Martingale approximations and anisotropic Banach spaces with an application to the time-one map of a Lorentz gas

Transcription

1 ariv: v1 [math.ds] 1 Jan 2019 Martingale approximations and anisotropic Banach spaces with an application to the time-one map of a Lorentz gas Mark Demers Ian Melbourne Matthew Nicol 30 December 2018 Abstract In this paper, we show how the Gordin martingale approximation method fits into the anisotropic Banach space framework. In particular, for the timeone map of a finite horizon planar periodic Lorentz gas, we prove that Hölder observables satisfy statistical limit laws such as the central limit theorem and associated invariance principles. 1 Introduction The traditional approach to proving decay of correlations and statistical limit laws for deterministic dynamical systems, following [7, 38, 39] and continuing with Young [43, 44], involves symbolic coding. In particular, by quotienting along stable leaves one passes from an invertible dynamical system to a one-sided shift. Decay of correlations is then a consequence of the contracting properties of the associated transfer operator. In addition, Nagaev perturbation arguments [20] and the martingale approximation method of Gordin [18] are available in this setting, leading to numerous statistical limit laws. These results on decay of correlations and statistical limit laws are then readily passed back to the original dynamical system. A downside to this approach is that geometric and smooth structures associated to the underlying dynamical system are typically destroyed by symbolic coding. In recent years, a method proposed by [6] and developed extensively by numerous authors (for Department of Mathematics, Fairfield University, Fairfield, CT 06824, USA. mdemers@fairfield.edu Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK. i.melbourne@warwick.ac.uk Department of Mathematics, University of Houston, Houston T , USA. nicol@math.uh.edu 1

2 recent articles with up-to-date references see [2, 16]) uses anisotropic Banach spaces of distributions to study the underlying dynamical system directly. In particular, the method does not involve quotienting along stable manifolds. This leads to results on rates of decay of correlations and also to various statistical limit laws via Nagaev perturbation arguments, see especially Gouëzel [19]. However, so far Gordin s martingale approximation argument has been absent from the anisotropic Banach space framework. This is the topic of the current paper. The utility of such an approach is illustrated by the following example. Example 1.1 The landmark result of Young [43] established exponential decay of correlations for the collision map corresponding to planar periodic dispersing billiards with finite horizon. The method, which involves symbolic coding, also yields the central limit theorem (CLT) for Hölder observables, recovering results of [8]. Turning to the corresponding flow, known as the finite horizon planar periodic Lorentz gas, the CLT follows straightforwardly from the result for billiards [8, 34]. However, decay of correlations for the Lorentz gas and the CLT for the time-one map of the Lorentz gas are much harder. Superpolynomial decay of correlations was established for sufficiently regular observables in [29] (see also [30]) using symbolic coding and Dolgopyat-type estimates [17]. This method also yields the CLT for the time-one map [1, 33], but again only for sufficiently regular observables. Here, regular means smooth along the flow direction, so this excludes many physically relevant observables such as velocity. The rate of decay of correlations was improved to subexponential decay[10] and finally in a recent major breakthrough to exponential decay [3]. Both references handle Hölder observables, suggesting that statistical limit laws such as the CLT for the time-one map should hold for general Hölder observables. Currently the Nagaev method is unavailable for Lorentz gases. We show that the Gordin approach is applicable and hence the CLT and related limit laws are indeed satisfied by Hölder observables for these examples. In particular, observables such as velocity are covered for the first time. Intheremainder oftheintroduction, we describesome ofthelimit lawsthatfollow from the methods in this paper. For definiteness, we focus on Example 1.1. Let be the three-dimensional phase space corresponding to a finite horizon planar periodic Lorentz gas, with invariant volume measure µ, and let T : be the time-one map of the Lorentz flow. Let φ : R be a Hölder observable with mean zero and define the Birkhoff sum φ n = n 1 j=0 φ Tj. It follows from [3, 10] that we can define σ 2 = lim n 1 φ n 2 n dµ = φφ T n dµ. n= By [1, Theorem B and Remark 1.1], typically σ 2 > 0 (the case σ 2 = 0 is of infinite codimension). We obtain the following results. 1 1 In what follows, d denotes convergence in distribution while w denotes weak convergence. 2

3 CLT: n 1/2 φ n d N(0,σ 2 ) as n. That is c lim µ(x : n n 1/2 φ n (x) c) = (2πσ 2 ) 1/2 e y2 /(2σ 2) dy for all c R. Weak invariance principle (WIP): Define W n (t) = n 1/2 φ nt for t = 0, 1 n, 2 n,... and linearly interpolate to obtain W C[0,1]. Then W n w W where W denotes Brownian motion with variance σ 2. Moment estimates: For every p 1 there exists C p > 0 such that φ n p C p n 1/2. Consequently, lim n n p/2 φ n p p = E Y p where Y = d N(0,σ 2 ). Homogenization: Now suppose that φ : R k. We continue to suppose that φ is C η for some η (0,1] and that φdµ = 0. Consider the fast-slow system x(n+1) = x(n)+ǫ 2 a(x(n))+ǫb(x(n))φ(y(n)), y(n+1) = Ty(n), (1.1) where x(0) = ξ R d and y(0) is drawn randomly from (,µ). We suppose that a : R d R d lies in C 1+η and b : R d R d k lies in C 2+η. Define ˆx ǫ (t) = x ǫ ([t/ǫ 2 ]), where x ǫ denotes the solution to (1.1). Then ˆx ǫ w Z as ǫ 0, where Z satisfies an Itô stochastic differential equation dz = ã(z)dt+b(z)dw, Z(0) = ξ, where W is a k-dimensional Brownian motion with covariance matrix Σ and ã(x) = a(x)+ d k α=1 β,γ=1 βγ bβ E (x)b αγ (x). x α Here, b β is the β th column of b and the matrices Σ,E R k k are given by Σ βγ = n= φ β φ γ T n dµ, E βγ = n=1 φ β φ γ T n dµ. The remainder of this paper is organized as follows. In Section 2, we recall background material on martingale-coboundary decompositions and statistical limit laws. In Section 3, we state an abstract theorem on obtaining martingale-coboundary decompositions for invertible systems with stable directions. In Section 4, we apply our results to the time-one map of the Lorentz gas. 2 Martingale-coboundary decompositions In this section, we review the approach going back to Gordin [18]. This method yields martingale-coboundary decompositions for observables of dynamical systems leading to various limit theorems. Related references include [4, 5, 15, 21, 26, 41, 42]. Let 3

4 (,µ) be a probability space, and let T : be an invertible ergodic measurepreserving transformation. Let F 0 be a sub-σ-algebra of the underlying σ-algebra on such that T 1 F 0 F 0. Consider an observable φ L 1 () with φdµ = 0. Definition 2.1 We say that φ admits a martingale-coboundary decomposition if φ = m+χ T χ, where m,χ L 1, m is F 0 -measurable, and E[m T 1 F 0 ] = 0. The conditions on m in Definition 2.1 mean that {m T n : n Z} is a sequence of martingale differences with respect to the filtration {T n F 0 : n Z}. Proposition 2.2 Let φ L p for some p 1. Suppose that n 1 E[φ T n F 0 ] p <, E[φ T n F 0 ] φ T n p <. (2.1) Then φ admits a martingale-coboundary decomposition with m, χ L p. Proof By (2.1), n 0 χ = n 0 (E[φ Tn F 0 ] φ T n )+ n 1 E[φ T n F 0 ] converges in L p. Define m = φ+χ χ T L p. Then m = n= (g n g n T) = n= (g n+1 g n T), (2.2) where g n = E[φ T n F 0 ]. Clearly, g n = E[φ T n F 0 ] is F 0 -measurable. Also, g n T is measurable with respect to T 1 F 0 F 0. Hence m is F 0 -measurable. Next, note that g n T = E[φ T n F 0 ] T = E[φ T n+1 T 1 F 0 ]. Hence E[g n T T 1 F 0 ] = E[φ T n+1 T 1 F 0 ] = E[E[φ T n+1 F 0 ] T 1 F 0 ] = E[g n+1 T 1 F 0 ], where we used that T 1 F 0 F 0. Substituting into (2.2), we obtain E[m T 1 F 0 ] = 0 as required. Corollary 2.3 Let φ L. Assume that φ admits a martingale-coboundary decomposition with m,χ L p for some p 1. Then (a) The moment bound maxk n k 1 j=0 φ Tj 2p = O(n 1/2 ) holds. (b) If p 2, then the CLT and WIP hold with σ 2 = m2 dµ. (c) If p 2, then lim n n q/2 φ n q q = E Y q for all q < 2p, where Y = d N(0,σ 2 ) Proof (a) This follows from [32, Eq. (3.1)]. (b) The CLT and WIP are standard applications of martingale limit theorems [18]. (c) This is an immediate consequence of parts (a) and (b), see for example [35, Lemma 2.1(e)]. 4

5 L 1 estimates and the WIP Somewhat surprisingly, by the results of [15], if φ L and conditions (2.1) hold for p = 1, then automatically m L 2 even though Proposition 2.2 only gives m,χ L 1. This suffices for the CLT. Related references for this phenomenon whereby m has extra regularity include [24, 26, 28, 37, 40]. Moreover, the following result holds: Theorem 2.4 Assume that φ L and conditions (2.1) are satisfied with p = 1. Then the CLT and WIP hold. Proof TheCLTandWIPinreverse time(asn )isanimmediate consequence of [15, Theorem 1] (see the discussion around [15, Condition (1.4)]). Passing from reverse time to forward time is standard (see for example [22, Section 4.2]). Homogenization As shown in [22, 23], rough path theory yields homogenization of fast-slow systems (1.1) provided iterated versions of the results in Corollary 2.3 (i.e. iterated moment estimates and iterated WIP) are satisfied. For recent refinements, we refer to [13, 14]. By [22, Theorem 4.3], the iterated WIP holds under (2.1) with p = 2. The required iterated moment control is given by[22, Proposition 7.1] provided p > 3. Hence we obtain homogenization results for p = Main abstract theorem Let T : be an invertible ergodic measure-preserving transformation on a probability space (,µ). We suppose that is covered by a collection W s of disjoint measurable subsets, called local stable leaves, such that TW s (x) W s (Tx) for all x, where W s (x) is the partition element containing x. Let F 0 denote the σ-algebra generated by W s. Note that W s (y) T 1 W s (x) for all y T 1 W s (x), so T 1 W s (x) is a union of elements of W s. Hence T 1 F 0 F 0. We denote by the L (F 0 ) the set of functions in L () that are F 0 -measurable. Theorem 3.1 Let φ L () be a mean zero observable. Assume that there exists β > 1 and C > 0 such that for all n 1, (a) φψ Tn dµ C ψ n β for all ψ L (F 0 ). (b) diam(φ(tn W s ))dµ Cn β. Then the conditions in (2.1) are satisfied for all 1 p < β. Proof Let where ξ = E[φ T n F 0 ] p 1 sgne[φ T n F 0 ] = ψ T n, ψ = E[φ T n F 0 ] p 1 sgne[φ T n F 0 ] L (F 0 ), 5

6 and ψ φ p 1. Then E[φ T n F 0 ] p p = E[φ T n F 0 ] p p = E[φ T n F 0 ]ξdµ = E[φξ T n F 0 ]dµ = φξdµ = By assumption (a), E[φ T n F 0 ] p p = φψ T n dµ C ψ n β C φ p 1 φψ T n dµ. n β, and the first part of condition (2.1) follows by taking pth roots and using the assumption p < β. Next, using the pointwise estimate E[φ T n F 0 ] φ diam(φ(t n W s )) and assumption (b), E[φ T n F 0 ] φ T n p p = E[φ Tn F 0 ] φ p p diam(φ(tn W s )) p p (2 φ ) p 1 diam(φ(t n W s )) 1 2 p 1 C φ p 1 n β. The second part of condition (2.1) follows. In the remainder of this section, we show that the conditions in Theorem 3.1 are satisfied in many standard situations. (The verifications below are not needed for our main example in Section 4.) 3.1 Verifying condition (b) in Theorem 3.1 Suppose that T : and W s are as above. Let Y be a positive measure subset that is a union of local stable leaves in W s. Define the first return time R : Y Z + and first return map F : Y Y, R(y) = inf{n 1 : T n y Y}, F(y) = T R(y) y. Let h n be the random variable on given by h n (x) = #{0 j n : T j x Y}. Lemma 3.2 Let φ : R be measurable. Suppose that µ(y Y : R(y) > n) = O(n (β+1) ) for some β > 1 and that there are constants C 1, γ (0,1) such that diam(φ(t n W s )) Cγ hn(x) for all W s W s, n 1. Then condition (b) in Theorem 3.1 holds. Proof We have n+1 n+1 diam(φ(t n W s ))dµ C γ k 1 {hn=k}dµ C γ k k=0 k=1 Y 1 {hn=k}rdµ. 6

7 Ify Y {h n = k}, then k 1 j=0 R Fj > n, andsor F j > n forsomej = 0,...,k 1. k Hence k 1 1 {hn=k}rdµ 1 {R F j n k } Rdµ. Y j=0 It follows from the tail assumption on R that there is a constant C 1 > 0 such that µ(y Y : R(y) > n) C 1 n (β+1) and 1 Y {R>n}Rdµ C 1 n β. Write R = 1 {R n} R+1 {R>n} R. Then 1 {R F j n k } Rdµ n1 {R F j n k } dµ+ 1 {R>n} Rdµ = nµ(r n)+ 1 k {R>n} Rdµ Y Y C 1 k β+1 n β +C 1 n β 2C 1 k β+1 n β. Therefore, 1 Y {h n=k}rdµ 2C 1 k β+2 n β, and diam(φ((t n W s ))dµ 2CC 1 n β γ k k β+2 = O(n β ), as required. Y Y k=1 Y 3.2 Verifying condition (a) in Theorem 3.1 For completeness, we show that Theorem 3.1 includes all examples that fit within the Chernov-Markarian-Zhang setup[12, 27] for Hölder mean zero observables φ : R. In particular, we recover limit theorems that have been obtained previously for such invertible examples [22, 31, 35, 36]. Since there are no new results here, we only sketch the construction from [12, 27]. It is part of the setup that is a metric space and T : has a first return map F = T R : Y Y with return tails µ(r > n) = O(n (β0+1) ), where we assume that β 0 > 1. Moreover, Y is modelled by a Young tower with exponential tails [43]. A standard argument (see for example [12, Theorem 4]) shows that T : is modelled by a Young tower f : with polynomial tails [44], with tail rate O(n (β+1) ) for all β < β 0. In particular, F = f R is the first return for f and there is a measure-preserving semiconjugacy π :, so we can work with f : instead of T : and observables ˆφ = φ π : R where φ : R is Hölder. The final part of the set up that we require is that is covered by stable leaves W s satisfying the contraction condition in Lemma 3.2. Hence f : satisfies condition (b) of Theorem 3.1 and it remains to verify condition (a). Let f : denote the quotient (one-sided) Young tower obtained by quotienting along stable leaves. Consider observables φ : R that are Lipschitz with respect to a symbolic metric on, with Lipschitz norm φ. By [44, Theorem 3], there is a constant C > 0 such that φ ψ f n d µ φd µ ψd µ C φ ψ n β, (3.1) 7

8 for all φ : R Lipschitz, ψ L ( ), n 1. (The dependence on φ and ψ is not stated explicitly in [44, Theorem 3] but follows by a standard argument using the uniform boundedness principle. Alternatively, see [25] for a direct argument.) Returning to the two-sided tower f : and the lifted observable ˆφ = φ π : R, it follows for instance from [25, Proposition 5.3] that there exists a choice (depending only onthehölder exponent of φ) of symbolic metric on andasequence of observables φ l L ( ), l 1, such that (i) φ l is F 0 -measurable and hence projects down to an observable φ l : R. (ii) sup l 1 L l φl <. (iii) lim l φ f l φ l 1 = 0. Here, F 0 is the σ-algebra generated by W s and L is the transfer operator corresponding to f :. Let ψ L (F 0 ) with projection ψ L ( ). Following [25, Proof of Corollary 5.4], φψ fn dµ = φ fl ψ f l+n dµ = I 1 +I 2 +I 3, where I 1 = I 3 = (φ f l φ l )ψ f l+n dµ, I 2 = φ l dµ φ l ψ f l+n dµ φ l dµ ψdµ. ψdµ, Now I 1 φ f l φ l 1 ψ. Also, I 2 = ( φ l φ f l )dµ ψdµ, so I 2 φ f l φ l 1 ψ 1. By (iii), lim l I j = 0 for j = 1,2. By (i), I 3 = φ l ψ fl+n d µ φ l d µ ψd µ = L l φl ψ fn d µ L l φl d µ ψd µ, so by (3.1) and (ii), I 3 C L l φl ψ n β ψ n β. Together, these estimates establish condition (a) in Theorem Application to Lorentz gases In this section, we use the results of [3] to show that the hypotheses of Theorem 3.1 (with β > 1 arbitrarily large) are satisfied for the time-one map corresponding to a finite horizon planar periodic Lorentz gas for all Hölder observables φ. Hence the results of Section 2 hold for all p, establishing the results listed in the introduction. 8

9 4.1 Setting and main result for Lorentz gases Let T 2 = R 2 /Z 2 denote the two-torus, and let B i T 2, i = 1,...,d, denote open convex sets such that their closures are pairwise disjoint and their boundaries are C 3 curves with strictly positive curvature. We refer to the sets B i as scatterers. The billiard flow Φ t is defined by the motion of a point particle in Q = T 2 \ d i=1 B i undergoing elastic collisions at the boundaries of the scatterers and moving at constant velocity with unit speed between collisions. Hence Φ t is defined on the three dimensional phase space = Q S 1, S 1 = [0,2π]/, where indicates that 0 and 2π are identified. Between collisions, Φ t (x 1,x 2,θ) = (x 1 + tcosθ,x 2 + tsinθ,θ), while at collisions the point (x,θ ) becomes (x,θ + ) where θ and θ + are the pre- and post-collisions angles, respectively. Defining 0 = /, where we identify (x,θ ) (x,θ + ) at collisions, we obtain a continuous flow Φ t : 0 0. Let M = d i=1 B i [ π/2,π/2]. The billiard map F : M M is the discretetime map which maps one collision to the next. Parametrizing each B i by an arclength coordinate r (oriented clockwise) and letting ϕ denote the angle that the post-collision velocity vector makes with the normal to the scatterer (directed inwards in Q), we obtain the standard coordinates (r,ϕ) on M. For x, define the collision time τ(x) to be the first time t > 0 that Φ t (x) M. Since the closures of the scatterers are disjoint, there exists τ min > 0 such that τ(x) τ min for all x M. In addition, we assume that the billiard has finite horizon so that there exists τ max < such that τ(x) τ max for all x. It is well known (see [11, Section 3.3]) that the flow preserves the contact form ω = cosθdx 1 +sinθdx 2, so that the contact volume is ω dω = dx 1 dθ dx 2. We denote by µ the normalized Lebesgue measure on, which by the preceding calculation is preserved by the flow. The main result of this section is the following. Theorem 4.1 Let T be the time-one map corresponding to a finite horizon Lorentz gas as described above, and let φ : R be a mean zero Hölder observable. Then conditions (a) and (b) of Theorem 3.1 hold for any β > 1. As a consequence, Proposition 2.2 holds for all p 1, and all the items of Corollary 2.3 as well as the homogenization results apply in this setting. 4.2 Proof of Theorem 4.1 The remainder of this section is devoted to the proof of Theorem 4.1, which consists of verifying the conditions of Theorem 3.1. First we recall some of the essential properties and main constructions used in [3]. 9

10 Hyperbolicity and singularities The singularities for both the collision map and the flow are created by tangential collisions with the scatterers. Let S 0 = {(r,ϕ) M : ϕ = ± π 2 }. Away from the set S 1 = S 0 F 1 S 0 (resp. S 1 = S 0 FS 0 ) the map F (resp. F 1 ) is uniformly hyperbolic: Letting Λ = 1+2τ min K min, (4.1) wherek min denotestheminimumcurvatureofthescatterers, thereexiststable C s and unstable C u cones in the tangent space of M such that stable and unstable vectors in these cones undergo uniform expansion and contraction at an exponential rate given by Λ. Flowing C s backward and C u forward between collisions allows us to define two families of stable C s and unstable C u cones for the flow that lie in the kernel of the contact form. (Hence they are flat two-dimensional cones in the tangent space of the flow; see [3, Sect. 2.1] for an explicit definition of these cones). Let P ± denote the projections from onto M under the forward and backward flow. Then C u is continuous on away from the surface S 1 = {x : P+ (x) S 1 }, and C s is continuous on away from the surface S + 1 = {x : P (x) S 1 }. To maintain control of distortion, we define the standard homogeneity strips H k = {(r,ϕ) M : π 2 1 k 2 ϕ π 2 1 (k+1) 2 }, k k 0, for some k 0 1 which is determined to ensure a one-step expansion condition. A similar set of of homogeneity strips H k, k k 0, is defined for ϕ near π 2. Following [3], we define a set of admissible stable curves A s for the flow. A C 2 curve W belongs to A s if the tangent vector at each point of W belongs to C s, and W has curvature bounded by B 0 and length W bounded by δ 0. Here, δ 0 > 0 is chosen to satisfy a complexity bound (see [3, Lemma 3.8]) and B 0 is chosen large enough that the family A s is invariant under Φ t, t 0. We call W A s homogeneous if P + (W) lies in a single homogeneity strip. We define W s to be the family of maximal C 2 connected homogeneous stable manifolds for the flow, such that each element of W s (up to subdivision due to the length δ 0 ) belongs to A s. When we define a homogeneous stable manifold W W s, we take into account cuts introduced at the boundary of the extended singularity set, which includes the boundaries of the homogeneity strips. Thus P + (Φ t W) lies in a single homogeneity strip for all t 0. 2 Norms and Banach spaces With the class of admissible stable curves defined, we can now describe the Banach spaces used to prove decay of correlations in [3]. Let α (0, 1 3 ]. For W As, let C α (W) denote the closure of C 1 functions in the Holder norm defined by ψ C α (W) = sup ψ(x) + sup ψ(x) ψ(x ) d W (x,x ) α, x W x,x W x x 2 Due to our definition of C s, if W A s, then P + (W) is a stable curve for the map; and if W W s, then P + (W) is a local homogeneous stable manifold for the map. 10

11 where d W is arclength distance along W. Define the weak norm of φ C 0 () by φ w = sup sup φψdm W, W A s ψ C α (W) ψ C α (W) 1 W where m W denotes arclength measure on W. The weak space B w is defined as the completion of the set {φ C 0 ( 0 ) : φ w < }. The strong norm φ B is defined as in [3, Section 2.3]. The space B is similarly definedasthecompletionofaclassofsmoothfunctions on 0 inthe B norm. Since we do not need the precise definition of B here, we omit its definition; however, the following lemma summarizes some of the important properties of these spaces. Lemma 4.2 ([3, Lemmas 3.9 and 3.10]) We have the inclusions C 1 () B B w (C α ()), where the first two inclusions are injective. Moreover, w B C C 1 () and the unit ball of B is compactly embedded in B w. When we refer to functions φ C 0 () as elements of B or B w, we identify φ with the measure φdµ. With this identification, the two definitions of L t φ given in the next section are reconciled. The following lemma is central to our verification of condition (a) in Theorem 3.1, and is a strengthening of [3, Lemma 2.11]. Let C α (W s ) denote those functions which are in C α (W) for all W W s. Define ψ C α (W s ) = sup W W s ψ C α (W). Lemma 4.3 There exists C > 0 such that for φ B w and ψ C α (W s ), φ(ψ) C φ w ψ C α (W s ). Again, due to our identification, when φ C 0 (), we intend φ(ψ) = φψdµ. Lemma 4.3 is proved at the end of this section. Transfer operator We define the transfer operator L t, for t 0, by L t φ = φ Φ t, for φ C 0 ( 0 ). This can be extended to any element of B w, and more generally a distribution of order α by L t φ(ψ) = φ(ψ Φ t ), for all ψ C α (A s ), φ (C α (A s )). By [3, Lemma 4.9], the map (t,φ) L t φ from [0, ) B to B is jointly continuous, so {L t } t 0 is a semi-group of bounded operators on B. L Define the generator of the semi-group by Zφ = lim tφ φ t 0 for φ C 1 (). t While Z is not a bounded operator on B, the strong continuity of L t implies that Z is closed with domain dense in B. Indeed, by [3, Lemma 7.5] the domain of Z contains C 2 () C 0 ( 0 ), and there is a constant C > 0 such that Zφ B C φ C 2 () for all φ C 2 () C 0 ( 0 ). (4.2) 11

12 Condition (b) of Theorem 3.1 Recall that T and F denote the time-one map and collision map. By the finite horizon condition, any W W s must undergo k n/τ max collisions after n iterates by T. By [3, Lemmas 3.3 and 3.4], diam(t n W) = Φ n W C F k 1 (P + (W)) CΛ (k 1) P + (W) C Λ n/τmax W, where Λ > 1isthehyperbolicity constant defined in(4.1). Wehave used herethatthe lengths of P + (W) and W are bounded multiples of one another (indeed the Jacobian of this map is C 1 2, see [3, Lemma 3.4]). Let φ : R be C η. Then diam(φ(t n W)) φ η diam(t n W) η CΛ nη/τmax. Hence condition (b) holds for any β > 1. Condition (a) of Theorem 3.1 By [3, Theorem 1.4], Z has a spectral gap on B and, using results of [9], L t admits the following decomposition: There exists ν > 0, a finite rank projector Π : B B and a family of bounded operators P t on B satisfying ΠP t = P t Π = 0, and a matrix Ẑ : Π(B) Π(B) with eigenvalues 0,z 1,...,z N C satisfying Rez j < ν for j = 1,...,N, such that L t = P t +e tẑ Π for all t 0. (4.3) Moreover, there exists C ν > 0 such that for all φ in Dom(Z) B, P t φ w C ν e νt Zφ B for all t 0. (4.4) Now suppose φ C 2 () C 0 ( 0 ) is of mean zero and ψ C α (W s ). By (4.3), φψ Φ tdµ = L tφψdµ = P tφψdµ+ etẑπφψdµ. Hence by Lemma 4.3, φψ Φ t dµ C { P t φ w + e tẑπφ } w ψ C α (W s ). (4.5) Letting Π 0 denote the projector corresponding to the simple eigenvalue 0, we see that Π 0 φ = φdµ = 0 since µ is the conformal probability measure with respect to L t. Hence by Lemma 4.2, By (4.2) and (4.4), e tẑπφ w = e tẑ(π Π 0 )φ w Ce νt φ w C e νt φ C 1 (). Substituting these estimates in (4.5), P t φ w C ν e νt Zφ B C e νt φ C 2 (). φψ Tn dµ = φψ Φ ndµ Ce νn φ C 2 () ψ C α (W s ) for all n 0. 12

13 The result extends to φ C η () as in [3] by a standard mollification argument. (Exponential contraction persists with a rate dependent on η.) In particular, for any β > 1, there is a constant C > 0 such that φψ Tn dµ Cn β φ C η () ψ C α (W s ) for all n 0, for all φ C η (), ψ C α (W s ). Let C 0 (F 0 ) denote the space of continuous functions that are constant along elements of W s. In particular, C 0 (F 0 ) C α (W s ) and C 0 (F 0 ) = C α (W s ). Hence φψ Tn dµ Cn β φ C η () ψ C 0 (F 0 ) for all n 0, for all φ C η (), ψ C 0 (F 0 ). Finally, let φ C η (), ψ L (F 0 ). For any ǫ > 0, there exists ψ C 0 (F 0 ) such that ψ C 0 (F 0 ) ψ and ψ ψ dµ < ǫ. Hence φψ Tn dµ φψ T n dµ + φ(ψ ψ ) T n dµ Cn β φ C η () ψ +ǫ φ. Since ǫ is arbitrary, this completes the verification of condition (a). As promised, we end this section by proving Lemma 4.3. Proof of Lemma 4.3 By density of C 0 ( 0 ) in B w, it suffices to prove the lemma for φ C 0 ( 0 ) and ψ C α (W s ). The normalized Lebesgue measure µ on projects to the measure µ = (2 Q ) 1 cosϕdrdϕ on M; this is the unique smooth invariant probability measure for the billiard map F. Let W s denote the set of maximal connected homogeneous stable manifolds for F. Note that P + (W s ) = W s. Indexing elements of W s, we write W s = {V ξ } ξ Ξ, which defines a (mod 0) partition of M. We disintegrate µ into conditional measures µ ξ on V ξ, ξ Ξ, and a factor measure λ on Ξ. Indeed, the conditional measures are smooth on each V ξ, and we can write d µ = ρ ξ d m ξ dλ(ξ), where m ξ is arclength measure along V ξ (in M), and log ρ ξ C 1 3(Vξ ) C, ρ ξ C 0 (V ξ ) C V ξ 1, (4.6) for some C > 0 depending only on the table Q. The exponent 1 comes from the 3 definition of the homogeneity strips. This is the standard 3 decomposition of µ into a proper standard family (see [11, Example 7.21]). We further subdivide Ξ = d i=1 Ξ i, where Ξ i is the index set corresponding to each component M i = B i [ π, π ] of M Standard families in [11] are standard pairs defined on local unstable manifolds, while here we use local stable manifolds. The decompositions of µ have equivalent properties due to the symmetry of the map F under time reversal. 13

14 Write = d i=1 i where i = {x : P + (x) M i }. On each i, we represent Lebesgue measure as dµ = ccosϕdrdϕds, where c is a normalizing constant, (r,ϕ) range over M i, and s ranges from 0 to the maximum free flight time under the backwards flow, which we denote by t i τ max. Next, for each ξ Ξ i, the flow surface V ξ = {x i : P + (x) V ξ } is smoothly foliated by elements of W s, which are simply flow translates of one another. For each s and V ξ, let W ξ,s = Φ t(s) V ξ, where t(s,z) is defined for z V ξ so that W ξ,s lies in the kernel of ω, i.e. it is an element of W s. Note that for s < δ 0, some points in V ξ may not have lifted off of M. For such small times, W ξ,s denotes only those points that have lifted off of M. Similarly, for s > τ min, some part of Φ t(s) V ξ may have collided with a scatterer. For such times, W ξ,s only denotes those points which have not yet undergone a collision. Thus s [0,t i ] W ξ,s = V ξ. Using this decomposition, we may represent Lebesgue measure on each i by dµ(x) = ρ ξ (x)dm Wξ,s (x)dλ(ξ)ds, where ρ ξ is smooth along each W ξ,s, satisfying analogous bounds to (4.6), since the contact form is C on i and the projection P + is sufficiently smooth (see [3, Lemma 3.4]), so that the arclength of W ξ,s varies smoothly with that of V ξ. Using the fact that each W ξ,s W s can be subdivided into at most Cδ0 1 elements of A s, we are ready to estimate φψdµ d φψdµ i i i=1 ti ti 0 Ξ i φψρ ξ dm Wξ,s dλ(ξ)ds W ξ,s Cδ0 1 φ w ψ C α (W ξ,s ) ρ ξ C α (W ξ,s ) dλ(ξ)ds i 0 Ξ i Cδ0 1 τ max φ w ψ C α (W s ) V ξ 1 dλ(ξ). This last integral is finite by [11, Exercise 7.15] since our decomposition of µ constitutes a proper standard family, completing the proof. Ξ Acknowledgements This research resulted from a Research in Pairs on Techniques des martingales et espaces de Banach anisotropes (Martingale techniques and anisotropic Banach spaces) at CIRM, Luminy, during August MD was supported in part by NSF Grant DMS IM was supported in part by European Advanced Grant StochExtHomog (ERC AdG ). MN was supported in part by NSF Grant DMS

15 References [1] V. Araújo, I. Melbourne and P. Varandas. Rapid mixing for the Lorenz attractor and statistical limit laws for their time-1 maps. Comm. Math. Phys. 340 (2015) [2] V. Baladi. The quest for the ultimate anisotropic Banach space. J. Stat. Phys. 166 (2017) [3] V. Baladi, M. F. Demers and C. Liverani. Exponential decay of correlations for finite horizon Sinai billiard flows. Invent. Math. 211 (2018) [4] P. Bálint and S. Gouëzel. Limit theorems in the stadium billiard. Comm. Math. Phys. 263 (2006) [5] P. Bálint and I. Melbourne. Statistical properties for flows with unbounded roof function, including the Lorenz attractor. J. Stat. Phys. 172 (2018) [6] M. Blank, G. Keller and C. Liverani. Ruelle-Perron-Frobenius spectrum for Anosov maps. Nonlinearity 15 (2002) [7] R. Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math. 470, Springer, Berlin, [8] L. A. Bunimovich, Y. G. Sinaĭ and N. I. Chernov. Statistical properties of twodimensional hyperbolic billiards. Uspekhi Mat. Nauk 46 (1991) [9] O. Butterley. A note on operator semigroups associated to chaotic flows. Ergodic Theory Dynam. Systems 36 (2016) [10] N. Chernov. A stretched exponential bound on time correlations for billiard flows. J. Stat. Phys. 127 (2007) [11] N. Chernov and R. Markarian. Chaotic billiards. Mathematical Surveys and Monographs 127, American Mathematical Society, Providence, RI, [12] N. I. Chernov and H.-K. Zhang. Billiards with polynomial mixing rates. Nonlinearity 18 (2005) [13] I. Chevyrev, P. K. Friz, A. Korepanov, I. Melbourne and H. Zhang. Multiscale systems, homogenization, and rough paths. Springer Volume Varadhan 75. To appear. [14] I. Chevyrev, P. K. Friz, A. Korepanov, I. Melbourne and H. Zhang. Determinstic homogenization for discrete time fast-slow systems. In preparation. 15

16 [15] J. Dedecker and E. Rio. On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) [16] M. F. Demers. A gentle introduction to anisotropic Banach spaces. Chaos Solitons Fractals 116 (2018) [17] D. Dolgopyat. Prevalence of rapid mixing in hyperbolic flows. Ergodic Theory Dynam. Systems 18 (1998) [18] M. I. Gordin. The central limit theorem for stationary processes. Soviet Math. Dokl. 10 (1969) [19] S. Gouëzel. Almost sure invariance principle for dynamical systems by spectral methods. Ann. Probab. 38 (2010) [20] H. Hennion and L. Hervé. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Math. 1766, Springer, Berlin, [21] C. C. Heyde. On the central limit theorem and iterated logarithm law for stationary processes. Bull. Austral. Math. Soc. 12 (1975) 1 8. [22] D. Kelly and I. Melbourne. Smooth approximation of stochastic differential equations. Ann. Probab. 44 (2016) [23] D. Kelly and I. Melbourne. Homogenization for deterministic fast-slow systems with multidimensional multiplicative noise. J. Funct. Anal. 272 (2017) [24] C. Kipnis and S. R. S. Varadhan. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986) [25] A. Korepanov, Z. Kosloff and I. Melbourne. Explicit coupling argument for nonuniformly hyperbolic transformations. Proc. Roy. Soc. Edinburgh A. To appear. [26] C. Liverani. Central limit theorem for deterministic systems. International Conference on Dynamical Systems (F. Ledrappier, J. Lewowicz and S. Newhouse, eds.), Pitman Research Notes in Math. 362, Longman Group Ltd, Harlow, 1996, pp [27] R. Markarian. Billiards with polynomial decay of correlations. Ergodic Theory Dynam. Systems 24 (2004) [28] M. Maxwell and M. Woodroofe. Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000)

17 [29] I. Melbourne. Rapid decay of correlations for nonuniformly hyperbolic flows. Trans. Amer. Math. Soc. 359 (2007) [30] I. Melbourne. Superpolynomial and polynomial mixing for semiflows and flows. Nonlinearity 31 (2018) R268 R316. [31] I. Melbourne and M. Nicol. Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260 (2005) [32] I. Melbourne and M. Nicol. Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360 (2008) [33] I. Melbourne and A. Török. Central limit theorems and invariance principles for time-one maps of hyperbolic flows. Comm. Math. Phys. 229 (2002) [34] I. Melbourne and A. Török. Statistical limit theorems for suspension flows. Israel J. Math. 144 (2004) [35] I. Melbourne and A. Török. Convergence of moments for Axiom A and nonuniformly hyperbolic flows. Ergodic Theory Dynam. Systems 32 (2012) [36] I. Melbourne and P. Varandas. A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion. Stoch. Dyn. 16 (2016) , 13 pages. [37] M. Peligrad and S. Utev. A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 (2005) [38] D. Ruelle. Thermodynamic Formalism. Encyclopedia of Math. and its Applications 5, Addison Wesley, Massachusetts, [39] Y. G. Sinaĭ. Gibbs measures in ergodic theory. Russ. Math. Surv. 27 (1972) [40] M. Tyran-Kamińska. An invariance principle for maps with polynomial decay of correlations. Comm. Math. Phys. 260 (2005) [41] M. Viana. Stochastic dynamics of deterministic systems. Col. Bras. de Matemática, [42] D. Volný. Approximating martingales and the central limit theorem for strictly stationary processes. Stochastic Process. Appl. 44 (1993) [43] L.-S. Young. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147 (1998) [44] L.-S. Young. Recurrence times and rates of mixing. Israel J. Math. 110 (1999)