Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards

Gouezel and Sarig introduced operator renewal theory as a method to prove sharp results on polynomial decay of correlations for certain classes of nonuniformly expanding maps. In this paper, we apply the method to planar dispersing billiards and multidimensional nonMarkovian intermittent maps.


Introduction
In two seminal papers, Young [53,54] obtained results on exponential and subexponential decay of correlations for nonuniformly hyperbolic dynamical systems. In the case of subexponential decay, a natural question is to establish that the decay rates obtained in this way are optimal. The first progress in this direction was by Sarig [47] who introduced the method of operator renewal theory. This method was extended and refined by Gouëzel [24] and gives optimal results for one-dimensional intermittent maps of Pomeau-Manneville type [46,51].
A challenge has been to extend the applicability of operator renewal theory to higher-dimensional examples. Two specific directions have required attention: (i) planar dispersing billiards, (ii) multidimensional nonMarkovian intermittent maps. For results in these directions, we mention [31,52].
In this paper, we extend the operator renewal theory of Gouëzel and Sarig [24,47] to provide lower bounds in general situations where the Young tower method [54] provides upper bounds. This includes directions (i) and (ii) above. In the case of lower bounds for dispersing billiards, these are the first results using operator renewal theory, and the first results by any methods for billiards with decay rates other than n −1 . For multidimensional intermittent maps, we obtain essentially optimal upper and lower bounds on decay of correlations.
Roughly speaking the result of Gouëzel and Sarig takes the following form. Let f : M → M be an ergodic measure-preserving transformation defined on a probability space (M, µ). The correlation function ρ v,w (n) is given by for L 2 observables v, w : M → R. For definiteness, as in [24,47] we consider onedimensional Markovian intermittent maps such as in [37] with f (x) ≈ x 1+1/β for x near zero, where β > 1, and unique absolutely continuous invariant probability measure µ. Fix η ∈ (0, 1). By [54], there is a constant C > 0 such that for all v ∈ C η (M), w ∈ L ∞ (M), n ≥ 1. Now fix a closed subset X ⊂ M with 0 ∈ X and let h : X → Z + denote the first return time to X. By [24,47], there exists a constant C > 0 such that for all v ∈ C η (M) supported in X, w ∈ L ∞ (X), n ≥ 1, where ζ β (n) =      n −β β > 2 n −2 log n β = 2 n −2(β−1) 1 < β < 2 . (1.3) Since µ(h > n) ∼ cn −β for some c > 0, this shows that the results in [54] are sharp. If in addition v dµ = 0, then ρ v,w (n) = O(n −β ) for all β > 1. (One consequence of the main result in this paper is that the latter estimate holds for all w ∈ L ∞ (M); this is not shown in previous papers. See Remark 3.3.) Abstract theorems for nonuniformly expanding and nonuniformly hyperbolic dynamical systems are stated in Sections 3 and 7 respectively. In common with the method of [24,47], we induce on a convenient subset Y ⊂ M with induced map F : Y → Y that is Gibbs-Markov for nonuniformly expanding maps and Gibbs-Markov after quotienting along local stable leaves for nonuniformly hyperbolic maps.
A key difference from [24,47] is that F need not be a first return map. As in [10], we are able to control the adverse effects associated with not being a first return and to obtain results that are essentially the same as those in [24,47]. Remark 1. 1 We note that the setting in [52] is currently restricted to planar timereversible systems.
In the remainder of the introduction, we focus on the applications to billiards and multidimensional intermittent systems.

Billiard examples
Markarian [38] and Chernov & Zhang [17] considered a general framework for analysing decay of correlations for diffeomorphisms with singularities, with special emphasis on slowly mixing planar dispersing billiards. All known results on upper bounds for decay of correlations for dispersing billiards fall within this framework. Within this framework, we obtain lower bounds.
The specific examples are described in more detail in Section 8. Here we summarize the results. All integrals are with respect to Liouville measure. Upper bounds are for general dynamically Hölder observables v and w. Lower bounds are for dynamically defined Hölder observables with nonzero mean supported in a suitable subset X of phase space.
• Bunimovich stadia, semidispersing billiards, billiards with cusps. In these examples, the correlation decay rate O(n −1 ) was established by [15,17,18,38]. By the argument in [7,Corollary 1.3] (see also [6,Corollary 1.1]), the result is essentially optimal in the sense that if v = w and if v is Hölder and satisfies a nondegeneracy condition, then nρ v,w (n) → 0 as n → ∞. However, for several years it remained an open question to obtain an asymptotic rate of the type (1.2).
We prove that for all three types of billiard there is a constant c > 0 such that ρ v,w (n) ∼ cn −1 v w. The constant c is given explicitly in terms of the billiard configuration space. For example, in the case of a Bunimovich stadium with straight sides of length ℓ, c = 4 + 3 log 3 4 − 3 log 3 ℓ 2 4(π + ℓ) .
(Throughout, log means logarithm to base e.) A similar result for semidispersing billiards and billiards with cusps (but not stadia) can be found in [52], though it is not clear that the asymptotic ρ v,w (n) ∼ const. n −1 is established there.

Hu-Vaienti maps
We consider a class of piecewise smooth multidimensional nonuniformly expanding intermittent maps f : M → M, M ⊂ R k compact, with a neutral fixed point. The case k = 1 is very well-understood. Upper bounds on decay of correlations were obtained by [29,54] and the results were shown to be sharp by [24,29,47]. Extending to multidimensional examples is relatively straightforward in the Markov case, but the nonMarkov case is very challenging because the standard symbolically Hölder spaces are unavailable for nonMarkov maps and there are difficulties using spaces of bounded variation in higher dimensions. Also, as shown in [30], such maps often have poor bounded distortion properties.
Hu & Vaienti [30] obtained results on existence of absolutely continuous ergodic invariant measures (both finite and infinite) for various classes of multidimensional nonMarkovian intermittent maps. In a subsequent paper [31], first results on upper and lower bounds on decay of correlations were obtained. As an application of the results in this paper, we obtain essentially optimal upper and lower bounds.
Moreover, our decay rate is essentially optimal. For v Hölder and w ∈ L ∞ with supports bounded away from 0 and nonzero mean, we show that for any ǫ > 0 The remainder of the paper is organized as follows. In Section 2, we recall background material on inducing, Gibbs-Markov maps, Young towers, and Chernov-Markarian-Zhang structures. Our main result for nonuniformly expanding maps is stated in Section 3 and proved in Section 4. In Section 5, we relate tail estimates for different return times. In Section 6, we apply our results to multidimensional nonuniformly expanding maps including those mentioned in Subsection 1.2.
In Section 7, we extend our main result to nonuniformly hyperbolic systems, including solenoidal versions of the maps in Section 6. Finally, in Section 8, we consider the examples from billiards mentioned in Subsection 1.1.
Notation We use the "big O" and ≪ notation interchangeably, writing a n = O(b n ) or a n ≪ b n if there is a constant C > 0 such that a n ≤ Cb n for all n ≥ 1. Also, we write a n ≈ b n if a n ≪ b n ≪ a n . As usual, a n ∼ b n as n → ∞ means that lim n→∞ a n /b n = 1.
Convolution of sequences a n , b n (n ≥ 0) is denoted (a ⋆ b) n = n j=0 a j b n−j . Often we use the abuse of notation a n ⋆ b n . If a 0 is undefined (as for example a n = n −2 log n) then we redefine a 0 = 1 without mentioning it. With these conventions we have the standard facts n −p ⋆ n −q = O(n −q ) and n −p ⋆ n −q log n = O(n −q log n) for all p > 1, q ∈ (0, p].

Preliminaries
In this section, we recall background material on (one-sided) Chernov-Markarian-Zhang structures.
Gibbs-Markov maps Let (Y, µ Y ) be a probability space with an at most countable measurable partition α, and let F : Y → Y be an ergodic measure-preserving transformation. For θ ∈ (0, 1), define the separation time s(y, y ′ ) to be the least integer n ≥ 0 such that F n y and F n y ′ lie in distinct partition elements in α. It is assumed that the partition α separates trajectories, so s(y, y ′ ) = ∞ if and only if y = y ′ ; then θ s is a metric.
A consequence is that there is a constant C > 0 such that for all y, y ′ ∈ a, a ∈ α.
Return maps Suppose that (M, µ) is a probability space and that f : M → M is an ergodic measure-preserving transformation. Fix a measurable subset X ⊂ M with µ(X) > 0 and h : Then h is called a return time and f h : X → X is called a return map.
If h is the first return time to X under f (i.e. h(x) = inf{n ≥ 1 : f n x ∈ X}), then f h : X → X is called the first return map and µ X = (µ| X )/µ(X) is an ergodic f h -invariant probability measure on X.
Young towers Let F : Y → Y be a full-branch Gibbs-Markov map on (Y, µ Y ) with partition α and let ϕ : Y → Z + be an integrable function constant on partition elements. We define the (one-sided) Young tower ∆ = Y ϕ and tower map f ∆ : ∆ → ∆ as follows: Now suppose that f : M → M is an ergodic measure-preserving transformation on a probability space (M, µ), and that Y ⊂ M is measurable with µ(Y ) > 0. Suppose that F : Y → Y is a full-branch Gibbs-Markov map with respect to a probability measure µ Y on Y , and that ϕ : Y → Z + is a return time, constant on partition elements, such that F = f ϕ . Form the tower ∆ = Y ϕ and tower map f ∆ : ∆ → ∆. The map π M : ∆ → M, π M (y, ℓ) = f ℓ y defines a semiconjugacy between f ∆ and f . We require moreover that (π M ) * µ ∆ = µ. Then we say that f is modelled by a Young tower.
Chernov-Markarian-Zhang structure Suppose that (M, µ) is a probability space and let f : M → M be an ergodic and mixing measure-preserving transformation. Roughly speaking, the map f admits a Chernov-Markarian-Zhang structure if there is an integrable first return time h : X → Z + such that the first return map f X = f h : X → X is modelled by a Young tower Y σ . The full map f : M → M is also modelled by a Young tower Y ϕ . We denote these towers by ∆ = Y ϕ and ∆ rapid = Y σ since in the applications that we have in mind either the tower ∆ rapid is exponential or for any q > 1 the subset Y ⊂ X can be chosen such that f X is modelled by a Young tower Y σ with µ Y (σ > n) = O(n −q ). In the latter case, we say that f X is modelled by Young towers with superpolynomial tails.
In more detail, suppose Y ⊂ X ⊂ M are Borel sets with µ(Y ) > 0. Define the first return time h : X → Z + and first return map f X = f h : X → X.
We assume that f X : X → X is modelled by a Young tower ∆ rapid = Y σ with return time σ : Y → Z + and return map F = f σ X : Y → Y . In particular, F = f σ X : Y → Y is a full-branch Gibbs-Markov map with ergodic invariant probability measure µ Y and partition α such that σ is constant on partition elements. We require in addition that h is constant on f ℓ X a for all a ∈ α, 0 ≤ ℓ ≤ σ(a) − 1. Define the induced return time Then ϕ is integrable with respect to µ Y and constant on partition elements. In particular, f : M → M is modelled by a Young tower ∆ = Y ϕ with the same Gibbs-Markov map F = f σ X = f ϕ .
We say that f : M → M satisfying these assumptions possesses a Chernov-Markarian-Zhang structure.
Remark 2.1 The method of choosing a first return map modelled by a Young tower with exponential tails arises in various contexts in the literature, see for example [9,10] in the noninvertible context. However, the method plays a special role in the context of billiards [17,38], see Remark 7.1 below.
Remark 2.2 It is part of our set up that µ is mixing, but in general the tower map f ∆ : ∆ → ∆ is mixing only up to a finite cycle d ≥ 1 where d is often unknown. As in [13, Theorem 2.1, Proposition 10.1], the a priori knowledge that µ is mixing ensures that for many purposes the value of d is irrelevant (in fact it suffices that µ is ergodic for all powers of f ).
We say that v is dynamically Hölder if v H < ∞ and denote by H(M) the space of such observables. Of particular interest are observables supported in X. We identify L ∞ (X) with It is standard that Hölder observables are dynamically Hölder for the classes of dynamical systems of interest in this paper, as we now recall. Given η ∈ (0, 1] and a metric d on M, define Let C η (M) be the space of bounded observables v : M → R for which |v| C η < ∞. Then Hence |v| H ≤ K η |v| C η and it follows that v ∈ H(M).

Statement of the main result
In this section, we state our main abstract result for maps f : M → M with a Chernov-Markarian-Zhang structure. Let Y ⊂ X ⊂ M denote the corresponding return map sets and recall that ϕ = h σ : Y → Z + is the induced return time. Throughout, we suppose that µ Y (ϕ > n) = O(n −β ′ ) for some β ′ > 1. (As discussed in Section 5, in our main examples any β ′ < β is permitted where µ X (h > n) = O(n −β ), and often we can take β ′ = β. However, h and β play no role in this section.) Define the correlation function ρ v,w (n) as in (1.1). It follows from Young [54] that We can now state our main theorem. Let Theorem 3.1 Let f : M → M be a map with a Chernov-Markarian-Zhang structure, and suppose that µ Y (ϕ > n) = O(n −β ′ ) for some β ′ > 1. Then there is a constant C > 0 such that for all n ≥ 1, for all v ∈ H(X), w ∈ L ∞ (X), Clearly n −β ′ ≤ σ n ≤ γ n . The sequences are readily estimated from above: Proof Suppose that σ has exponential tails, and fix K > 0. Then for some c > 0. Choosing K = (β ′ /c) log n, we obtain σ n = O(n −β ′ log n). Also The other cases are similar and hence omitted.

Remark 3.3
In particular, if σ is bounded, then we are back in the situation of [24,47] and our estimates reduce to theirs. Note that we have the slight improvement in Theorem 3.1(b) that w is an arbitrary L ∞ function, not necessarily supported in X. Such a result does not seem to have been noted before. When σ is unbounded, [24,47] does not apply directly since the estimates required for applying operator renewal theory are problematic on X, while the dynamics on Y is not given by a first return map, so it is necessary to incorporate arguments from [10].
Remark 3.4 As in [10], we can incorporate observables supported on the whole of M that decay sufficiently quickly off X. Letσ : Then Theorem 3.1 holds with σ n defined usingσ instead of σ.
In contrast to [10], we do not require that the Hölder constants of v decay off X.

Proof of the main theorem
In this section we prove Theorem 3.1. We continue to suppose that f : M → M possesses a Chernov-Markarian-Zhang structure and that µ Y (ϕ > n) = O(n −β ′ ) for some β ′ > 1. In Subsection 4.

Tower reformulation
Recall that M is modelled by a Young tower ∆ = Y ϕ and that F = f ϕ : Y → Y is a full-branch Gibbs-Markov map. Let d = gcd{ϕ(a) : a ∈ α}. The tower map f ∆ : ∆ → ∆ is mixing if and only if d = 1. To deal with the cases d = 1 and d ≥ 2 uniformly, we set Φ = d −1 ϕ. Replace ∆ by ∆ = Y Φ and redefine f ∆ : ∆ → ∆ accordingly. Also, define Then π M is a semiconjugacy between f ∆ and g = f d , and (π M ) * µ ∆ is an ergodic ginvariant probability measure on M. It is an easy consequence of the definitions that (π M ) * µ ∆ is absolutely continuous with respect to the original measure µ. Moreover, µ is mixing for f by assumption and so is also ergodic for g. Hence π M is a measurepreserving semiconjugacy between (∆, f ∆ , µ ∆ ) and (M, g, µ). Observables Since h : X → Z + is the first return time to X under the map f : M → M we have that f ℓ y ∈ X for some ℓ ≥ 0 precisely when ℓ = h j (y) for some j ≥ 0. Since ϕ(y) = h σ(y) (y), there are precisely σ(y) returns of y to X under f by time ϕ(y). Hence ϕ(y)−1 ℓ=0 and the result follows.
Fix θ ∈ (0, 1) and define . We conclude this subsection by showing that Theorem 3.1 is a direct consequence of Theorem 4.2.
Using the measure-preserving semiconjugacy π M : ∆ → M, π M (y, ℓ) = g ℓ y, we can write Hence it follows from (4.2) and Theorem 4.2(a) that Summing over j yields . This completes the proof of part (a). Similarly, in the context of part (b),

Operator renewal theory on
and the corresponding Fourier series R(z), T (z) : A calculation shows that Also, for z ∈ D, we define

Proof of Theorem 4.2
The following formula is a discrete time analogue of a formula in [45]. (The proof is in the Appendix.) Proof The estimates for |V (n)| 1 and |W (n)| 1 are immediate. Let y, y ′ ∈ a, a ∈ α.
Proof By Proposition 4.5 and the estimate for H(n) in Lemma 4.3(a), the first integral is estimated by The second integral is estimated in the same way using Lemma 4.3(b).
For n ≥ 0, define Proof Sinceṽ is supported in X, |ṽ(y, ℓ)| and it again follows that The Fourier series for V (n), W (n) are given by Proof We have The calculation for A 2 is similar.
Proof of Theorem 4.2 By Lemma 4.3(a) and Proposition 4.4, By Corollary 4.6(a) and Lemma 4.7(a), . Hence it follows from Proposition 4.5 and Lemma 4.7 that and we obtain Also, and similarly P W (1) = ∆ w dµ ∆ . This completes the proof of part (a). The proof of part (b) proceeds in much the same way but with b(n) and H(n) replaced byb(n) and H(n) from Lemma 4.3(b). Using Corollary 4.6(b) instead of Corollary 4.6(a), we obtain where E(n) =b(n)⋆P V (n)⋆P W (n). Calculating as in part (a) and using P V (1) = 0, yielding the desired estimate in part (b). Note also that the terms involving A 2 (n) are no longer present. The estimate for A 2 (n) in Lemma 4.7(c) was the only one that required w to be supported in X, so part (b) holds for all w ∈ L ∞ (∆).

Tail estimates
In applications, we are often given information about the first return time h : X → Z + . To apply Theorem 3.1, it is necessary to translate this into information about the tails µ Y (ϕ > n) of the induced return time ϕ : Y → Z + . We begin with a rough estimate of this type.
Suppose that f : M → M possesses a Chernov-Markarian-Zhang structure and that ∆ rapid = Y σ has exponential tails.
Now suppose that ∆ rapid has polynomial tails with µ Y (σ > n) = O(n −q ) for q sufficiently large (depending on β and ǫ).
(c,d) These arguments are similar and hence omitted.
Next, we consider a sharper estimate following [44]. First we collect some special cases of existing results about limit laws. Assume that f X : X → X is modelled by a Young tower ∆ rapid = Y σ with σ ∈ L 2 (Y ). In particular, Proof (a) Since F is Gibbs-Markov, the condition σ ∈ L 2 ensures that n −1/2 (σ n −nσ) converges in distribution (to a possibly degenerate normal distribution) and hence that b −1 n (σ n − nσ) converges in probability to zero.
6 Piecewise smooth multidimensional nonMarkovian nonuniformly expanding maps In this section, we show how to combine the methods in this paper with a result of Alves et al. [1] to treat a large class of multidimensional examples. In particular, we obtain essentially optimal upper and lower bounds, as well as strong statistical properties, for Hu-Vaienti maps [30].

Existence of Chernov-Markarian-Zhang structures in arbitrary dimensions
Let M ⊂ R k be compact. We consider local diffeomorphisms f : M → M with finitely many branches. That is, there are disjoint open subsets U 1 , . . . , U K ⊂ M with M = K i=1 U i , and there exists η ∈ (0, 1) and for i = 1, . . . , K there exist Next we specify a compact first return set X ⊂ M with int X = X. (We could take X to be the closure of one of the U i but this need not be the case.) For simplicity, we suppose that the boundaries of U 1 , . . . , U K and X are piecewise smooth (with finitely many pieces). Let S 0 ⊂ M denote the singularity set S 0 = ∂X ∪ K i=1 ∂U i for f . Now define the first return time h : X → Z + and first return map f X = f h : X \ S → X \ S with singularity set A result of Alves et al. [1] guarantees under very mild conditions that f X is modelled by Young towers with superpolynomial tails if and only if f X has superpolynomial decay of correlations. We verify these conditions for a large class of nonuniformly expanding maps.
Define X m = {x ∈ X \ S : h(x) = m}. Let denote the Euclidean norm on R k and on k × k matrices. We suppose that there are constants λ ∈ (0, 1), δ > 0 and C, q > 1 such that then (iii) is automatic with Cm q replaced by 1 and (iv) is automatic by (ii) with δ = 1.
Lemma 6.2 Suppose that f : M → M is a nonuniformly expanding map satisfying conditions (i)-(iv). Let µ be an absolutely continuous mixing f -invariant probability measure on M and define µ X = µ(X) −1 µ| X . Suppose further that Then f possesses a Chernov-Markarian-Zhang structure and the map f X : X → X is modelled by Young towers with superpolynomial tails.
Proof To prove that f X is modelled by Young towers with superpolynomial tails, we apply [1, Theorem C]. Since there are some small inaccuracies in the statement there, we refer to [5, Theorem A.1] for a corrected version. It suffices to verify that µ X is an expanding measure and to verify conditions (C0)-(C3) in [5,Appendix A].
By condition (ii), log (Df X ) −1 ≤ log λ < 0 and This is the definition for µ X to be an expanding measure. Assumption (i) is precisely condition (C0). By (ii) and definition of S, For conditions (C2) and (C3), we consider a pair of points x, y ∈ M \ S with dist(x, y) < dist(x, S)/2. in particular, x, y ∈ X m for some m ≥ 1 and f j x, f j y lie in common open sets U i(j) for each 0 ≤ j ≤ m − 1. On by (6.2) and (iv). This verifies (C3). Also, by (iii). By (iv), Hence by (ii), This verifies condition (C2). Hence we conclude from [5, Theorem A.1] that for any q > 1 the map f X : X → X is modelled by a Young tower ∆ rapid = Y σ with µ Y (σ > n) = O(n −q ).
Finally, we note that the construction in [1] uses [2, Main Theorem 1] where it is made explicit that Y together with its partition elements a ∈ α are diffeomorphic to open balls in R k with the property that f σ X maps each a diffeomorphically onto Y . In particular, the connected set f ℓ X a lies in one of the subsets X m for each 0 ≤ ℓ < σ(a) − 1, so h is constant on f ℓ X a. Hence f possesses a Chernov-Markarian-Zhang structure.

Remark 6.3
In the situation of Lemma 6.2, suppose in addition that condition (vi) is improved to stretched exponential decay of correlations and that dµ X /d Leb is bounded below. Then part (2) of [1, Theorem C] yields Young towers ∆ rapid with stretched exponential tails. In particular, if the rate of decay of correlations is exponential and dµ X /d Leb is bounded below, then for every γ ∈ (0, 1 9 ), there exists Y ⊂ X and c > 0 such that f X : X → X is modelled by a Young tower Y σ with µ Y (σ > n) = O(e −cn γ ).
A standard argument (see for example [30, Theorem A and p. 1210]) shows that dµ X /d Leb is bounded below whenever f is noncontracting and topologically exact.

Upper bounds and limit laws
Although the emphasis in this paper is on lower bounds, we obtain essentially optimal upper bounds and many strong statistical properties as a consequence of Lemma 6.2.
Homogenization (convergence of fast-slow systems to a stochastic differential equation) when the fast dynamics is one of these maps f : M → M follows from [19,23,34]. Convergence rates in the WIP and homogenization are obtained in [4].

Application to Hu-Vaienti maps
We continue to consider local diffeomorphisms f : M → M, where M ⊂ R k is compact, with finitely many branches as in Subsection 6.1. We now specialize to intermittent maps with a neutral fixed point at 0 as described in Section 1.2. These maps are piecewise C 1+η for some η ∈ (0, 1) with finitely many branches, noncontracting everywhere (so Df (x)v ≥ v for all x ∈ R k , v ∈ R k ), and expanding everywhere except at 0 (so Df (x)v > v for all v ∈ R k if and only if x = 0).
The existence of absolutely continuous invariant probability measures for onedimensional intermittent maps was studied by [51] when the maps are Markov and by [57] in the nonMarkov case. In [30], a Banach space of quasi-Hölder observables studied by [33,49] was used to establish existence of σ-finite absolutely continuous ergodic f -invariant measures µ on M for multidimensional nonMarkov nonuniformly expanding maps. The cases µ(M) < ∞ and µ(M) = ∞ are considered equally in [30]; here we focus on the case of finite measures. The results in [30] require a delicate analysis taking into account poor distortion properties of multidimensional nonuniformly expanding maps. In [31], the quasi-Hölder space was used further to analyze upper and lower bounds on decay of correlations. Here we show how to combine [30] and Lemma 6.2 to obtain the essentially optimal results mentioned in Section 1.2.
To fix ideas, we focus on [31, Example 5.1], setting for x close to 0 where γ ∈ (0, k) and γ ′ > γ. Recall that the domains of the branches are denoted U 1 , . . . , U K and have piecewise smooth boundaries; we assume that 0 ∈ int U 1 and f −1 0∩ ∂U j = ∅. Choose an open ball R with 0 ∈ R ⊂ U 1 such thatR ⊂ f R and f R ⊂ U 1 . Set X = M \ R. Proposition 6.4 There is a unique absolutely continuous invariant probability measure µ X on X. The assumptions of Lemma 6.2 are satisfied and µ X (h > n) ≈ n −β where β = k/γ.
Proof The singular set ∂S is a countable union of piecewise smooth submanifolds limiting on finitely many piecewise smooth submanifolds, so condition (i) is satisfied.
Since the first return set X is bounded away from 0, it is immediate from noncontractivity on M and uniform expansion on X that (Df X ) −1 ≤ λ < 1. The remaining estimates in (ii) are established in [30,31]. (A big advantage here is that δ can be taken arbitrarily small and q arbitrarily large, so the fine details in [30,31] such as unbounded distortion are not an issue.) Since f is noncontracting, conditions (iii) and (iv) hold by Remark 6.1.
A key step in [30] is to establish quasicompactness of the transfer operator for the first return map f X : X → X. Assumptions 1-3 of [30, Theorem A] are mentioned explicitly above. As noted in [31, Example 5.1], Assumption 4 is automatic. Hence [30, Theorem A] guarantees the existence of an absolutely continuous invariant probability measure µ X on X. The density is quasi-Hölder and hence lies in L ∞ (X) verifying condition (v) of Lemma 6.2. By Remark 6.3, the density is also bounded below and hence µ X is unique.
Moreover, [30, Theorem A] establishes quasicompactness in the quasi-Hölder space and hence µ X is mixing up to a finite cycle. Since the support of a nonvanishing quasi-Hölder function has nonempty interior [49, Lemma 3.1], it follows from topological exactness that µ X is mixing. Condition (vi) is now an immediate consequence of quasicompactness.
By [30], each X m is a finite union of approximately spherical shells bounded by hypersurfaces S m and S m+1 where S m is approximately a sphere of radius ≈ m −1/γ . It follows that µ X (X m ) ≪ Leb(X m ) ≪ m −k/γ so condition (vii) is satisfied. By Remark 6.3, topological exactness ensures that µ X (h > n) ≈ Leb(h > n). Moreover, {h > n} = m>n X m is a finite union of balls of radius ≈ n −1/γ so µ X (h > n) ≈ n −β .
Hence for γ ∈ (0, k), we can apply the results in Subsections 6.2 and 6.3 to obtain the upper and lower bounds in (1.4), as well as the limit laws mentioned in Section 6.2.

Two-sided version of the main result
In this section, we extend Theorem 3.1 to invertible maps. A two-sided analogue of the Chernov-Markarian-Zhang structure is described in Subsection 7.1. The main result of this section, Theorem 7.4, is stated in Subsection 7.2 and reformulated for towers in Subsection 7.3. In Subsection 7.4, we show how to approximate two-sided observables by one-sided observables. In Subsection 7.5, we complete the proofs.

Preliminaries
We describe a two-sided (invertible) analogue of the structures discussed in Section 2. Throughout, f : M → M, f X : X → X, f ∆ : ∆ → ∆ and F : Y → Y are all twosided versions of the maps from Section 2, and the one-sided versions are denoted F :Ȳ →Ȳ and so on. We continue to writeφ = Y ϕ dµ Y but as will become clear this does not cause any confusion.
Two-sided Gibbs-Markov maps Let (Y, d) be a bounded metric space with Borel probability measure µ Y and let F : Y → Y be an ergodic measure-preserving transformation. LetF :Ȳ →Ȳ be a full-branch Gibbs-Markov map with partition α and ergodic invariant probability measureμ Y .
Now suppose that f : M → M is an ergodic measure-preserving transformation on a probability space (M, µ), and that Y ⊂ M is measurable with µ(Y ) > 0. Suppose that F : Y → Y is a two-sided Gibbs-Markov map with respect to a probability µ Y on Y , and that ϕ : Y → Z + is a return time as above. Form the tower ∆ = Y ϕ and tower map f ∆ : ∆ → ∆. The map π M : ∆ → M, π M (y, ℓ) = f ℓ y defines a semiconjugacy between f ∆ and f . We require moreover that (π M ) * µ ∆ = µ. Then we say that f is modelled by a two-sided Young tower.
We require that f X : X → X is modelled by a two-sided Young tower ∆ rapid = Y σ with return time σ : Y → Z + and return map F = f σ X : Y → Y . Here, F = f σ X : Y → Y is a two-sided Gibbs-Markov map with ergodic invariant probability measure µ Y and partition α such that σ is constant on partition elements. We require in addition that h is constant on f ℓ Xπ −1 a for all a ∈ α, 0 ≤ ℓ ≤ σ(a) − 1. Define the induced return time ϕ = h σ : Y → Z + as in (2.2). Then ϕ is an integrable return time (constant onπ −1 a for a ∈ α). In particular, f : M → M is modelled by a Young tower ∆ = Y ϕ with the same two-sided Gibbs-Markov map F = f σ X = f ϕ . We say that f : M → M satisfying these assumptions possesses a two-sided Chernov-Markarian-Zhang structure.
Remark 7.1 Young [53] introduced Young towers with exponential tails as a general method for dealing with diffeomorphisms with singularities; the initial landmark application was to prove exponential decay of correlations for planar finite horizon dispersing billiards. Chernov [13] simplified the construction of exponential Young towers and used this to prove exponential decay of correlations for planar dispersing billiards with infinite horizon. Then Young [54] studied examples with subexponential decay of correlations using Young towers with subexponential tails. Markarian [38], noting that Chernov's simplification no longer applies in the subexponential case, devised the method outlined in this section: namely to construct a first return map for which Chernov [13] applies. This was used to prove the decay of correlations bound O(1/n) for Bunimovich stadia. The method was extended and simplified by Chernov & Zhang [17] who applied it to a large class of billiard examples. Subsequent applications of the method include [15,16,56].

Remark 7.2
We have omitted much of the structure often associated with Young towers, mentioning only those properties required in the sequel. For instance, we have not made any explicit mention of a product structure, though we make use of condition (7.1) which is a consequence. Similarly, we have not made explicit the quotienting procedure (along local stable leaves) that passes from F toF .
Hence |v| H ≪ |v| C η and it follows that v ∈ H(M).

Statement of the main result
As in Section 3, we provide an abstract result for maps f : M → M with a Chernov-Markarian-Zhang structure under the assumption that µ Y (ϕ > n) = O(n −β ′ ) for some β ′ > 1. In this situation it follows from Young [54] that ρ v,w (n) = O(n −(β ′ −1) ) for dynamically Hölder observables. (The result in [54] is formulated for one-sided systems; see [36,Theorem 2.10] or [42,Appendix B] for the two-sided case.) We obtain a lower bound for dynamically Hölder observables supported in X.
Theorem 7.4 Let f : M → M be a map with a two-sided Chernov-Markarian-Zhang structure, and suppose that µ Y (ϕ > n) = O(n −β ′ ) for some β ′ > 1. Then there is a constant C > 0 such that for all n ≥ 1, for all v, w ∈ H(X),  [3,43] are written down for one-dimensional maps but apply equally to multidimensional maps.) The resulting solenoidal intermittent maps fall within the two-sided Chernov-Markarian-Zhang framework and have stable and unstable directions of any specified dimension. Our results yield essentially optimal upper and lower bounds on decay of correlations for these examples. Again, the lower bounds are realized by Hölder observables that are supported away from the neutral fixed point.
The counterpart of Theorem 4.2 is: There is a constant C > 0 such that for all n ≥ 1, with ∆ṽ dµ ∆ = 0. Theorem 7.4 is a direct consequence of Theorem 7.6 in exactly the same way that Theorem 3.1 was a direct consequence of Theorem 4.2. Hence we omit the details except to mention that we make use of the estimate The key steps in the proof of Theorem 7.6 are contained in the following result.
Lemma 7.7 There is a constant C > 0 such that for all n ≥ 1: with ∆ṽ dµ ∆ = 0. The proof of Lemma 7.7 takes up most of the remainder of this section. Assuming this result, we can complete the proof of Theorem 7.6.
Since h is the first return time under f to X, we have g ℓ y ∈ X d if and only if ℓd ≤ k−1 j=0 h(f j X y) < (ℓ + 1)d for some k = 0, . . . , σ(a) − 1. Now use that h is constant on f j Xπ −1 (πy). By Proposition 7.8, we can write X =π −1Z whereZ =π X ⊂∆. By Proposition 4.1,

Approximation by one-sided observables
In this subsection, we show how to approximate two-sided observables by one-sided observables, broadly following the method used in [42,Appendix B] which was in turn based on a private communication by Gouëzel. Using this we prove Lemma 7.7(b). Extend the separation time s on Y to ∆ by setting s((y, ℓ), (y ′ , ℓ ′ )) = s(y, y ′ ) when ℓ = ℓ ′ and 0 otherwise. Let ψ n = n−1 j=0 1 Y • f j ∆ be the number of entries to Y . For v ∈ L ∞ (∆), we approximateṽ • f n ∆ bỹ v n : ∆ → R,ṽ n (p) = inf{ṽ • f n ∆ (q) : s(p, q) ≥ 2ψ n (p)}, n ≥ 1.
(b) If suppṽ ⊂ X, then suppṽ n ⊂ f −n ∆ X and supp L nv n ⊂Z.
Proof Note that θ ψn is well-defined on∆. By Proposition 7.9(b,c), Hence the result follows from Lemma 7.11.

It follows that
Since w has mean zero, we can apply Theorem B.1 to obtain We conclude that for all j < k, and hence completing the proof.
Proof of Lemma 7.7(a) Let k ≥ 1 and write ρ * 1 X , w (n) = I 2 (k, n) + I 4 (k, n), where By Corollary 7.12, |I 2 (k, n)| ≪ | w| θ σ kd . Note that I 4 (k, n) is defined on∆ and in the notation of Section 4.1, I 4 (k, n) = ρ * 1Z ,w k (n − k). Hence we can proceed almost as in the proof of Theorem 4.2(a), withṽ and w replaced by 1Z andw k respectively. Let m = n − k. Following Proposition 4.4, we write By Proposition 7.9(a), |w k | ∞ ≤ | w| ∞ . Proceeding exactly as in Section 4.3, we obtain the estimates Moreover, V (m)(y) = V (m)(y ′ ) for y, y ′ ∈ a, a ∈ α. Hence following the proof of By assumption, ∆ w dµ ∆ = 0. Hence by Lemma 7.13, We have now estimated all the expressions arising in the proof of Theorem 4.2(a). Continuing as in that proof, we obtain (cf. (4.3)) Recall that the Fourier coefficients of (z − 1)b(z) are O(m −β ′ ). Also, Hence Combining this with the estimate for I 2 (k, n) and taking k = [n/2] yields the desired result.

Billiard examples
In this section, we provide details and proofs for the examples considered in Section 1.1. For background material on billiards, we refer to [14]. The billiard domain, denoted by Q, is a compact connected subset of R 2 or T 2 with piecewise smooth boundary and the billiard flow is defined on Q × S 1 . Fix a point q ∈ Q and a unit vector v ∈ S 1 . Then q moves in straight lines with unit speed in direction v until reflecting (angle of reflection equalling the angle of incidence) off the boundary ∂Q. This defines a volume-preserving flow. A natural Poincaré section is given by M = ∂Q × [−π/2, π/2] corresponding to collisions with ∂Q (with outgoing velocities in [−π/2, π/2]). The Poincaré map f : M → M is called the collision map or the billiard map. It preserves a probability measure µ, equivalent to Lebesgue, called Liouville measure. Part of the framework in [17,38] is that the billiard map f : M → M has a (twosided) Chernov-Markarian-Zhang structure as defined in Section 7. In particular, f has a suitably chosen first return map f X = f h : X → X modelled by a Young tower ∆ rapid = Y ϕ with exponential tails. Roughly speaking, X is chosen to be a subset of phase space bounded away from the regions where hyperbolicity is expected to break down, e.g. for billiards with cusps, X excludes a neighborhood of each cusp. Since the specific choice of X involves notation which is not required for understanding the results, we mainly point the reader to the original references for the precise definitions. (An exception is Example 8.3 below, where no extra notation is needed.) Example 8.1 (Bunimovich stadia [12]) These are convex billiard domains Q ⊂ R 2 where ∂Q is a simple closed curve consisting of two parallel line segments and two semicircles. By [38], the billiard map f : M → M falls within the Chernov-Markarian-Zhang framework with µ X (h > n) = O(n −2 ). By [18, Theorem 1.1], µ Y (ϕ > n) = O(n −2 ) and hence ρ v,w (n) = O(n −1 ) for dynamically Hölder observables.
Here, we improve the estimate on µ Y (ϕ > n) and use this to obtain lower bounds on decay of correlations. In addition ρ v,w (n) = O(n −2 log n) for all v ∈ H(X) with v dµ = 0 and all w ∈ H(X).
Proof In the proof of [7, Theorem 1.1] (see in particular [7, page 504, line 11]) it is shown for h : X → Z + (denoted there by ϕ + ) that (n log n) −1/2 ( n−1 j=0 h • f j X − n X h dµ X ) converges to a nondegenerate normal distribution. Hence the first statement follows from Corollary 5.3 and the second statement from Theorem 7.4(a).
Finally, γ n = O(n −2 log n) by Proposition 3.2, so the final statement follows from Theorem 7.4(b).

Example 8.3 (Semidispersing billiards)
The billiard domain is given by Q = R \ S k where R is a rectangle and there are finitely many disjoint convex scatterers S k ⊂ R with C 3 boundaries of nonvanishing curvature.
Example 8.7 (Billiards with cusps at flat points [55]) These are billiard domains Q ⊂ R 2 where ∂Q is a simple closed curve consisting of finitely many convex inwards C 3 curves such that the interior angles at one of the corner points is zero. Moreover the curves have nonvanishing curvature except at this corner point where ∂Q has the form ±x b for some b > 2.
In addition ρ v,w (n) = O(n −β log n) for all v ∈ H(X) with v dµ = 0 and all w ∈ H(X).
Proof By [32, Theorem 3.1], n −1/β ( n−1 j=0 h • f j X − n X h dµ X ) converges to a nondegenerate β-stable law. Hence the first statement follows from Corollary 5.3 and the second statement from Theorem 7.4(a).
Finally, γ n = O(n −β log n) by Proposition 3.2, so the final statement follows from Theorem 7.4(b).
(r, φ) with r close to the beginning of S and φ close to π/2. Since S is circular, the angles at successive collisions remain close to this initial value of φ so it is clear that {h > n} contains a set of the form E n = {(r, φ) : 0 ≤ r ≤ a n , π 2 − b n ≤ φ ≤ π 2 }, where a and b are constants independent of n. Hence µ X (h > n) ≥ r 0 0 π/2 −π/2 1 En cos φ dφ dr ∼ 1 2 ab 2 n −3 .

A Formula for the correlation function
In this appendix, we prove Proposition 4.4. One method would be to check equality of coefficients directly, but we choose to convert all sequences into Fourier series. Recall that V (n)(y) = 1 {Φ≥n}ṽ (y, Φ(y) − n), W (n)(y) = 1 {Φ>n} w(y, n), with Fourier series for allṽ, w ∈ F θ (∆), n ≥ 1.