sequence of partitions, ξ 2 , ξ 3 , . . ., by ξ n = F −(n−1) ξ 1 , for n ≥ 2. Note that T n is constant on each element of ξ n and F n maps each element of ξ n bijectively to ∆ 0 × ∆ 0 . Consider the measure m × m for the dynamical system F and the Jacobian, J F , of F with respect to m × m. Define a separation time s : (∆ × ∆) × (∆ × ∆) → N 0 as s(z, w) = min n ∈ N 0 : F z and F w belong to different elements of ξ 1 .
Lemma A.1. For z, w ∈ ∆ × ∆ such that s(z, w) ≥ n, for some n ∈ N, we have Proof. Let z = (x, x ′ ) and w = (y, y ′ ). Then, since log x ≤ x − 1 for x ∈ R + and ϕ, ϕ ′ ∈ G + θ , There exists a constant C > 0 depending only on D ϕ and D ϕ ′ , such that, for Proof. Take z 0 , w 0 ∈ Γ such that F i (z 0 ) = z and F i (w 0 ) = w. As s(z 0 , w 0 ) ≥ i, using Lemma A.1 and Lemma A.2, we get Recalling Lemma A.1 we define C F = 2C F . We take and From here on we assume that θ > 2e K .
Proposition A.4. There exists ε 0 > 0 such that, for all i ≥ 2 and Γ ∈ ξ i with T |Γ > τ i−1 , we have The constant ε 0 depends only on D ϕ , D ϕ ′ and, if we choose i ≥ i 0 (D ϕ , D ϕ ′ ), the dependence can be removed.
Proposition A.5. There exists k 0 > 0 such that, for all i ∈ N 0 , Γ ∈ ξ i and n ∈ N 0 , The constant k 0 depends only on D ϕ , D ϕ ′ and, if we choose i ≥ i 0 (D ϕ , D ϕ ′ ), the dependence can be removed.
The proofs of these two propositions follow the same steps of the proofs of (E1) and (E2) in [1, Subsections A.3.1 and A.3.2]. We only need to adapt the proof of [1, Lemma A.2] to our case, which we do next.
The dependence of C 0 on D ϕ may be removed if we assume that the number of visits j ≤ k of A to ∆ 0 is bigger then a certain j 0 = j 0 (D ϕ ).
Proof. Given x 0 , y 0 ∈ A such that F k (x 0 ) = x and F k (y 0 ) = y then, as ϕ ∈ G + θ and using Lemma 3.2, The following proposition, whose proof can be found in [1, Subsection A.2.1], follows from Propositions A.4 and A.5.
Proposition A.7. Let C > 0 and ζ > 1 be such that Leb{R > n} ≤ Cn −ζ . Then, there exists C ′ > 0 such that We want to define a sequence of densities ( Φ i ) in ∆ × ∆ such that Φ 0 ≥ Φ 1 ≥ · · · and for all i ∈ N and Γ ∈ ξ i , Take ζ as in Theorem 3.5. Noting that 1 < ζ < θ e K − 1, we fix ρ such that Take for i ≥ i 0 , where i 0 is such that ε i 0 < 1. Further restrictions on i 0 will be imposed during the proof of Lemma A.8. Define Φ i ≡ Φ for i < i 0 , and for i ≥ i 0 where ξ i (z) is the element of ξ i which contains z. It is easy to verify that the sequence Φ i (z) satisfies condition (40). For z ∈ ∆ × ∆, let and Lemma A.8. There exists i 0 ∈ N such that, for i ≥ i 0 and for all z, w ∈ ∆ × ∆ with w ∈ ξ i (z), we have Proof. We divide this proof into several steps.
Step 1: By the definition of Ψ i and Lemma A.1, Step 2: . (43) We may assume that Ψ i (w) ≤ Ψ i (z). Otherwise, we can swap the positions of z and w. We can easily verify that, for all 0 < a ≤ b < 1, we have Taking a = ε i Ψ i (z) and b = ε i Ψ i (w) and recalling the definition of ε i , we obtain Gathering the expressions (43) and (44), we obtain .
Step 3: Note that and so, using step 2, Lemma A.1 and Lemma A.2, Then log Ψ i 0 (z) and so we can choose i 0 sufficiently large such that obtaining the conclusion of the Lemma for i = i 0 .
Step 4: Using steps 2 and 1, we obtain Step 5: Using the equality s( F i−j z, F i−j w) = s( F i z, F i w) + j and the inequalities in steps 3 and 4, we get, for i ≥ i 0 + 1, In the next two steps we will control the two terms of the previous expression.
Step 6: Recalling that ε ′ i = ε i 1−ε i and ε i = e K 1 − i−1 i ρ , it is easy to check that Remember that, in (41), we chose ρ such that θ > e K ρ. So, for i 0 sufficiently large and i ≥ i 0 , we have ε ′ i < θ i . As log(1 + x) ≤ x for x > 0, then Step 7: We may choose i 0 sufficiently large such that (1 + ε ′ i 0 )β < 1. Note that we will later impose additional restrictions on i 0 . So, recalling that (ε ′ i ) is a decreasing sequence converging to zero, then, for all i ≥ i 0 , Step 8: Replacing the conclusions of steps 6 and 7 on the expression in step 5, we obtain, for i ≥ i 0 + 1, Observing that we chose C > C F 1−β , then there exists i 0 large enough such that, for i ≥ i 0 + 1, Recalling that we proved the same result for i = i 0 in step 3, this concludes the proof.
Lemma A.9. There exists i 0 ∈ N such that, for all i ≥ i 0 and Γ ∈ ξ i , The constant k 1 depends only on D ϕ and D ϕ ′ .
We will now verify that the other terms vanish. Let A k,i = {z ∈ ∆ × ∆ : k = T i (z)} and A k = A k,i . Note that each of the sets A k,i is a union of elements of Γ ∈ ξ i and A k,i = A k,j for i = j. By (51) we have Φ k−1 −Φ k = Φ i−1 − Φ i on Γ ∈ ξ i |A k,i and Φ k−1 = Φ k on ∆×∆\A k . Given k ∈ N and remembering that, from (40), ) . This completes the proof of (52). As a consequence, we have For the first term of this expression we have and for each of the others, using Lemma A.10, we obtain So, replacing the previous two expressions in (53), we get On the other hand, Gathering the last two inequalities we conclude that where k 1 depends only on i 0 . Fixing i 0 sufficiently large, from Lemma A.10 we obtain the dependence of k 1 on ϕ and ϕ ′ .
The proof of the following proposition can be found in [1,Subsection A.3.4]. We remark that though it uses [1, Lemma A.6], whose proof does not necessarily follow for functions in G + θ , we obtained the same conclusion in Lemma A.3.
Proposition A.12. There exists a constant k 2 > 0 such that, for n ∈ N and i ∈ N 0 , The constant k 2 depends only on D ϕ and D ϕ ′ .
We are now ready to prove Theorem 3.5.
Proof of Theorem 3.5. Given i ∈ N, we have The last inequality is true because there exists j ≤ i such that T j+1 − T j > n i+1 . In fact, if we assume the opposite, then n i i ≥ i j=1 T j+1 − T j = T i+1 , which contradicts the assumption. It follows, respectively from Proposition A.11, (54) and Proposition A.12 that We know from Proposition A.7 that P {T > n} ≤ C/n ζ−1 . So, taking P = m × m we obtain Since, in (41), we chose ρ > ζ + 1, we obtain F n * λ − F n * λ ′ ≤ C ′ 1 n ζ−1 .