Local limit theorem for randomly deforming billiards

TL;DR

This paper establishes limit theorems for randomly deformed dispersing billiards, including CLT and local limit theorems, using anisotropic Banach spaces, in both finite and infinite measure settings.

Contribution

It extends limit theorems to randomly perturbed billiard systems, introducing new techniques for the random setting with anisotropic Banach spaces.

Findings

Proved a Central Limit Theorem for cell index in finite horizon billiards.

Established a mixing local limit theorem for observables.

Derived limit theorems for visits to obstacles and self-intersections in infinite measure systems.

Abstract

We study limit theorems in the context of random perturbations of dispersing billiards in finite and infinite measure. In the context of a planar periodic Lorentz gas with finite horizon, we consider random perturbations in the form of movements and deformations of scatterers. We prove a Central Limit Theorem for the cell index of planar motion, as well as a mixing Local Limit Theorem for the cell index with piecewise H\"older continuous observables. In the context of the infinite measure random system, we prove limit theorems regarding visits to new obstacles and self-intersections, as well as decorrelation estimates. The main tool we use is the adaptation of anisotropic Banach spaces to the random setting.

Equations

$$M_0:=\{(q,\vec{v})\in {\mathbb{R}}^2\times {\mathbb{R}}^2:q\in \partial Q,\Vert \vec{v}\Vert =1,⟨\vec{n}(q),\vec{v}⟩>0\},$$

$$T_0(x+\ell )={\bar{T}}_0(x)+\ell +{\Phi }_0(x).$$

$$\bar{M}:={\bar{M}}_0\times E^{\mathbb{N}},\bar{\mu }:={\bar{\mu }}_0\otimes {\eta }^{\otimes \mathbb{N}},\bar{T}(x,({\omega }_k{)}_{k\ge 0}):=({\bar{T}}_{{\omega }_0}x,({\omega }_{k+1}{)}_{k\ge 0}).$$

$$M:=M_0\times E^{\mathbb{N}},\mu :={\mu }_0\otimes {\eta }^{\otimes \mathbb{N}},T\left((x,\ell ,({\omega }_k{)}_k{)}_{k\ge 0}\right):=(T_{{\omega }_0}(x,\ell ),({\omega }_{k+1}{)}_{k\ge 0}).$$

$$\Phi (x,({\omega }_k{)}_{k\ge 0})={\Phi }_{{\omega }_0}(x)).$$

$$T^n\left((x,\ell ,({\omega }_k{)}_k{)}_{k\ge 0}\right)$$

$$S_n(x,({\omega }_k{)}_k):=\sum _{k=0}^{n-1}\Phi \circ {\bar{T}}^k(x,({\omega }_k{)}_k)=\sum _{k=0}^{n-1}{\Phi }_{{\omega }_k}\circ {\bar{T}}_{{\omega }_{k-1}}\circ \cdots \circ {\bar{T}}_{{\omega }_0}(x),$$

$$\mathbb{H}_{k}=\big{\{}(r,\varphi)\in\bar{M}_{0}:\tfrac{\pi}{2}-\tfrac{1}{k^{2}}<\varphi<\tfrac{\pi}{2}-\tfrac{1}{(k+1)^{2}}\big{\}},$$

$$\Vert D{F}^n(x)v\Vert \ge C_e^{-1}{\Lambda }^n\Vert v\Vert ,\forall v\in C^u(x),\text{mbox}and\Vert D{F}^{-n}(x)v\Vert \ge C_e^{-1}{\Lambda }^n\Vert v\Vert ,\forall v\in C^s(x),$$

$$C_a\Vert v\Vert \le \Vert D{F}^{-1}(x)v\Vert \cos \phi (F^{-1}x)\le C_a^{-1}\Vert v\Vert ,\forall v\in C^s(x),$$

$$C_a\le \Vert D^2{F}^{-1}(x)\Vert {\cos }^3\phi (F^{-1}x)\le C_a^{-1}.$$

$$\underset{\delta \to 0}{\mathrm{limsup}}\underset{F\in \bar{\mathcal{F}}}{\sup }\underset{\begin{array}{c}W\in {\hat{\mathcal{W}}}^s\\ |W|<\delta \end{array}}{\sup }\sum _i|J_{V_i}F{|}_{\ast }<1,$$

$$\underset{F\in \bar{\mathcal{F}}}{\sup }\underset{\begin{array}{c}W\in {\hat{\mathcal{W}}}^s\\ |W|<\delta \end{array}}{\sup }\sum _i|J_{V_i}F{|}_{{\mathcal{C}}^0(V_i)}^{\zeta }\le C_1.$$

$$\left|\frac{J_W{F}^n(x)}{J_W{F}^n(y)}-1\right|\le C_d{d}_W(x,y{)}^{1/3},$$

$${\bar{\mu }}_0(A)={\int }_{\alpha \in \mathcal{A}}{\int }_{x\in W_{\alpha }}{1}_A(x)d{\mu }_{\alpha }d\lambda (\alpha ),$$

$${\bar{\mathcal{F}}}_{{\vartheta }_0}(F_0)=\{F\in \bar{\mathcal{F}}:d_{\bar{\mathcal{F}}}(F,F_0)<{\vartheta }_0\},$$

$${\tau }_{\min }(\bar{Q})=\inf \{\tau (x):\tau (x)\text{mbox}isdefinedfortheconfiguration\bar{Q}\}.$$

$${\tau }_{\ast }\le {\tau }_{\min }(\bar{Q})\le {\tau }_{\max }(\bar{Q})\le {\tau }_{\ast }^{-1},{\mathcal{K}}_{\ast }\le {\mathcal{K}}_{\min }(\bar{Q})\le K_{\max }(\bar{Q})\le {\mathcal{K}}_{\ast }^{-1},E_{\max }(\bar{Q})\le E_{\ast }.$$

$$\Delta (\bar{Q},{\bar{Q}}_0)=\sum _{i=1}^l|u_i-{\tilde{u}}_i{|}_{C^2(J_i,{\mathbb{R}}^2)}.$$

$${\Sigma }^2:={\left({\mathbb{E}}_{\bar{\mu }}\left[{\Phi }^{(i)}.{\Phi }^{(j)}\right]+\sum _{k\ge 1}{\mathbb{E}}_{\bar{\mu }}\left[{\Phi }^{(i)}.{\Phi }^{(j)}\circ {\bar{T}}^k+{\Phi }^{(j)}.{\Phi }^{(i)}\circ {\bar{T}}^k\right]\right)}_{i,j=1,2},$$

$$H_g(\cdot ):={\mathbb{E}}_{\bar{\mu }}[g.\cdot ].$$

$${\mathbb{E}}_{\bar{\mu }}\left[f.1_{\{S_n=\ell \}}.g\circ {\bar{T}}^n\right]=\frac{\exp \left(-\frac{{\Sigma }^{-2}\ell \cdot \ell }{2n}\right)}{2\pi n\sqrt{\det {\Sigma }^2}}{\mathbb{E}}_{\bar{\mu }}[f]{\mathbb{E}}_{\bar{\mu }}[g]+O\left(n^{-\frac{3}{2}}\Vert f{\Vert }_{\tilde{\mathcal{B}}}\Vert H_g{\Vert }_{{\tilde{\mathcal{B}}}^{\prime }}\right).$$

$$({\ell }_0+S_k(\bar{x},\underline{\omega }),{\mathcal{I}}_k(\bar{x},\underline{\omega }))=(\ell ,i),$$

$$B_n:=\{\forall k=1,...,n:({\mathcal{I}}_k,S_k)\ne ({\mathcal{I}}_0,(0,0))\}\subset \bar{M}.$$

$$B_n^{\prime }:=\{\forall k=0,...,n-1:({\mathcal{I}}_k,S_k)\ne ({\mathcal{I}}_n,S_n)\}\subset \bar{M}.$$

$$\bar{\mu }(B_n)=\bar{\mu }(B_n^{\prime })=\frac{2I\pi \sqrt{\det {\Sigma }^2}}{\log n}+O\left((\log n{)}^{-\frac{4}{3}}\right),\text{mbox}asn\to +\infty .$$

$${\mathcal{V}}_n:=\sum _{i,j=1}^n{1}_{\{S_j=S_i,{\mathcal{I}}_j={\mathcal{I}}_i\}}.$$

$$\underset{n\to \infty }{\lim }\frac{{\mathcal{V}}_n}{n\log n}=\frac{1}{\pi \sqrt{\det {\Sigma }^2}}\frac{{\sum }_{a=1}^I|\partial O_a{|}^2}{{\left({\sum }_{b=1}^I|\partial O_a|\right)}^2}.$$

$${\tilde{\mathcal{Z}}}_n(t)={\mathcal{Z}}_{\lfloor nt\rfloor }+(nt-\lfloor nt\rfloor )({\mathcal{Z}}_{\lfloor nt\rfloor +1}-{\mathcal{Z}}_{\lfloor nt\rfloor }).$$

$$\mathbb{E}[B_1^2]=\frac{{\sigma }_{\xi }^2}{\pi \sqrt{\det {\Sigma }^2}}\frac{{\sum }_{a=1}^I|\partial O_a{|}^2}{{\left({\sum }_{b=1}^I|\partial O_b|\right)}^2}.$$

Full text

AMS classification numbers: 37D50, 37A25

1. 1 Notation and assumptions

2. 1.1 Deterministic billiard systems.

3. 1.2 Random perturbations of the initial billiard system

4. 1.3 A uniform family of maps

5. 1.4 Distance in the class $\bar{\mathcal{F}}$

6. 1.5 Applications – Deterministic perturbations

7. 2 Main results

8. 2.1 Local Limit Theorem

9. 2.2 Return time, visit to new obstacles and self intersections

10. 2.3 Billiard in random scenery

11. 2.4 Limit theorems in infinite measure

12. 3 Transfer Operators

13. 3.1 Banach spaces $\mathcal{B}$ and $\mathcal{B}_{w}$

14. 3.2 Banach spaces $\widetilde{\mathcal{B}}$ and $\widetilde{\mathcal{B}}_{w}$

15. 3.3 View $P_{u}$ as a perturbation of a quasicompact operator

16. 3.4 Doeblin-Fortet-Lasota-Yorke type inequality for $P_{u}$

17. 3.5 Quasicompactness of $P$ and of $P_{u}$

18. 4 Limit theorems under general assumptions

19. 4.1 Proof of the Local Limit Theorem – Theorem 2.2

20. 4.2 Return time to the original obstacle: Proof of Theorem 2.4

21. 4.3 Number of self-intersections: Proof of Theorem 2.5

22. 4.4 Random scenery: Proof of Theorem 2.6

23. A Proof of Lemma 4.5

24. B Proof of Lemma 4.10.

25. C Spectrum of $\mathcal{P}_{u}$

Index

Introduction

The Lorentz process is a physically interesting mechanical system modeled by mathematical billiards with chaotic behavior. Introduced by Sinai in [36], it has been studied extensively by many authors, see [8, 9, 12] and other related references. It is the deterministic motion of a point particle starting from a random phase point and undergoing specular reflections on the boundaries of strictly convex scatterers. Throughout this paper we will consider a $\mathbb{Z}^{2}$-periodic random configuration of scatterers, with finite horizon. The diffusion limit of the planar Lorentz process can be described by a Wiener process [9], and is thus closely related to the Central Limit Theorem (CLT) and Local Limit Theorem (LLT).

The history of the LLT goes back to the historic De Moivre Laplace theorem for independent identically distributed (iid) Bernoulli random variables. It has then been generalized in many contexts. The CLT appears as a consequence of the LLT. In the context of dynamical systems, the first LLT was established by Guivarc’h and Hardy for subshifts of finite type [19]. The method they used, also used by Nagaev in [26], was based on perturbations of an associated transfer operator and has since been used for many expanding and hyperbolic dynamical systems. This method is now often called the Nagaev-Guivarc’h method. For the Sinai billiard (with fixed scatterers), the LLT was proved by Szász and Varjú in [34] using Young towers and the Nagaev-Guivarc’h method. Also using Young towers, Pène established and used in [28, 29, 30] some precise versions of the LLT to prove further limit theorems for the Sinai billiard (see also her works with Saussol [32] and with Thomine [33] for other applications of the LLT).

The goal of this article is to prove the LLT, as well as several of its applications, in the context of randomly deforming scatterers in a dispersing Lorentz gas with finite horizon. In this context the use of Young towers does not appear very adequate, since a different tower is associated to every different $\mathbb{Z}^{2}$-periodic configuration of scatterers. It is therefore much more natural to work directly with the billiard transformations since these transformations act on the same space $\bar{M}_{0}$ and preserve the same measure. To this end, we will work with the spaces considered in [14, 15, 16], which are spaces $\mathcal{B},\mathcal{B}_{w}$ made of distributions instead of being spaces of functions contained in $L^{p}$ for some $p>1$ as in [19, 34]. This will complicate our study. One advantage of the approach used by Demers and Zhang is that the Banach spaces they construct in [15] are the same for natural families of billiard transformations.

Since we are interested in random iterations of billiard transformations, we will consider the full random billiard system corresponding to the skew product transformation which takes in account both the billiard configuration (position and speed) and the randomness of the configuration of scatterers. Let us mention that Aimino, Nicol and Vaienti established in [2] an LLT (together with other limit theorems) for random iterations of expanding dynamical systems. Their approach was based on the Nagaev-Guivarc’h method applied to the restriction of the transfer operator of the full random system to functions depending only on the phase space coordinate (and not on the random coordinate). The advantage of their method is that they worked on a simple Banach space (in which the randomness of the transformations is not taken into account). But the disadvantage is that they had to reprove for this restricted operator theorems that were already known for transfer operators. In the present paper, we apply directly the Nagaev-Guivarc’h method to the transfer operator of the full random system acting on suitable Banach spaces $\widetilde{\mathcal{B}},\widetilde{\mathcal{B}}_{w}$ which are easily defined using ${\mathcal{B}},{\mathcal{B}}_{w}$. As a consequence, our results apply to observables that may depend on both the position and speed of the billiard, as well as the random coordinate.

This article is organized as follows. In Section 1, we specify our assumptions and notation. In Section 2, we state our main limit theorems: LLT, asymptotic estimate of the return time to the initial scatterer, asymptotic behavior of the number of self-intersections, annealed and quenched limit theorem for a random billiard in random scenery, limit theorems for some ergodic sums of the planar random billiard (in infinite measure), mixing and decorrelation for the planar random billiard (in infinite measure). In Section 3, we study the spectral properties of the transfer operator of the full random system. Section 4 is devoted to the proof of our main results under general spectral assumptions.

1. Notation and assumptions

1.1. Deterministic billiard systems.

Let $I\geq 1$ and let $O_{1},...,O_{I}$ be $I$ convex open subsets of $\mathbb{R}^{2}$, having $\mathcal{C}^{3}$ boundary with strictly positive curvature, and such that the closure of the sets $(U_{i,\ell}:=\ell+O_{i})_{i=1,...,I;\ \ell\in\mathbb{Z}^{2}}$ are pairwise disjoint. We consider the $\mathbb{Z}^{2}$-periodic billiard table $Q:=\mathbb{R}^{2}\setminus\bigcup_{\ell\in\mathbb{Z}^{2}}\bigcup_{i=1}^{I}(U_{i,\ell})$. We assume moreover that every line meets $\partial Q$ (i.e. that the horizon is finite). We are interested in the behavior of a point particle moving in $Q$ at unit speed, going straight inside $Q$, and reflecting elastically off $\partial Q$ (the reflected direction being the symmetric of the incident one with respect to the normal line to $Q$ at the reflection point).

We consider the planar billiard system $(M_{0},\mu_{0},T_{0})$ modeling the behavior of the point particle at reflection times. A configuration is given by a pair $(q,\vec{v})\in M_{0}$ representing position and velocity, and corresponding to a reflected vector off $\partial Q$, with

[TABLE]

where $\vec{n}(q)$ is the unit vector, normal to $\partial Q$ at $q$ and directed into $Q$. The transformation $T_{0}$ maps a reflected vector to the reflected vector at the next reflection time. This transformation preserves the measure $\mu_{0}$ given by $d\mu_{0}=\tilde{c}\cos\varphi\,dr\,d\varphi$ (where $r$ is the parametrized arclength coordinate on $\partial Q$ corresponding to $q$ and $\varphi$ is the algebraic measure of the angle $\widehat{(\vec{n}(q),\vec{v})}$ and where $\tilde{c}=1/(2\sum_{i=1}^{I}|\partial O_{i}|)$, the reason for the choice of $\tilde{c}$ will be clear in a few lines).

For every $i\in\{1,...,I\}$ and every $\ell\in\mathbb{Z}^{2}$, we define $M_{i,\ell}:=\{(q,\vec{v})\in M_{0}\ :\ q\in\partial U_{i,\ell}\}$ for the set of reflected vectors based on the obstacle $U_{i,\ell}$. For every $\ell\in\mathbb{Z}^{2}$, we will call an $\boldsymbol{\ell}$-cell the set $\bar{M}_{\ell}:=\bigcup_{i=1}^{I}M_{i,\ell}$ .

Identifying the boundary of each scatterer $\partial O_{i}$ with a circle $\mathbb{S}_{i}$ of length $|\partial O_{i}|$, we define $\bar{M}_{0}:=\cup_{i=1}^{I}\mathbb{S}_{i}\times[-\pi/2,\pi/2]$. Thus $\bar{M}_{0}$ is a parametrization of $\bar{M}_{(0,0)}$ in the coordinates $(r,\varphi)$ introduced above. Note that may configurations of obstacles $O_{i}$ result in the same parametrized space $\bar{M}_{0}$. We shall exploit this fact when defining the classes of random perturbations that we shall consider.

Because of its $\mathbb{Z}^{2}$ periodicity, the planar billiard system can be identified with a $\mathbb{Z}^{2}$-cylindrical extension over a dynamical system $(\bar{M}_{0},\bar{\mu}_{0},\bar{T}_{0})$. Indeed, using the notation $x+\ell=(q+\ell,\vec{v})$ for every $x=(q,\vec{v})\in M_{0}$ and every $\ell\in\mathbb{Z}^{2}$, we observe that there exists a transformation $\bar{T}_{0}:\bar{M}_{0}\rightarrow\bar{M}_{0}$ (corresponding to the billiard map modulo $\mathbb{Z}^{2}$) and a function $\Phi_{0}:\bar{M}_{0}\rightarrow\mathbb{Z}^{2}$ called a cell-change) such that

[TABLE]

This transformation $\bar{T}_{0}$ preserves the probability measure $\bar{\mu}_{0}(\cdot):=\mu_{0}(\cdot\cap\bar{M}_{0})$ (the fact that $\bar{\mu}_{0}$ is a probability comes from our choice for the normalizing constant $\tilde{c}$).

In the following, identifying a couple $(x,\ell)\in\bar{M}_{0}\times\mathbb{Z}^{2}$ with $x+\ell\in M_{0}$, we identify $(M_{0},\mu_{0},T_{0})$ with the $\mathbb{Z}^{2}$-cylindrical extension of $(\bar{M}_{0},\bar{\mu}_{0},\bar{T}_{0})$ by $\Phi_{0}$, i.e. we identify $M_{0}$ with $\bar{M}_{0}\times\mathbb{Z}^{2}$, $\mu_{0}$ with $\bar{\mu}_{0}\otimes\mathfrak{m}$, where $\mathfrak{m}:=\sum_{k\in\mathbb{Z}^{2}}\delta_{k}$ is the counting measure on $\mathbb{Z}^{2}$.

1.2. Random perturbations of the initial billiard system

Before describing the random perturbations we shall consider, we describe a class of maps $\bar{\mathcal{F}}$ on $\bar{M}_{0}$ with uniform properties from which we will draw random sequences of maps. The class $\bar{\mathcal{F}}$ we will use is a slightly simplified version of the one introduced in [15]. The perturbations in [15] allowed billiards with infinite horizon, while for the present work we will assume a finite horizon condition and that the invariant measure is absolutely continuous with respect to the Lebesgue measure, which simplifies several of our assumptions.

We consider a probability space $(E,\mathfrak{T},\eta)$ containing 0 and a family $(T_{\omega})_{\omega\in E}$ of $\mathbb{Z}^{2}$-periodic planar Sinai billiard systems (with finite horizon) defined on $M$, the quotient billiard maps (modulo $\mathbb{Z}^{2}$ for the position) $\bar{T}_{\omega}$ of which are in $\bar{\mathcal{F}}$, and below we will choose $\bar{\mathcal{F}}_{\vartheta_{0}}(\bar{T}_{0})$ as a small $\vartheta_{0}$-neighbourhood of our original map $\bar{T}_{0}$, see (5).

For any $\underline{\omega}\in E^{\mathbb{N}}$, we will consider random iterations $T^{k}_{\underline{\omega}}:=T_{\omega_{k-1}}\circ...\circ T_{\omega_{0}}$, where $\omega_{k}$ are chosen independently with respect to $\eta$. Here $\underline{\omega}=(\omega_{k})_{k\geq 0}$, and $T_{\omega_{k}}\in{\mathcal{F}}$, for any $k\geq 0$, where ${\mathcal{F}}$ is a collection of $\mathbb{Z}^{2}$ extensions of $\bar{\mathcal{F}}$. This will be formalized below. In our model, the modification of environment is applied during the reflection time of the particle; the particle stays on the obstacle and moves with it during the modification of the billiard system. At its $k$-th reflection time, the particle arrives on an obstacle in an environment parametrized by $\omega_{k-1}$, but when it leaves it sees the environment $\omega_{k}$.

We identify $(M_{0},\mu_{0},T_{\omega})$ with the $\mathbb{Z}^{2}$-extension of $(\bar{M}_{0},\bar{\mu}_{0},\bar{T}_{\omega})$ by some function $\Phi_{\omega}:\bar{M}_{0}\rightarrow\mathbb{Z}^{2}$ which is constant on each connected component of continuity of $\bar{T}_{\omega}$. We define the random billiard system $(\bar{M},\bar{\mu},\bar{T})$, corresponding to random iterations of maps in $\bar{\mathcal{F}}$, by setting:

[TABLE]

We also define the planar random billiard system $(M,\mu,T)$ with:

[TABLE]

This dynamical system is a $\mathbb{Z}^{2}$-extension of $\bar{M}$ by $\Phi:\bar{M}\rightarrow\mathbb{Z}^{2}$ given by:

[TABLE]

Observe that

[TABLE]

with

[TABLE]

corresponding to the cell change, starting from $x$, after $n$ iterations of maps labeled successively by $\omega_{0},\dots,\omega_{n-1}$.

Notation 1.1**.**

As exemplified by the definitions above, we will use overlines such as $\bar{\mu}$, $\bar{M}$, $\bar{T}$ to denote objects associated with the quotient random system, defined in finite measure. When we introduce a subscript such as $\bar{\mu}_{0}$, $\bar{M}_{0}$, $\bar{T}_{\omega}$, these denote objects which are not functions of the random coordinate, but are still defined on the quotient space.

1.3. A uniform family of maps

We fix the phase space $\bar{M}_{0}=\cup_{i=1}^{I}\mathbb{S}_{i}\times[-\pi/2,\pi/2]$ as described above, and denote by $m$ the normalized Lebesgue measure on $\bar{M}_{0}$. Define $\mathcal{S}_{0}=\{\varphi=\pm\frac{\pi}{2}\}$ and for a fixed $k_{0}\in\mathbb{N}$ with value to be chosen in (3), for $k\geq k_{0}$ we define the homogeneity strips,

[TABLE]

and the strips $\mathbb{H}_{-k}$ are defined similarly in a neighborhood of $\varphi=-\pi/2$. For the class of maps defined below, we will work with the extended singularity set $\mathcal{S}_{0,H}=\mathcal{S}_{0}\cup(\cup_{k\geq k_{0}}\partial\mathbb{H}_{\pm k})$. Thus for any $F\in\bar{\mathcal{F}}$, the set $\mathcal{S}_{\pm n}^{F}:=\cup_{i=0}^{n}F^{\mp i}\mathcal{S}_{0,H}$ represents the singularity set for $F^{\pm n}$.

We suppose there exists a class $\bar{\mathcal{F}}$ of maps $F:\bar{M}_{0}\circlearrowleft$ such that each $F\in\bar{\mathcal{F}}$ is a $C^{2}$ diffeomorphism of $\bar{M}_{0}\setminus\mathcal{S}_{1}^{F}$ onto $\bar{M}_{0}\setminus\mathcal{S}_{-1}^{F}$ and satisfies the following properties.

(H1) Hyperbolicity and Singularities. There exist continuous families of stable and unstable cones, $C^{s}(x)$ and $C^{u}(x)$, which are strictly invariant in the following sense: $DF(x)C^{u}(x)\subset C^{u}(Fx)$ and $DF^{-1}(x)C^{s}(x)\subset C^{s}(F^{-1}x)$ for all $F\in\bar{\mathcal{F}}$ wherever $DF$ and $DF^{-1}$ are defined.

The sets $\mathcal{S}_{\pm n}^{F}$ comprise finitely many smooth curves for each $n\in\mathbb{N}$. $\mathcal{S}_{n}^{F}$ is uniformly transverse111The uniformity is assumed to be a lower bound on the angle between these curves and the relevant cone, which is indepedent of $x\in\bar{M}_{0}$, $n\in\mathbb{N}$ and $F\in\bar{\mathcal{F}}$. to $C^{u}(x)$ and $\mathcal{S}_{-n}^{F}$ is uniformly transverse to $C^{s}(x)$ for each $n\geq 0$. Moreover, $C^{s}(x)$ and $C^{u}(x)$ are uniformly transverse on $\bar{M}_{0}$ and $C^{s}(x)$ is uniformly transverse to the horizontal and vertical directions on all of $\bar{M}_{0}$.222This is not a restrictive assumption for perturbations of the Lorentz gas since the standard cones for the associated billiard map satisfy this property [12, Section 4.5].

We assume there exist constants $C_{e}>0$ and $\Lambda>1$ such that for all $F\in\bar{\mathcal{F}}$ and $n\geq 0$,

[TABLE]

where $\|\cdot\|$ is the Euclidean norm on the tangent space to $\bar{M}_{0}$.

Finally, near singularities, we assume the maps in $\bar{\mathcal{F}}$ behave like billiards: there exists $C_{a}>0$ such that

[TABLE]

where $\varphi(z)$ denotes the angle $\varphi$ at the point $z=(r,\varphi)\in\bar{M}_{0}$. We also require that the second derivative is bounded by,

[TABLE]

(H2) Families of stable and unstable curves. We call a $\mathcal{C}^{2}$ curve $W\subset\bar{M}_{0}$ a stable curve with respect to the class $\bar{\mathcal{F}}$ if the unit tangent to $W$ lies in $C^{s}(x)$ for all $x\in W$. We say $W$ is homogeneous if it lies in a single homogeneity strip $\mathbb{H}_{k}$. We define homogeneous unstable curves analogously.

Let $\widehat{\mathcal{W}}^{s}$ denote the set of $\mathcal{C}^{2}$ homogeneous stable curves in $\bar{M}_{0}$ whose curvature is bounded above by a constant $B>0$. We assume there exists $B$ large enough that $F^{-1}W$ is a union of elements of $\widehat{\mathcal{W}}^{s}$ for all $W\in\widehat{\mathcal{W}}^{s}$ and $F\in\bar{\mathcal{F}}$. A family $\widehat{\mathcal{W}}^{u}$ of unstable curves is defined analogously.

(H3) One-step Expansion. Assume there exists an adapted norm $\|\cdot\|_{*}$ on the tangent space to $\bar{M}_{0}$, equivalent to $\|\cdot\|$, in which the constant $C_{e}$ in (2) can be taken to be 1. This yields a uniform expansion and contraction in one step for maps in the class $\bar{\mathcal{F}}$.

Let $W\in\widehat{\mathcal{W}}^{s}$. For $F\in\bar{\mathcal{F}}$, we subdivide $F^{-1}W$ into maximal homogeneous curves $V_{i}=V_{i}(F)\in\widehat{W}^{s}$. We denote by $|J_{V_{i}}F|_{*}$ the minimum contraction on $V_{i}$ under $F$ in the metric induced by the adapted norm $\|\cdot\|_{*}$. We assume that $k_{0}$ in (1) can be chosen sufficiently large that,

[TABLE]

where $|W|$ denotes the arclength of $W$.

In addition, if we weaken the power of the Jacobian slightly, we assume that the sum above still converges (although it need not be a contraction). There exists $\zeta_{0}\in(0,1)$ and $C_{1}>0$ such that for each $\delta>0$ and $\zeta\in[\zeta_{0},1]$,

[TABLE]

(H4) Bounded distortion. There exists a constant $C_{d}>0$ with the following properties. Let $W^{\prime}\in\widehat{\mathcal{W}}^{s}$ and for $F\in\bar{\mathcal{F}}$, $n\in\mathbb{N}$, let $x,y\in W\subset F^{-n}W^{\prime}$ such that $F^{i}W$ is a homogeneous stable curve for each $0\leq i\leq n$. Then,

[TABLE]

where $J_{W}F^{n}$ denotes the (stable) Jacobian of $F^{n}$ along $W$ with respect to arclength.

(H5) Invariant measure. All the maps $F\in\bar{\mathcal{F}}$ have the same invariant measure $\bar{\mu}_{0}$.

Remark 1.2**.**

Assumption (H5) can be replaced more generally with the requirement that all $F\in\bar{\mathcal{F}}$ preserve the same measure $\bar{\mu}$ which is absolutely continuous with respect to $m$ and mixing. In addition, $\bar{\mu}$ should satisfy the following technical assumptions: For $k\geq k_{0}$, $\bar{\mu}(\mathbb{H}_{k})=O(k^{-5q})$ for some $q\in(\frac{4}{5},1)$; also, $\bar{\mu}$ can be disintegrated into measures $\mu_{\alpha}$ along any measurable foliation of $\bar{M}_{0}$ into stable manifolds $\{W_{\alpha},\alpha\in\mathcal{A}\}$, with a factor measure $\lambda$, such that

[TABLE]

where $d\mu_{\alpha}=\rho_{\alpha}dm_{\alpha}$ satisfies a regularity condition: $|\ln\rho_{\alpha}(x)-\ln\rho_{\alpha}(y)|\leq C_{F}d_{W_{\alpha}}(x,y)^{1/3}$, for some constant $C_{F}\geq C_{d}$, $d_{W}(x,y)$ is the distance of $x$ and $y$ measured along the curve $W$, and $m_{\alpha}$ is arclength measure on $W_{\alpha}$.

This generalization to other smooth invariant measures is of interest, for example, when considering perturbations in the form of certain soft potentials rather than hard scatterers, or the case of external forces due to gradient fields. See for instance [3, 11] and their inclusion in a similar perturbative framework [15].

A crucial lemma, which will allow us to draw random sequences from the class $\bar{\mathcal{F}}$, is the following.

Lemma 1.3**.**

Fix a class $\bar{\mathcal{F}}$ satisfying (H1)-(H5)* with uniform constants. Let $\underline{\omega}\in E^{\mathbb{N}}$, and suppose $\bar{T}_{\omega_{k}}\in\bar{\mathcal{F}}$ for all $k\geq 0$.*

Then for all $n\in\mathbb{N}$, the composition $\bar{T}^{n}_{\underline{\omega}}:=\bar{T}_{\omega_{n-1}}\circ\cdots\circ\bar{T}_{\omega_{0}}$ satisfies assumptions (H1)-(H5), with possibly larger constants (that are nonetheless independent of $n$ and $\underline{\omega}$), and with respect to the singularity sets $\mathcal{S}_{n}^{\bar{T}_{\underline{\omega}}}=\cup_{k=0}^{n-1}\bar{T}_{\omega_{0}}^{-1}\circ\cdots\circ\bar{T}_{\omega_{k}}^{-1}\mathcal{S}_{0}$.

Lemma 1.3 is proved in [15, Section 5.3].

1.4. Distance in the class $\bar{\mathcal{F}}$

To define a notion of distance $d_{\bar{\mathcal{F}}}(\cdot,\cdot)$ in the class of maps $\bar{\mathcal{F}}$, let $F_{1},F_{2}\in\bar{\mathcal{F}}$ and for $\epsilon>0$, let $N_{\epsilon}(\mathcal{S}_{-1}^{F_{i}})$ denote the $\epsilon$-neighborhood of the singularity set $\mathcal{S}_{-1}^{F_{i}}$. We say $d_{\bar{\mathcal{F}}}(F_{1},F_{2})\leq\epsilon$ if for all $x\notin N_{\epsilon}(\mathcal{S}_{-1}^{F_{1}}\cup\mathcal{S}_{-1}^{F_{2}})$:

• (C1)

$d((F_{1})^{-1}(x),(F_{2})^{-1}(x))\leq\epsilon$;

• (C2)

$\displaystyle\left|\frac{J_{\bar{\mu}_{0}}F_{i}(x)}{J_{\bar{\mu}_{0}}F_{j}(x)}-1\right|\leq\epsilon$, $i,j=1,2$;

• (C3)

$\displaystyle\left|\frac{J_{W}F_{i}(x)}{J_{W}F_{j}(x)}-1\right|\leq\epsilon$, for all $W\in\widehat{\mathcal{W}}^{s}$ and $x\in W$, $i,j=1,2$;

• (C4)

$\|D(F_{1})^{-1}(x)v-D(F_{2})^{-1}(x)v\|\leq\sqrt{\epsilon}$, for any unit vector $v$ tangent to $W\in\widehat{\mathcal{W}}^{s}$ at $x$.

For $F_{0}\in\bar{\mathcal{F}}$ and $\vartheta_{0}>0$, define

[TABLE]

to be the $\theta_{0}$ neighborhood of $F_{0}$ in $\bar{\mathcal{F}}$.

We remark that is definition of distance does not require the sets $\mathcal{S}_{-1}^{1}$ and $\mathcal{S}_{-1}^{2}$ to be close in any sense, only that the maps are $\mathcal{C}^{1}$-close outside an $\epsilon$-neighborhood of the union of the two singularity sets. Next, we describe a perturbation family of billiards that satisfying assumptions (H1)-(H5), to illustrate that these assumptions are reasonable.

1.5. Applications – Deterministic perturbations

Given $I$ intervals $J_{1},\ldots J_{I}$, we fix the phase space $\bar{M}_{0}=\bigcup_{i=1}^{I}J_{i}\times[-\pi/2,\pi/2]$ on which the maps in class $\bar{\mathcal{F}}$ are defined. We use the notation $\bar{Q}=\bar{Q}(\{O_{i}\}_{i=1}^{I};\{J_{i}\}_{i=1}^{I})$ to denote the configuration of scatterers $O_{1},\ldots,O_{l}$ placed on the billiard table such that $|\partial O_{i}|=|J_{i}|$, $i=1,\ldots,I$. We identify the endpoints of $J_{i}$ so that each $J_{i}$ can be identified with a circle and each component of $\bar{M}_{0}$ is a cylinder. Since we have fixed $J_{1},\ldots,J_{I}$, $\bar{M}_{0}$ remains the same for all configurations $\bar{Q}$ that we consider. For each such configuration, we define

[TABLE]

Similarly, we define $\tau_{\max}$, as well as $\mathcal{K}_{\min}(\bar{Q})$ and $\mathcal{K}_{\max}(\bar{Q})$, which denote the minimum and maximum curvatures respectively of the $\partial O_{i}$ in the configuration $\bar{Q}$. The constant $E_{\max}(\bar{Q})$ denotes the maximum $C^{3}$ norm of the $\partial O_{i}$ in $\bar{Q}$.

For each fixed $\tau_{*},\mathcal{K}_{*},E_{*}>0$, define $\mathcal{Q}_{1}(\tau_{*},\mathcal{K}_{*},E_{*})$ to be the collection of all configurations $\bar{Q}$ such that:

[TABLE]

Let $\bar{\mathcal{F}}_{1}(\tau_{*},\mathcal{K}_{*},E_{*})$ be the corresponding set of billiard maps induced by the configurations in $\mathcal{Q}_{1}$. The following lemma is proved in [15].

Lemma 1.4**.**

([15, Theorem 2.7])*

Fix intervals $J_{1},\ldots,J_{I}$ and let $\tau_{*},\mathcal{K}_{*},E_{*}>0$. The family $\bar{\mathcal{F}}_{1}(\tau_{*},\mathcal{K}_{*},E_{*})$ satisfies (H1)-(H5) with uniform constants depending only on $\tau_{*}$, $\mathcal{K}_{*}$ and $E_{*}$. *

We fix an initial configuration of scatterers $\bar{Q}_{0}\in\mathcal{Q}_{1}(\tau_{*},\mathcal{K}_{*},E_{*})$ and consider configurations $\bar{Q}$ which alter each $\partial O_{i}$ in $\bar{Q}_{0}$ to a curve $\partial\tilde{O}_{i}$ having the same arclength as $\partial O_{i}$. We consider each $\partial O_{i}$ as a parametrized curve $u_{i}:J_{i}\to\mathbb{R}^{2}$ and each $\partial\tilde{O}_{i}$ as parametrized by $\tilde{u}_{i}$. Define

[TABLE]

The following is proved in [15] .

Lemma 1.5**.**

([15, Theorem 2.8])*

Choose $\vartheta_{0}\leq\min\{\tau_{*}/2,\mathcal{K}_{*}/2\}$ and let $\bar{\mathcal{F}}_{A}(\bar{Q}_{0},E_{*};\vartheta_{0})$ be the set of all billiard maps corresponding to configurations $\bar{Q}$ such that $\Delta(\bar{Q},\bar{Q}_{0})\leq\vartheta_{0}$ and $E_{\max}(\bar{Q})\leq E_{*}$. Then $\bar{\mathcal{F}}_{A}(\bar{Q}_{0},E_{*};\vartheta_{0})\subset\bar{\mathcal{F}}_{1}(\tau_{*}/2,\mathcal{K}_{*}/2,E_{*})$ and $d_{\bar{\mathcal{F}}}(\bar{F}_{1},\bar{F}_{2})\leq C|\vartheta_{0}|^{1/3}$ for any $\bar{F}_{1},\bar{F}_{2}\in\bar{\mathcal{F}}_{A}(Q_{0},E_{*};\vartheta_{0})$. *

The importance of these results is that together, they will imply that the transfer operators associated to maps in the neighborhood $\bar{\mathcal{F}}_{\vartheta_{0}}(\bar{T}_{0})$ have a uniform spectral gap if the transfer operator associated with $\bar{T}_{0}$ has a spectral gap. Moreover, small changes in the configuration of scatterers are seen to generate small differences in the distance $d_{\bar{\mathcal{F}}}(\cdot,\cdot)$.

2. Main results

In this section, we consider all $\bar{T}_{\omega}\in\bar{\mathcal{F}}_{\vartheta_{0}}(\bar{T}_{0})$, for some $\vartheta_{0}>0$ small enough and a fixed map $\bar{T}_{0}:\bar{M}_{0}\circlearrowleft$.

2.1. Local Limit Theorem

Adapting the proof of [15, Corollary 2.4] (with the slight difference that, here, the observable $\Phi(x,\underline{\omega})$ we are interested in depends also on $\underline{\omega}$), we will prove the following central limit theorem.

Theorem 2.1** (Central Limit Theorem for the cell index).**

With respect to $\bar{\mu}$, the covariance matrix of $(S_{n}/\sqrt{n})_{n}$ converges to a non-negative symmetric function

[TABLE]

*where, for every $j=1,2$, $\Phi^{(j)}$ is the $j$-th coordinate of $\Phi$, and using . to denote multiplication.

Moreover $(S_{n}/\sqrt{n})_{n}$ converges in distribution to a centered Gaussian distribution with covariance matrix $\Sigma^{2}$.*

The fact that $\Sigma^{2}$ is positive if $\vartheta_{0}$ is small enough will be proved in Lemma 3.18 (using a continuity argument). In Section 3.2, we will define a Banach space $\widetilde{\mathcal{B}}$, containing a class of distributions on $\bar{M}$, and its dual $\widetilde{\mathcal{B}}^{\prime}$. For a function $g:\bar{M}\to\mathbb{R}$, define the functional $H_{g}$, by

[TABLE]

Remark 3.1 and Lemma 3.3 will give conditions on $g$ that guarantee that $H_{g}\in\widetilde{\mathcal{B}}^{\prime}$.

Theorem 2.2** (Local limit theorem).**

For every $f,g:\bar{M}\rightarrow\mathbb{R}$ such that $H_{g}\in\widetilde{\mathcal{B}}^{\prime}$ and such that $f\in\widetilde{\mathcal{B}}$,

[TABLE]

Remark 2.3**.**

Due to Lemma 3.3 and Remark A.1, it suffices for the conclusion of Theorem 2.2 that $f(\cdot,\underline{\omega})$ and $g(\cdot,\underline{\omega})$ be piecewise Hölder continuous on $\bar{M}_{0}$ (with Hölder bounds that are uniform in $\underline{\omega}$). For instance, the coordinates $\Phi^{(i)}$ of the displacement function $\Phi$ satisfy these conditions, as well as the free flight function for the billiard map $\bar{T}_{\omega}$, $\tau(\cdot,\omega)$.

2.2. Return time, visit to new obstacles and self intersections

We define ${\mathcal{I}}_{0}(x,\underline{\omega}):=i$ if $x\in\bigcup_{\ell\in\mathbb{Z}^{2}}M_{i,\ell}$ as the index in $\{1,...,I\}$ of the obstacle on which the particle is at time [math] and ${\mathcal{I}}_{k}:={\mathcal{I}}_{0}\circ T^{k}$. Since the quantity ${\mathcal{I}}_{0}(x,\underline{\omega})$ does not depend on $\underline{\omega}$, we will also write $\mathcal{I}_{0}(x)$ for this quantity. Note that $\mathcal{I}_{k}(x,\underline{\omega})$ does not depend on the index $\ell$ of the cell containing $x$, this allows us to define also $\mathcal{I}_{k}$ on $\bar{M}$ (by projection).

Observe that the fact that the point particle is on the obstacle $(i,\ell)$ at the $k$-th reflection time (i.e. $T^{k}(x,\underline{\omega})\in M_{i,\ell}$) can be rewritten:

[TABLE]

if $x=(\bar{x},\ell_{0})\in\bar{M}_{0}\times\mathbb{Z}^{2}$. We are interested here in the study of the probability that a point particle starting333Throughout the paper, we shall use the notation $\boldsymbol{0}=(0,0)$ as an element of $\mathbb{Z}^{2}$. from $\bar{M}\times\{\boldsymbol{0}\}$ does not come back to its original obstacle until time $n$, that is in $\bar{\mu}(B_{n})$ with

[TABLE]

We also study the probability that the obstacle visited at time $n$ has not been visited before, that is $\bar{\mu}(B^{\prime}_{n})$ with

[TABLE]

Observe that, because of the reversibility of our model, $\mu(B_{n})=\mu(B^{\prime}_{n})$.

Theorem 2.4**.**

We have the following asymptotics

[TABLE]

In Section 4.2, we give a proof of the above asymptotic estimates of $\mu(B_{n})$ and $\mu(B^{\prime}_{n})$ for in a more general context. This result will appear as an easy and direct consequence of the local limit theorem, Theorem 2.2. We now consider the number of couples of times at which the point particle hits the same obstacle:

[TABLE]

Theorem 2.5**.**

$\bar{\mu}$-almost surely, we have:

[TABLE]

The proof of the previous result is delicate as it uses a precise estimate of the variance of $\mathcal{V}_{n}$. As can be seen from the works by Bolthausen [5] and by Deligiannidis and Utev [13], going from a rough to a precise estimate of the variance of the number of self intersections requires important additional work. In section 4.3, we give a proof of this result under general spectral assumptions. Our argument provides, in the case of random walks, an alternative argument to the one given by Deligiannidis and Utev in [13]. Let us indicate that even if we use the general sheme of the previous unpublished paper [30] (in which an analogous result is proved for a single billiard map), this general scheme being just the natural decomposition already used by Bolthausen in [5] to get a non-optimal estimate of the variance, the method we use in the present paper to establish our crucial estimates is different from [30]. In particular our method enables us to get rid of some assumptions (bounded cell change function, Banach spaces continuously injected in some $L^{p}$) that were satisfied and used in [30].

The two previous results (probability to visit a new site, precise asymptotics for the number self-intersections), in addition to being interesting in their own right, will greatly help us to prove the result of the next section.

2.3. Billiard in random scenery

To each obstacle $(i,\ell)$, we associate a random variable $\xi_{(i,\ell)}$. We assume that these random variables are i.i.d., centered and square integrable and independent of the dynamic of the billiard. We assume that, each time the point particle hits the obstacle $(i,\ell)$, it wins the value $\xi_{(i,\ell)}$. Let $\mathcal{Z}_{n}$ be the total amount won by the particle up to the $n$-th reflection. For every $n$, we consider the linearized process $(\widetilde{\mathcal{Z}}_{n}(t))_{t\geq 0}$ defined by

[TABLE]

Theorem 2.6**.**

The sequence of processes $((\widetilde{\mathcal{Z}}_{n}(t)/\sqrt{n\log n})_{t\geq 0})_{n}$ converges in distribution, with respect to the uniform norm on $[0,T]$ for every $T>0$, to a Brownian motion $B=(B_{t})_{t\geq 0}$ such that

[TABLE]

If, moreover, there exists $\chi>0$ such that $\mathbb{E}[|\xi_{(1,0)}|^{2}(\log^{+}|\xi_{(1,0)}|)^{\chi}|)]<\infty$, then, for almost every realization of $(\xi_{i,\ell})_{i,\ell}$, $(\widetilde{\mathcal{Z}}_{n})_{n}$ converges in distribution to the same Brownian motion $B$.

Let us say a few words about the historical background of this result. Limit distributional theorems of analogous processes when $S_{n}$ is replaced by a random walk on $\mathbb{Z}^{d}$ were first established at the end of the 70’s by Borodin in [6, 7] and by Kesten and Spitzer in [25], by Boltausen [5] in dimension 2 ten years later, and more recently by Deligiannidis and Utev in [13] and by Castell, Guillotin-Plantard and the second author in [10]. Let us also remark that when the random walk is the one dimensional simple symmetric random walk on $\mathbb{Z}$, the random walk in random scenery corresponds to an ergodic sum of a dynamical system, the so-called $T,T^{-1}$-transformation. This dynamical system has been introduced in a list of open problems by Weiss [37, problem 2, p. 682] in the early 1970’s. This dynamical system is a famous natural example of a $K$-transformation which is not Bernoulli and even not loosely Bernoulli as has been shown by Kalikow in [22].

We prove Theorem 2.6 in a more general context in Section 4.4. As noticed by Deligiannidis and Utev in [13] in the context of random walks, the estimate provided by Theorem 2.5 simplifies greatly the proof of Theorem 2.6 compared to [5, 29] ([29] contained a proof of this result for a single billiard map, with the use of the properties of Young towers). Furthermore, we simplify also the tightness argument used by Bolthausen in [5].

2.4. Limit theorems in infinite measure

The following results are consequences of our perturbation result (Proposition 3.17), combined with the general results of [33] and of [31].

Our next result deals with the asymptotic behavior of additive functionals of $S_{n}$, that is of quantities of the form $\sum_{k=0}^{n-1}g(S_{k})$, for summable functions $g:\mathbb{Z}^{2}\rightarrow\mathbb{R}$. This can be seen as the ergodic sum $\sum_{k=0}^{n-1}G\circ T^{k}$ with $G(x,\ell,\underline{\omega}):=g(\ell)$.

Theorem 2.7** (Additive functionals of $S_{n}$).**

If $g$ is a summable (i.e. $\sum_{\ell\in\mathbb{Z}^{2}}|g(\ell)|<\infty$), then

[TABLE]

convergence in distribution, where $\mathcal{E}$ is an exponential random variable with expectation 1 and where $\Longrightarrow$ means the convergence in distribution with respect to any probability measure absolutely continuous with respect to $\bar{\mu}$.

If moreover $\sum_{\ell\in\mathbb{Z}^{2}}g(\ell)=0$ and $\sum_{\ell\in\mathbb{Z}^{2}}|\ell|^{\varepsilon}|g(\ell)|<\infty$, for some $\varepsilon>0$, then

[TABLE]

convergence in distribution, where $\mathcal{E}$ is as above and where $\mathcal{N}$ is a standard Gaussian random variable independent of $\mathcal{E}$ and where

[TABLE]

For $g:M_{0}\to\mathbb{R}$, define $H_{g,\ell}:\mathcal{B}\to\mathbb{R}$ by $H_{g,\ell}(h)=\mathbb{E}_{\bar{\mu}_{0}}[g(\cdot,\ell)h]$. We also obtain the decay rates of correlations for the process generated by our random systems in infinite measure:

Theorem 2.8** (Mixing and decorrelation in infinite measure).**

Let $K\geq 1$. Let $f,g:M_{0}\rightarrow\mathbb{R}$ be two functions such that

[TABLE]

Then, there exist real numbers $C_{0}(f,g),...,C_{K}(f,g)$ such that

[TABLE]

with $C_{0}(f,g)=\frac{1}{2\pi\sqrt{\det\Sigma^{2}}}\int_{M_{0}}f\,d\mu_{0}\,\int_{M_{0}}g\,d\mu_{0}$.

3. Transfer Operators

In order to prove our main limit theorems, we will study the transfer operators associated with the random maps $T$ and $\bar{T}$ as perturbations of the transfer operator associated with a fixed quotient billiard map $\bar{T}_{0}$.

In this section, we fix a class of maps $\bar{\mathcal{F}}$ satisfying (H1)-(H5) with uniform constants. $\bar{T}$ denotes the quotient of the full random map $T$, while $\bar{T}_{\omega}$, $\omega\in E$ denotes a quotient billiard map belonging to $\bar{\mathcal{F}}$, following the notation defined in Section 1.2.

Using (H3), choose $\delta_{0}>0$ for which there exists $\theta<1$ so that (3) satisfies,

[TABLE]

We then define $\mathcal{W}^{s}\subset\widehat{\mathcal{W}}^{s}$ to be those stable curves in $\widehat{\mathcal{W}}^{s}$ whose length is at most $\delta_{0}$.

Following [14], for any $\bar{T}_{\omega}\in\bar{\mathcal{F}}$ and $n\geq 0$, define $\bar{T}_{\omega}^{-n}\mathcal{W}^{s}\subset\mathcal{W}^{s}$ to be the set of homogeneous stable curves $W\in\mathcal{W}^{s}$ whose images $\bar{T}_{\omega}^{i}W\in\mathcal{W}^{s}$ for $0\leq i\leq n$. For $p\in[0,1]$ and letting $\mathcal{C}^{p}(\bar{T}_{\omega}^{-n}\mathcal{W}^{s})$ denote those functions $\psi$ which are Hölder continuous on elements of $\bar{T}_{\omega}^{-n}\mathcal{W}^{s}$, it follows from (H1) that $\psi\circ\bar{T}_{\omega}\in\mathcal{C}^{p}(\bar{T}^{-n-1}\mathcal{W}^{s})$. Thus if $f\in(\mathcal{C}^{p}(\bar{T}_{\omega}^{-n-1}\mathcal{W}^{s}))^{\prime}$ is an element of the dual of $\mathcal{C}^{p}(\bar{T}_{\omega}^{-n-1}\mathcal{W}^{s})$, then $\mathcal{L}_{\bar{T}_{\omega}}:(\mathcal{C}^{p}(\bar{T}_{\omega}^{-n-1}\mathcal{W}^{s}))^{\prime}\to(\mathcal{C}^{p}(\bar{T}_{\omega}^{-n}\mathcal{W}^{s}))^{\prime}$ is defined by

[TABLE]

If in addition, $f$ is a measure absolutely continuous with respect to $\bar{\mu}_{0}$, then we identify $f$ with its density in $L^{1}(\bar{\mu}_{0})$, which we shall also denote $f$, i.e. $f(\psi)=\int_{\bar{M}_{0}}\psi\,f\,d\bar{\mu}_{0}$. With this identification, we write $L^{1}(\bar{\mu}_{0})\subset(\mathcal{C}^{p}(\bar{T}_{\omega}^{-n}\mathcal{W}^{s}))^{\prime}$ for each $n\in\mathbb{N}$. Then acting on $L^{1}(\bar{\mu}_{0})$, $\mathcal{L}_{\bar{T}_{\omega}}$ has the following familiar expression,

[TABLE]

For brevity, sometimes we will denote $\mathcal{L}_{\bar{T}_{\omega}}$ by $\mathcal{L}_{\omega}$.

Let $P$ be the transfer operator of $\bar{T}$ with respect to $\bar{\mu}:=\bar{\mu}_{0}\otimes\eta^{\otimes\mathbb{N}}$. This operator is given by

[TABLE]

Let us write $\cdot$ for the usual scalar product on $\mathbb{R}^{2}$. We consider the family of operators $(P_{u})_{u\in\mathbb{R}^{2}}$ given by

[TABLE]

where

[TABLE]

Note that

[TABLE]

Using results of [15], we will see that if we restrict $\bar{T}_{\omega}$ to a neighborhood $\bar{\mathcal{F}}_{\vartheta_{0}}(\bar{T}_{0})$ according to (5), then $P$ is a small (depending on $\vartheta_{0}$) perturbation of the transfer operator $\mathcal{P}_{0}$ of the product system $(\bar{M},\bar{\mu}:=\bar{\mu}_{0}\times\eta^{\otimes\mathbb{N}},\bar{T}_{0}\times\sigma)$, with $\sigma$ is the shift over $E^{\mathbb{N}}$ (i.e. $\sigma((\omega_{k})_{k\geq 0})=(\omega_{k+1})_{k\geq 0}$) and where $(\bar{T}_{0}\times\sigma)(x,\underline{\omega}):=(\bar{T}_{0}(x),\sigma(\underline{\omega}))$.

3.1. Banach spaces $\mathcal{B}$ and $\mathcal{B}_{w}$

We start by defining Banach spaces $\mathcal{B}\subset\mathcal{B}_{w}$ of distributions on $\bar{M}_{0}$, on which the transfer operators $\mathcal{L}_{\omega}$ associated to $\bar{T}_{\omega}\in\bar{\mathcal{F}}$ are well-behaved.

In order to define our norms, we first require a notion of distance $d_{\mathcal{W}^{s}}(\cdot,\cdot)$ between stable curves as well as a distance $d(\cdot,\cdot)$ defined among functions supported on these curves.

Due to the transversality condition on the stable cones $C^{s}(x)$ given by (H1), each $W\in\mathcal{W}^{s}$ can be viewed as the graph of a function $\varphi_{W}(r)$ of the arc length parameter $r$. For each $W\in\mathcal{W}^{s}$, let $J_{W}$ denote the interval on which $\varphi_{W}$ is defined and set $G_{W}(r)=(r,\varphi_{W}(r))$ to be its graph so that $W=\{G_{W}(r):r\in J_{W}\}$. We let $m_{W}$ denote the unnormalized arclength measure on $W$, defined using the Euclidean metric.

Let $W_{1},W_{2}\in\mathcal{W}^{s}$ and let $\varphi_{W_{i}}$, $G_{W_{i}}$ denote the corresponding functions defined above, for $i=1,2$. Denote by $\ell(J_{W_{1}}\triangle J_{W_{2}})$ the length of the symmetric difference between $J_{W_{1}}$ and $J_{W_{2}}$. If $W_{1}$ and $W_{2}$ belong to the same homogeneity strip, we define the distance between them to be,

[TABLE]

otherwise, we set $d_{\mathcal{W}^{s}}(W_{1},W_{2})=\infty$.

For $0\leq p\leq 1$, let $\tilde{\mathcal{C}}^{p}(W)$ denote the set of continuous complex-valued functions on $W$ with Hölder exponent $p$, measured in the Euclidean metric. Denote by $\mathcal{C}^{p}(W)$ the closure of $\mathcal{C}^{\infty}(W)$ in the $\tilde{\mathcal{C}}^{p}$-norm444While $\mathcal{C}^{p}(W)$ is smaller than $\tilde{\mathcal{C}}^{p}(W)$, it does contain $\mathcal{C}^{p^{\prime}}\!(W)$ for all $p^{\prime}>p$.:

[TABLE]

where $C^{(p)}_{W}(\psi)$ is the Hölder constant of $\psi$ along $W$. It is remarkable to note that that with this definition,

[TABLE]

$\tilde{\mathcal{C}}^{p}(\bar{M}_{0})$ and $\mathcal{C}^{p}(\bar{M}_{0})$ can be defined similarly.

Given two curves $W_{1},W_{2}\in\mathcal{W}^{s}$ with $d_{\mathcal{W}^{s}}(W_{1},W_{2})<\infty$, and two test functions $\psi_{i}\in\mathcal{C}^{p}(W_{i},\mathbb{C})$, * the distance between* $\psi_{1}$, $\psi_{2}$ is defined as:

[TABLE]

We will define the relevant Banach spaces by closing $\mathcal{C}^{1}(\bar{M}_{0})$ with respect to the following set of norms. Fix $0<p\leq\frac{1}{3}$. Given a function $f\in\mathcal{C}^{1}(\bar{M}_{0})$, define the weak norm of $f$ by

[TABLE]

Choose555The restrictions on the constants are placed according to the dynamical properties summarized in (H1)-(H5). For example, $p\leq 1/3$ due to the distortion bounds in (H4), while $\varsigma\leq 1-\zeta_{0}$ due to (H3), which is relevant for the uniform Lasota-Yorke inequalities (Lemma 3.14). $q$, $\gamma$, $\varsigma>0$ such that $\varsigma\leq 1-\zeta_{0}$, $q<p$ and $\gamma\leq\min\{\varsigma,p-q\}$. We define the strong stable norm of $f$ as

[TABLE]

and the strong unstable norm as

[TABLE]

where $\omega_{0}>0$ is chosen less than $\delta_{0}$, the maximum length of $W\in\mathcal{W}^{s}$ which is determined by (9). The strong norm of $f$ is defined by

[TABLE]

where $c_{0}$ is a small constant chosen so that the uniform Lasota-Yorke inequalities in [15, Theorem 2.2] hold.

We define $\mathcal{B}$ to be the completion of $\mathcal{C}^{1}(\bar{M}_{0})$ in the strong norm666As a measure, $f\in\mathcal{C}^{1}(\bar{M}_{0})$ is identified with $fd\bar{\mu}_{0}$ according to our earlier convention. As a consequence, Lebesgue measure $dm=(\cos\varphi)^{-1}d\bar{\mu}_{0}$ is not automatically included in $\mathcal{B}$ since $(\cos\varphi)^{-1}\notin\mathcal{C}^{1}(\bar{M}_{0})$. It follows from [15, Lemma 3.5] that in fact, $m\in\mathcal{B}$ (and $\mathcal{B}_{w}$). and $\mathcal{B}_{w}$ to be the completion of $\mathcal{C}^{1}(\bar{M}_{0})$ in the weak norm.

Remark 3.1**.**

Due to [15, Lemma 3.4], we have for $f\in\mathcal{B}_{w}$,

[TABLE]

This permits us to extend $\mathbb{E}_{\bar{\mu}_{0}}[\cdot]$ to a linear continuous form on $\mathcal{B}_{w}$ (and so on $\mathcal{B}$) since

[TABLE]

We begin by recalling some properties of $\mathcal{B}$ and $\mathcal{B}_{w}$ proved in [14, 15, 16].

Lemma 3.2**.**

• a)

[14, Lemma 3.7]** $\mathcal{B}$ contains piecewise Hölder continuous functions $f$ with exponent $\zeta>\gamma/(1-\gamma)$ as described in Lemma 3.3 below.

• b)

[15, Lemma 3.5]** $(\cos\varphi)^{-1}\in\mathcal{B}$. Thus, Lebesgue measure $m=(\cos\varphi)^{-1}\bar{\mu}_{0}\in\mathcal{B}$ and so is $fm$ for any $f$ as in item (a) above.

• c)

[14, Lemma 2.1]** $\mathcal{L}_{\omega}$ is well-defined as a continuous linear operator on both $\mathcal{B}$ and $\mathcal{B}_{w}$ for any $\bar{T}_{\omega}\in\bar{\mathcal{F}}$. Moreover, there exists a sequence of continuous777The first three of these are also injective. The fourth can be made injective by introducing a weight $|W|^{-\eta}$ for test functions $\psi$ in the weak norm (as appears in the definition of $\|\cdot\|_{s}$) and requiring $\eta>p$ (see, for example, [16, Lemma 3.8]).* inclusions $\mathcal{C}^{\zeta}(\bar{M}_{0})\hookrightarrow\mathcal{B}\hookrightarrow\mathcal{B}_{w}\hookrightarrow(\mathcal{C}^{p}(\bar{M}_{0}))^{\prime}$, for all $\zeta>\gamma/(1-\gamma)$.*

• d)

[14, Lemma 3.10]** The unit ball of $(\mathcal{B},\|\cdot\|_{\mathcal{B}})$ is compactly embedded in $(\mathcal{B}_{w},|\cdot|_{w})$.

The following lemma is crucial for describing the types of discontinuities allowed in elements of $\mathcal{B}$ and for proving that the operator $\mathcal{L}_{u,\omega}$ is analytic in $u$.

Lemma 3.3**.**

Let $\mathfrak{P}$ be a (mod 0) countable partition of $\bar{M}_{0}$ into open, simply connected sets such that: (1) for each $k\in\mathbb{N}$, there is an $N_{k}<\infty$ such that at most $N_{k}$ elements $Z\in\mathfrak{P}$ intersect $\mathbb{H}_{k}$; (2) there are constants $K,C_{0}>0$ such that for each $Z\in\mathfrak{P}$ and $W\in\mathcal{W}^{s}$, $Z\cap W$ comprises at most $K$ connected components and for any888In fact, Lemma 3.5 of [16] allows a nondegenerate tangency between $\partial\mathfrak{P}$ and the stable cone: $m_{W}(N_{\varepsilon}(\partial Z)\cap W)\leq C_{0}\varepsilon^{t_{0}}$, for some $t_{0}>0$. But we will not need this weaker condition here so we assume $t_{0}=1$ in order to simplify the proofs and also the statement of the norms (which otherwise would depend on $t_{0}$). $\varepsilon>0$, $m_{W}(N_{\varepsilon}(\partial Z)\cap W)\leq C_{0}\varepsilon$.

• a)

[16, Lemma 3.5]** Let $\zeta>\gamma/(1-\gamma)$. If $f\in\mathcal{C}^{\zeta}(Z)$ for each $Z\in\mathcal{Z}$ and $\sup_{Z\in\mathfrak{P}}|f|_{\mathcal{C}^{\zeta}(Z)}<\infty$, then $f\in\mathcal{B}$ and $\|f\|_{\mathcal{B}}\leq C\sup_{Z\in\mathfrak{P}}|f|_{\mathcal{C}^{\zeta}(Z)}$, for some $C>0$ independent of $f$. In particular, $\mathcal{C}^{\zeta}(\bar{M}_{0})\subset\mathcal{B}$ for each $\zeta>\gamma/(1-\gamma)$.

• b)

[16, Lemma 5.3]** Suppose in addition that $\zeta>\max\{p,\gamma/(1-\gamma)\}$ and there is a uniform bound on the $N_{k}$ above. If $g$ satisfies $\sup_{Z\in\mathfrak{P}}|g|_{\mathcal{C}^{\zeta}(Z)}<\infty$ and $f\in\mathcal{B}$, then $fg\in\mathcal{B}$ and $\|fg\|_{\mathcal{B}}\leq C\|f\|_{\mathcal{B}}\sup_{Z\in\mathfrak{P}}|g|_{\mathcal{C}^{\zeta}(Z)}$ for some $C>0$ independent of $f$ and $g$.

3.2. Banach spaces $\widetilde{\mathcal{B}}$ and $\widetilde{\mathcal{B}}_{w}$

In this section, we introduce the associated Banach spaces $\widetilde{\mathcal{B}}_{w}$ and $\widetilde{\mathcal{B}}$ on $\bar{M}$ on which $P$ acts suitably. $\widetilde{\mathcal{B}}$ will correspond to a set of Lipschitz functions from $E^{\mathbb{N}}$ to $\mathcal{B}$ and $\widetilde{\mathcal{B}}_{w}$ will correspond to the set of uniformly bounded functions from $E^{\mathbb{N}}$ to $\mathcal{B}_{w}$. For convenience, we will identify elements of $\mathcal{B}^{E^{\mathbb{N}}}$ with distributions $f$ on $\bar{M}_{0}\times E^{\mathbb{N}}$ such that $f(\cdot,\underline{\omega})\in\mathcal{B}$ for all $\underline{\omega}\in E^{\mathbb{N}}$.

Let $\varkappa>\sup_{\omega\in E}\|\mathcal{L}_{\omega}\|_{L(\mathcal{B},\mathcal{B})}\geq 1$. Let us define

[TABLE]

with

[TABLE]

and with

[TABLE]

It is immediate from this definition and the definition of $\mathcal{B}$, that $\widetilde{\mathcal{B}}$ is the completion in the $\|\cdot\|_{\widetilde{\mathcal{B}}}$ norm of the set of functions

[TABLE]

In particular, $\widetilde{\mathcal{B}}$ is a Banach space.

Remark 3.4**.**

It will be worthwhile to notice that, due Lemma 3.3(a), for every $\omega\in E$, the coordinates of $\Phi_{\omega}$ belong to $\mathcal{B}$, so that the coordinates of $\Phi$ are in $\widetilde{\mathcal{B}}$.

We also define

[TABLE]

with $|f|_{\widetilde{\mathcal{B}}_{w}}:=\sup_{{\underline{\omega}}\in E^{\mathbb{N}}}|f(\cdot,{\underline{\omega}})|_{w}\,.$ As with $\widetilde{\mathcal{B}}$, the space $\widetilde{\mathcal{B}}_{w}$ can also be realized as the completion of ${(C^{1}(\bar{M}_{0}))^{E^{\mathbb{N}}}}$ in the $|\cdot|_{\widetilde{\mathcal{B}}_{w}}$ norm.

Remark 3.5**.**

Using Remark 3.1, we extend $\mathbb{E}_{\bar{\mu}}[\cdot]$ to a continuous linear form on $\widetilde{\mathcal{B}}_{w}$ (and so on $\widetilde{\mathcal{B}}$) by setting

[TABLE]

It follows from Lemma 3.3(a) that for any obstacle $O_{a}$, $1_{O_{a}}\in\mathcal{B}$, and from Lemma 3.3(b) that $f(\cdot,\underline{\omega})\mapsto 1_{O_{a}}f(\cdot,\underline{\omega})$ is a bounded linear operator on $\mathcal{B}$ for each $\underline{\omega}\in E^{\mathbb{N}}$ and $f\in\widetilde{\mathcal{B}}$. Thus $f\mapsto 1_{O_{a}}f$ is a bounded linear operator on $\widetilde{\mathcal{B}}$ as well.

We introduce the following notation for convenience.

Notation 3.6**.**

For any positive integer $m$, any $\tilde{\underline{\omega}}_{m}\in E^{m}$ and any $\underline{\omega}\in E^{\mathbb{N}}$, we will write $(\tilde{\underline{\omega}}_{m},\underline{\omega})$ as the element of $E^{\mathbb{N}}$ obtained by concatenation; i.e. such that the first $m$ terms correspond to those of $\tilde{\underline{\omega}}_{m}$ and that the term of order $m+k$ corresponds to the term of order $k$ of $\underline{\omega}$.

Lemma 3.7**.**

• (a)

Let $n$ be a positive integer. Denote the norm $\|\cdot\|_{\sigma}$, for $\sigma\in\{w,s,u\}$. If $(f(\cdot,\tilde{}\underline{\omega}_{n}))_{\tilde{}\underline{\omega}_{n}\in E^{n}}$ is a measurable (in $\tilde{}\underline{\omega}_{n}$) family of elements of $\mathcal{B}_{w}$ such that

[TABLE]

then

[TABLE]

• (b)

If $(H_{\omega})_{\omega\in E}$ is a measurable (in $\omega$) family of uniformly bounded operators on $\mathcal{B}$ (resp. $\mathcal{B}_{w}$), then $H:f(x,\underline{\omega})\mapsto\int_{E}H_{\tilde{\omega}}(f(x,(\tilde{\omega},\underline{\omega})))\,d\eta(\tilde{\omega})$ defines a continuous linear operator on $\widetilde{\mathcal{B}}$ (resp. $\widetilde{\mathcal{B}}_{w}$) with operator norm dominated by $\sup_{\omega\in E}\|H_{\omega}\|_{L(\mathcal{B},\mathcal{B})}$.

Proof.

(a) is just the triangle inequality. Let us prove Item (b). Let $f\in\widetilde{\mathcal{B}}$ or in $\widetilde{\mathcal{B}}_{w}$ and writing $\|\cdot\|_{\sigma}$ for the associated norm, due to (a), for every $\underline{\omega},\underline{\omega}^{\prime}\in E^{\mathbb{N}}$, we have

[TABLE]

which proves (b) if $f\in\widetilde{\mathcal{B}}_{w}$. If, in addition, $f\in\widetilde{\mathcal{B}}$, then

[TABLE]

where $\tilde{}\underline{\omega}=(\omega,\underline{\omega})$ and $\tilde{}\underline{\omega}^{\prime}=(\omega,\underline{\omega}^{\prime})$. ∎

Remark 3.8**.**

The previous lemma ensures in particular that $P$ acts continuously $\widetilde{\mathcal{B}}$ and on $\widetilde{\mathcal{B}}_{w}$ since $\mathcal{L}_{\omega}$ acts uniformly continuously on $\mathcal{B}$ and on $\mathcal{B}_{w}$.

A key step in our proof is the study the spectral properties on $\widetilde{\mathcal{B}}$ of $P$ and of the family of operators $P_{u}$ defined by

[TABLE]

The next lemma ensures, in particular, that $P_{u}$ is a linear operator on $\widetilde{\mathcal{B}}$. Denote by $\Phi^{(1)}$ and $\Phi^{(2)}$ the components of the vector $\Phi$.

Lemma 3.9**.**

For every $u\in\mathbb{R}^{2}$, any positive integer $m$ and any $i_{1},...,i_{m}\in\{1,2\}$, $P(\Phi^{(i_{1})}...\Phi^{(i_{m})}e^{iu\cdot\Phi}\cdot)$ is a linear operator on $\widetilde{\mathcal{B}}$ and on $\widetilde{\mathcal{B}}_{w}$, with operator norms uniformly in $O(\sup_{\omega\in E}\|\Phi\|^{m}_{\infty})$.

Proof.

This proof is a variation of the argument used in [15, Section 5.2]. Recall that $Pg(\cdot,\underline{\omega})=\int_{E}f(\bar{T}_{\omega}^{-1}(\cdot),(\omega,\underline{\omega}))\,d\eta(\omega)$, so that

[TABLE]

Let $\omega\in E$. The singularity set for $\Phi_{\omega}$ is $\mathcal{S}_{1}^{\bar{T}_{\omega}}$, which does not satisfy the hypotheses of Lemma 3.3; however, the singularity set for $\Phi_{\omega}\circ\bar{T}_{\omega}^{-1}$ is $\mathcal{S}_{-1}^{\bar{T}_{\omega}}$, which satisfies the hypotheses of Lemma 3.3(b) since there are only finitely many such curves and they lie in the unstable cone. Let $\mathcal{Z}$ denote the (finite) partition of $\bar{M}_{0}\setminus\mathcal{S}_{-1}^{\bar{T}_{\omega}}$ into its maximal connected components. Note that $\Phi_{\omega}\circ\bar{T}_{\omega}^{-1}$ is constant on each element of $\mathcal{Z}$. We use Lemma 3.3(b) to estimate, for every $f\in\widetilde{\mathcal{B}}$,

[TABLE]

Analogously, for every $f\in\widetilde{\mathcal{B}}_{w}$,

[TABLE]

and we conclude by Item (b) of Lemma 3.7. ∎

3.3. View $P_{u}$ as a perturbation of a quasicompact operator

For the remainder of Section 3, we fix a billiard map $\bar{T}_{0}$, and for $\vartheta_{0}>0$, define $\bar{\mathcal{F}}_{\vartheta_{0}}(\bar{T}_{0})$ as in (5). Our main results in this setting will be that for $\vartheta_{0}$ sufficiently small, both $P$ and $P_{u}$ are quasi-compact and have a spectral gap in $\widetilde{\mathcal{B}}$. These statements are contained in Proposition 3.15 and Theorem 3.17.

Recall $P_{u}:=P(e^{iu\cdot\Phi}\cdot)$. Our next result states that $P_{u}$ is a small perturbation (as $\vartheta_{0}\rightarrow 0$) of $\mathcal{P}_{u}:=\mathcal{P}(e^{iu\cdot\Phi_{0}}\cdot)$, where $\mathcal{P}$ is the transfer operator $\mathcal{P}_{0}$ of the direct product $(\bar{M},\bar{\mu},\bar{T}^{\prime}_{0}:=\bar{T}_{0}\times\sigma)$, i.e.

[TABLE]

and

[TABLE]

Here, $\mathcal{L}_{0}=\mathcal{L}_{\bar{T}_{0}}$ and $\mathcal{L}_{u,0}=\mathcal{L}_{\bar{T}_{0}}(e^{iu\cdot\Phi_{0}}\cdot)$.

Proposition 3.10**.**

There exists $C>0$ such that for every $u\in\mathbb{R}^{2}$ and every $f\in\widetilde{\mathcal{B}}$,

[TABLE]

Before proving this proposition, we state the following lemma.

Lemma 3.11**.**

There exists $C>0$ such that for all $\omega\in E$ and $u\in\mathbb{R}^{2}$,

[TABLE]

Proof.

This lemma for $u=0$ is proved in [15, Theorem 2.3]. As in the proof of Lemma 3.14, we must show that the relevant estimates are independent of $u$. The relevant estimate is eq. (5.1) of [15]: Let $W\in\mathcal{W}^{s}$, $f\in\mathcal{C}^{1}(\bar{M}_{0})$ and $\psi\in\mathcal{C}^{p}(W)$ with $|\psi|_{\mathcal{C}^{p}(W)}\leq 1$. Then,

[TABLE]

Following [15], we split $\bar{T}_{\omega}^{-1}W$ and $\bar{T}_{0}^{-1}W$ into matched pieces (which can be connected by a transverse foliation of unstable curves) and unmatched pieces which cannot. On the unmatched pieces, we use (16) to note that the estimate [15, eq. (5.3)] remains identical. Similarly, since matched pieces lie in the same connected component of $\bar{M}_{0}\setminus(\mathcal{S}_{1}^{\bar{T}_{\omega}}\cup\mathcal{S}_{1}^{\bar{T}_{0}})$, we have $\Phi_{\omega}=\Phi_{0}$ on such components. Thus factoring out the common constant $e^{iu\cdot\Phi_{0}}$ from the difference on matched pieces, we see that the estimate [15, eq. (5.4)] holds without any changes. Thus the final estimate [15, eq. (5.9)] holds with the same constants, independently of $u\in\mathbb{R}^{2}$. ∎

Proof of Proposition 3.10.

This comes directly from Lemmas 3.11 and 3.7. Indeed, for every $f\in\widetilde{\mathcal{B}}$, we have

[TABLE]

since $\bar{T}_{\omega}\in\bar{\mathcal{F}}_{\theta_{0}}(\bar{T}_{0})$. ∎

Lemma 3.12**.**

${\mathcal{P}}$* is quasicompact, 1 its is only dominating eigenvalue and it is a simple eigenvalue (with eigenspace $\mathbb{C}.\bar{\mu}$). In particular, there exists $\tilde{C}>0$ and $\tilde{\alpha}\in(0,1)$ such that*

[TABLE]

Proof.

Due to [15, Theorem 2.2 and Corollary 2.4]) $\mathcal{L}_{0}$ is quasicompact, 1 is its only dominating eigenvalue and it is a simple eigenvalue (with eigenspace $\mathbb{C}.\mathbf{1}_{\bar{M}_{0}}$). In particular, there exists $\tilde{C}>0$, $\tilde{\alpha}_{0}\in(0,1)$ such that

[TABLE]

Let $f\in\widetilde{\mathcal{B}}$. Observe that

[TABLE]

and that

[TABLE]

First, setting $\tilde{\omega}_{n}=(\omega_{-n},\ldots,\omega_{-1})$, we have, using Lemma 3.3(a),

[TABLE]

since $\mathbf{1}_{\bar{M}_{0}}$ is in $\mathcal{B}$ and $\mathbb{E}_{\bar{\mu}_{0}}[\cdot]$ is in the dual of $\mathcal{B}$ by Remark 3.1.

Second, for every $\underline{\omega}$ and $\underline{\omega}^{\prime}$ in $E^{\mathbb{N}}$, we have

[TABLE]

This proves the lemma with $\tilde{\alpha}=\max\{\tilde{\alpha}_{0},\varkappa^{-1}\}$. ∎

3.4. Doeblin-Fortet-Lasota-Yorke type inequality for $P_{u}$

We next establish the spectral properties of $P$ and $P_{u}$ on $\widetilde{\mathcal{B}}$.

Proposition 3.13**.**

If $\vartheta_{0}$ is small enough, there exist $\tilde{C}>0$ and $\tilde{\tau}\in(0,1)$, such that for every $n\geq 1$, $f\in\widetilde{\mathcal{B}}$, $u\in\mathbb{R}^{2}$ and $n\geq 0$,

[TABLE]

This result will follow directly from the next lemma.

Lemma 3.14**.**

If $\vartheta_{0}$ is small enough, there exist $C>0$ and $\tau\in(0,1)$, such for every $n\geq 1$, $\varepsilon_{1},...,\varepsilon_{n}\in E$, $f\in\mathcal{B}$, $u\in\mathbb{R}^{2}$ and $n\geq 0$,

[TABLE]

Proof.

Here we denote $\mathcal{L}_{u,\underline{\omega}}^{n}:=\mathcal{L}_{u,\omega_{1}}\cdots\mathcal{L}_{u,\omega_{n}}$, and $\bar{T}^{n}_{\underline{\omega}}=\bar{T}_{\omega_{n}}\circ\cdots\circ\bar{T}_{\omega_{1}}$. The above Lasota-Yorke inequalities are proved999The estimates in [15, Proposition 5.6] include a factor $\eta\geq 1$, which comes from the Jacobian of $\bar{T}_{\omega}$ with respect to $\bar{\mu}_{0}$. Since we have assumed that $J_{\bar{\mu}_{0}}\bar{T}_{\omega}=1$ in our simplified version of (H5), we have $\eta=1$ in the present setting. Also note that the density function $g$ for the random perturbation in [15] is identically 1 in our setting as well. for $\mathcal{L}^{n}_{\underline{\omega}}$ as long as each $\bar{T}_{\omega_{k}}\in\bar{\mathcal{F}}$ by [15, Proposition 5.6], with $\underline{\omega}=(\omega_{k})_{k\geq 1}$. We must show that the constants appearing in the inequalities are independent of $u\in\mathbb{R}^{2}$, and all $\underline{\omega}\in E^{\mathbb{N}}$. We will use the fact that $S_{n}\Phi_{\underline{\omega}}$ is constant on elements of $\bar{M}_{0}\setminus\mathcal{S}_{n}^{\bar{T}_{\underline{\omega}}}$.

For $f\in\mathcal{B}$, $W\in\mathcal{W}^{s}$ and appropriate test functions $\psi$, we must estimate expressions of the form,

[TABLE]

where $\mathcal{G}_{n}(W)$ are the components of $\bar{T}_{\underline{\omega}}^{-n}W$, subdivided so that they each belong to $\mathcal{W}^{s}$.

For example, for the weak norm estimate,

[TABLE]

which is the same as [15, eq. (4.5)], except that the test function is $e^{iu\cdot S_{n}\Phi_{\underline{\omega}}}\psi\circ\bar{T}_{\underline{\omega}}^{n}$ rather than simply $\psi\circ\bar{T}_{\underline{\omega}}^{n}$. (See also the generalization of this inequality for the random sequence of maps in [15, eq. (5.26)].) But since $e^{iu\cdot S_{n}\Phi_{\underline{\omega}}}$ is constant on each $W_{i}$, we have

[TABLE]

so that the estimate is precisely the same and independent of $u$, as (H1) and (H5) are used for $|J_{W_{i}}\bar{T}_{\underline{\omega}}^{n}|_{\mathcal{C}^{p}(W_{i})}$ to get the uniform bound independent of ${\underline{\omega}}$.

The same observation is true for the strong norm estimates for precisely the same reason. For the unstable norm estimate, we must compare values of test functions on two stable curves $W^{1},W^{2}$ that lie close together. But the ‘matched’ pieces101010These are curves $U^{1}_{j}\in\mathcal{G}_{n}(W^{1})$ and $U^{2}_{j}\in\mathcal{G}_{n}(W^{2})$ that can be connected by a transverse foliation of unstable curves. See [15, Section 4.3] for a precise definition. of $\bar{T}_{\underline{\omega}}^{-n}W^{1}$ and $\bar{T}_{\underline{\omega}}^{-n}W^{2}$ lie in the same component of $\bar{M}_{0}\setminus\mathcal{S}_{n}^{\bar{T}_{\underline{\omega}}}$ so that $S_{n}\Phi_{\underline{\omega}}$ is the same constant on both curves and does not affect any of the constants appearing in the Lasota-Yorke inequalities. For the ‘unmatched’ pieces of $\bar{T}_{\underline{\omega}}^{-n}W^{1}$ and $\bar{T}_{\underline{\omega}}^{-n}W^{2}$, the estimate is precisely the same as in [15, eq. (4.14)] due to (16). ∎

Proof of Proposition 3.13.

Observe that

[TABLE]

Due to Lemma 3.7 and to the first inequality of Lemma 3.14, for any $f\in\widetilde{\mathcal{B}}_{w}$ and $n\geq 1$,

[TABLE]

Analogously, using again Lemma 3.7 and, this time, the second inequality of Lemma 3.14, we obtain, for any $f\in\widetilde{\mathcal{B}}$ and $n\geq 1$,

[TABLE]

Finally, using Lemma 3.7,

[TABLE]

since $\varkappa>\sup_{\omega\in E}\|\mathcal{L}_{\omega}\|_{L(\mathcal{B},\mathcal{B})}$, we obtain that $P_{u}$ satisfies Doeblin-Fortet-Lasota-Yorke conditions for $(\widetilde{\mathcal{B}}$ and $\widetilde{\mathcal{B}}_{w})$. ∎

3.5. Quasicompactness of $P$ and of $P_{u}$

Proposition 3.15**.**

If $\vartheta_{0}$ and $\varkappa^{-1}$ are small enough, ${P}$ is quasicompact on $\widetilde{\mathcal{B}}$, 1 its is only dominating eigenvalue and it is a simple eigenvalue (with eigenspace $\mathbb{C}.\mathbf{1}_{\bar{M}}$). In particular, there exists $\tilde{C}>0$ and $\tilde{\alpha}\in(0,1)$, such that

[TABLE]

Proof.

This comes from the Keller-Liverani perturbation theorem [24, Corollary 1] thanks to Lemma 3.12 and Propositions 3.10 and 3.13. Observe moreover that, since $P$ is the dual operator of $f\mapsto f\circ\bar{T}$, the spectral radius of $P$ is 1 and 1 is an eigenvalue of $P$. We conclude that 1 is the dominating eigenvalue and that it is simple. ∎

Proposition 3.16**.**

$P_{u}$, as an operator acting on $\widetilde{\mathcal{B}}$, is an analytic perturbation of $P$.

Proof.

Observe that the $n$-th derivative of $u\mapsto P_{u}$ is the operator defined by

[TABLE]

Due to Lemma 3.9 and to classical results on analytic functions, we conclude that, in $L(\widetilde{\mathcal{B}},\widetilde{\mathcal{B}})$, $u\mapsto P_{u}$ is analytic on $\mathbb{R}^{2}$ and that

[TABLE]

where $\mathcal{A}_{n}f(u)$ is $n$-linear in $u$. ∎

Our main results will follow from the following technical result.

Theorem 3.17**.**

*The function $\mathbf{1}_{\bar{M}}$ is in $\widetilde{\mathcal{B}}$ and $\mathbb{E}_{\bar{\mu}}[\cdot]$ is a continuous linear form on $\widetilde{\mathcal{B}}$ and $\widetilde{\mathcal{B}}_{w}$.

If $\vartheta_{0}$ and $\varkappa^{-1}$ are small enough, there exist $\beta\in(0,\pi)$, $C>0$ and $\alpha\in(0,1)$, three analytic maps $u\mapsto\lambda_{u}$ from $[-\beta,\beta]^{2}$ to $\mathbb{C}$, $u\mapsto N_{u}$ and $u\mapsto\Pi_{u}$ from $[-\beta,\beta]^{2}$ to $L(\widetilde{\mathcal{B}},\widetilde{\mathcal{B}})$ such that*

• a)

$\lambda_{0}=1$, $\Pi_{0}:=\mathbb{E}_{\bar{\mu}}[\cdot]\mathbf{1}_{\bar{M}}$,

• b)

for every $u\in[-\beta,\beta]^{2}$ and every integer $n\geq 1$, $P_{u}^{n}=\lambda_{u}^{n}\Pi_{u}+N_{u}^{n}$, $\Pi_{u}N_{u}=N_{u}\Pi_{u}=0$, $\Pi_{u}^{2}=\Pi_{u}$, and $\|N_{u}^{n}\|_{L(\widetilde{\mathcal{B}},\widetilde{\mathcal{B}})}\leq C\alpha^{n}$.

Moreover, for every integer $k\geq 0$, $\|(N_{u}^{n})^{(k)}\|_{L(\widetilde{\mathcal{B}},\widetilde{\mathcal{B}})}=O(\alpha^{n})$, where $(N_{u}^{n})^{(k)}$ means the $k$-th derivative of $N_{u}^{n}$.

• c)

for every $u\in[-\pi,\pi]^{2}\setminus[-\beta,\beta]^{2}$ and every integer $n\geq 1$, we have $\|P_{u}^{n}\|_{L(\widetilde{\mathcal{B}},\widetilde{\mathcal{B}})}\leq C\alpha^{n}$.

• d)

There exists a positive symmetric matrix $\Sigma^{2}$ such that $\lambda_{u}=1-\frac{1}{2}(\Sigma^{2}u\cdot u)+O(|u|^{3})$.

Proof of Theorem 3.17.

The fact that $\mathbf{1}_{\bar{M}}$ is in $\widetilde{\mathcal{B}}$ comes from the fact that $\mathbf{1}_{\bar{M}_{0}}$ is in $\mathcal{B}$.

As seen in Remark 3.5, $\mathbb{E}_{\bar{\mu}}[\cdot]$ is a continuous form on $\widetilde{\mathcal{B}}$. The proof of the remaining part of the theorem relies on Propositions 3.10, 3.13 and 3.15 and 3.16.

Propositions 3.13, 3.15 and 3.16 immediately imply the existence of a spectral gap for $P_{u}$ for $|u|$ sufficiently small, using standard perturbation theory [23]. This yields the analyticity and items (a) and (b) of the proposition with $\beta$ depending on $\vartheta_{0}$ and the uniform constants depending on the family $\bar{\mathcal{F}}_{\vartheta_{0}}$, but not on the probability measure $\eta$.

For item (c), due to [1, Lemma 4.3], it is enough to prove that, if $\vartheta_{0}$ is small enough, then for every $u\in[-\pi,\pi]^{2}\setminus[-\beta,\beta]^{2}$, $P_{u}$ admits no eigenvalue of modulus 1. Assume the contrary. There would exist a sequence of operators $(P_{u_{k}}^{(k)})_{k}$ corresponding to a sequence of vanishing neighbourhoods $(E_{k})_{k}$ of $\bar{T}_{0}$ in $\mathcal{F}$ and with $\beta\leq|u_{k}|\leq\pi$ and $\rho(P_{u_{k}}^{(k)})=1$, where $\rho(\cdot)$ denotes the spectral radius. Up to extracting a subsequence, we also have $\lim_{k\rightarrow+\infty}u_{k}=u_{\infty}$. But, due to Proposition 3.10 and since $u\mapsto\mathcal{L}_{u,0}$ is continuous from $\mathbb{R}^{2}$ to $L(\mathcal{B},\mathcal{B})$, we would deduce that

[TABLE]

Combining this with Proposition 3.13 and with the perturbation theorem of [24], this would imply that $\rho(\mathcal{P}_{u_{\infty}})=1$, which would contradict Proposition C.2. We conclude that, as soon as $\vartheta_{0}$ is sufficiently small, $\sup_{\beta\leq|u|\leq\pi}\rho(P_{u})<1$ as claimed.

It remains to prove item (d). Due to [15, Corollary 2.4], for any initial probability measure $\nu\in\mathcal{B}$, $(S_{n}/\sqrt{n})_{n}$ converges in distribution to a (possibly generalized) centered Gaussian random variable with variance $\Sigma^{2}$. Moreover, due to item (b) of the present theorem,

[TABLE]

and so

[TABLE]

with uniform convergence on any compact set of $\mathbb{R}^{2}$. This implies that

[TABLE]

On the other hand, $\log(\lambda_{t/\sqrt{n}})\sim(\lambda_{t/\sqrt{n}}-1)$ as $n\rightarrow+\infty$. Hence

[TABLE]

Setting $u=t/\sqrt{n}$, we can then deduce the stated Taylor expansion since $u\mapsto\lambda_{u}$ is analytic. The positivity of $\Sigma^{2}$ follows from the next lemma. ∎

Lemma 3.18**.**

If $\vartheta_{0}$ is small enough, $\Sigma^{2}$ is positive.

Proof.

Recall that $\Sigma^{2}$ has been defined in (6). We consider $\Sigma_{0}^{2}$ being defined by

[TABLE]

It is enough to prove that $\Sigma^{2}$ converges to $\Sigma_{0}^{2}$ as $\vartheta_{0}$ goes to [math]. We use (6) together with the fact that $\Sigma^{2}_{0}$ satisfies an analogous formula (with $\Phi(x,\underline{\omega})$ replaced by $\Phi_{0}(x)$ and with $\bar{T}(x,\underline{\omega})$ replaced by $\bar{T}_{0}(x)$. Therefore

[TABLE]

with $A_{k}:=\mathbb{E}_{\bar{\mu}}\left[\Phi.\Phi\circ\bar{T}^{k}]-\mathbb{E}_{\bar{\mu}_{0}}[\Phi_{0}.\Phi_{0}\circ\bar{T}_{0}^{k}\right].$ Extending the definition of $\Phi_{0}$ on $\bar{M}$ by setting $\Phi_{0}(x,\underline{\omega}):=\Phi_{0}(x)$, we obtain

[TABLE]

The two first terms of the right hand side of this formula are less than

[TABLE]

which goes to 0 as $\vartheta_{0}\rightarrow 0$. The third term is dominated by

[TABLE]

We deduce that this quantity goes to 0 using Remark 3.4, Lemma 3.9, and Proposition 3.10, and since $\mathbb{E}_{\bar{\mu}}[\Phi_{0}\cdot]=\mathbb{E}_{\bar{\mu}}[\mathcal{P}(\Phi_{0}\cdot)]$ (applying Lemma 3.9 with $E=\{0\}$).

We conclude with the use of the dominated convergence theorem, since

[TABLE]

where we used Proposition 3.15 since $\mathbb{E}_{\bar{\mu}}[\Phi]=\mathbf{0}$, and a similar bound holds for $\mathbb{E}_{\bar{\mu}_{0}}[\Phi_{0}.\Phi_{0}\circ\bar{T}_{0}^{k}]$. ∎

4. Limit theorems under general assumptions

We start with the proof of our results which are direct consequences of Proposition 3.17 and of general results existing in the literature.

Proofs of Theorems 2.1, 2.7 and 2.8.

Theorem 2.1 is a direct corollary of Theorem 3.17 by standard arguments (see [26, 19, 20]) since $\mathbb{E}_{\bar{\mu}}\left[e^{iu\cdot S_{n}}\right]=\mathbb{E}_{\bar{\mu}}\left[P_{u}^{n}\mathbf{1}_{\bar{M}}\right]$.

Due to Theorem 3.17, our dynamical system $(\bar{M},\bar{\mu},\bar{T})$ satisfies the general assumptions of [33, Theorem 1.4] and [31, Theorem 3.2, Remark 3.3] and so Theorems 2.7 and 2.8 follow. ∎

We will prove the other results in a general context. About these results, let us mention that Theorem 2.4 and the first part of Theorem 2.6 have been proved in [17, 28] and in [29] for a single billiard map. We give here the proof in a more general context with a significant simplification in the proof of Theorem 2.6 due to the better estimate of the variance of the auto-intersection and to some simplification in the Bolthausen tightness argument. The second part of Theorem 2.6 uses a general argument from [18]. Theorem 2.5 exists for a single billiard map, but only in an unpublished paper by the second author [30]. Let us indicate that the generality of the proof we give in the present paper is possible due to important modifications of the proof (we do not assume that $\Phi$ is bounded, nor that the Banach space we consider is continuously injected in $L^{p}$ for a suitable $p>1$; both of these conditions were used in [30]).

We will prove the limit theorems we are interested in under the following general hypothesis.

Assumption 4.1**.**

*Let $(M,\mu,T)$ be a $\mathbb{Z}^{2}$-extension of a probability preserving dynamical system $(\bar{M},\bar{\mu},\bar{T})$ by a function $\Phi:\bar{M}\rightarrow\mathbb{C}$. Let $P$ be the transfer operator associated with $\bar{T}$ with respect to $\bar{\mu}$ and let $(P_{u}:=P(e^{iu\cdot\Phi}\cdot))_{u\in\mathbb{R}^{2}}$. We assume that these operators act on two Banach spaces $\widetilde{\mathcal{B}}_{1}$ and $\widetilde{\mathcal{B}}_{2}$ such that $\mathbf{1}_{\bar{M}}\in\widetilde{\mathcal{B}}_{1}\hookrightarrow\widetilde{\mathcal{B}}_{2}$ (continuous inclusion) and that $\mathbb{E}_{\bar{\mu}}[\cdot]$ is a continuous linear form111111up to extending by continuity the definition of $\mathbb{E}_{\bar{\mu}}[\cdot]$ on $\widetilde{\mathcal{B}}_{2}$.

Assume that there exist $\beta\in(0,\pi)$, $C>0$ and $\alpha\in(0,1)$, three continuous maps $u\mapsto\lambda_{u}$ from $[-\beta,\beta]^{2}$ to $\mathbb{C}$, $u\mapsto N_{u}$ and $u\mapsto\Pi_{u}$ from $[-\beta,\beta]^{2}$ to $L(\widetilde{\mathcal{B}}_{1},\widetilde{\mathcal{B}}_{2})$ such that*

• (A1)

for every $u\in[-\beta,\beta]^{2}$ and every integer $n\geq 1$,

[TABLE]

and $\|N_{u}^{n}\|_{L(\widetilde{\mathcal{B}}_{1},\widetilde{\mathcal{B}}_{1})}\leq C\alpha^{n}$.

• (A2)

for every $u\in[-\pi,\pi]^{2}\setminus[-\beta,\beta]^{2}$ and every integer $n\geq 1$, we have $\|P_{u}^{n}\|_{L(\widetilde{\mathcal{B}}_{1},\widetilde{\mathcal{B}}_{1})}\leq C\alpha^{n}$.

• (A3)

$u\mapsto\Pi_{u}$, seen as a $L(\widetilde{\mathcal{B}}_{1},\widetilde{\mathcal{B}}_{2})$-valued function, is differentiable at 0 and that $\Pi_{0}:=\mathbb{E}_{\bar{\mu}}[\cdot]\mathbf{1}_{\bar{M}}$,

• (A4)

There exists a positive symmetric matrix $\Sigma^{2}$ such that $\lambda_{u}=1-\frac{1}{2}(\Sigma^{2}u\cdot u)+O(|u|^{3})$.

We will also use the following notation and considerations. We write $S_{n}$ for the ergodic sum $S_{n}:=\sum_{k=0}^{n-1}\Phi\circ\bar{T}^{k}$. It will be crucial to notice that $P_{u}^{n}=P^{n}(e^{iu\cdot S_{n}}\cdot)$.

We consider a partition of $\bar{M}$ in $I$ subsets $\bar{O}_{1},...,\bar{O}_{I}$ of $\bar{\mu}$ positive measure (corresponding to $(\partial O_{i}\times S^{1})\times E^{\mathbb{N}}$ in our example). We consider the function $\mathcal{I}_{0}$ which, at every $x\in\bar{M}$, associates the index $\mathcal{I}_{0}(x)$ of the atom $\bar{O}_{\mathcal{I}_{0}(x)}$ of the partition containing $x$. We also define $\mathcal{I}_{k}:=\mathcal{I}_{0}\circ\bar{T}^{k}$.

We remark that our random map $\bar{T}$ with $\bar{T}_{\omega}\in\bar{\mathcal{F}}_{\vartheta_{0}}(\bar{T}_{0})$ for all $\omega\in E$ satisfies all the items of Assumption 4.1 due to Theorem 3.17.

4.1. Proof of the Local Limit Theorem – Theorem 2.2

For every $n\in\mathbb{N}^{*}$, $\ell\in\mathbb{Z}^{2}$ and $h\in\widetilde{\mathcal{B}}_{1}$, we set:

[TABLE]

Recall that

[TABLE]

where $du$ is understood as $du_{1}du_{2}$ for $u=(u_{1},u_{2})\in\mathbb{R}^{2}$ (integral with respect to the Lebesgue measure), which leads us to the following formula

[TABLE]

Theorem 4.2**.**

Assume Assumptions 4.1. Then

[TABLE]

Moreover, there exists $K_{0}\geq 1$ such that for every integer $n\geq 0$ and every $\ell\in\mathbb{Z}^{2}$,

[TABLE]

Proof.

Up to a change of $\beta$, there exists $a>0$ such that, for every $u\in[-\beta,\beta]^{2}$, $|\lambda_{u}|\leq\exp(-a|u|^{2})$. Hence, using Assumption 4.1, we have the following equalities in $L(\widetilde{\mathcal{B}}_{1},\widetilde{\mathcal{B}}_{2})$:

[TABLE]

where we have changed variables, $v=u\sqrt{n}$, and the $O$ are in $L(\widetilde{\mathcal{B}}_{1},\widetilde{\mathcal{B}}_{2})$ with uniform bound. This bound is in $L(\widetilde{\mathcal{B}}_{1},\widetilde{\mathcal{B}}_{2})$ and not in $L(\widetilde{\mathcal{B}}_{1},\widetilde{\mathcal{B}}_{1})$ because according to Assumption (A3), the map $u\mapsto\Pi_{u}$ is differentiable from $[-\beta,\beta]$ to $L(\widetilde{\mathcal{B}}_{1},\widetilde{\mathcal{B}}_{2})$ and a priori not from $[-\beta,\beta]$ to $L(\widetilde{\mathcal{B}}_{1},\widetilde{\mathcal{B}}_{1})$.

For the second estimate, we write

[TABLE]

using again the change of variable $v=u\sqrt{n}$. ∎

Due to Theorem 3.17, Theorem 2.2 is a direct consequence of the following result.

Corollary 4.3**.**

Assume Assumption 4.1. Let $f,g:\bar{M}\rightarrow\mathbb{R}$ such that $H_{g}(\cdot):=\mathbb{E}_{\bar{\mu}}[g\cdot]\in\widetilde{\mathcal{B}}^{\prime}_{2}$ and such that $f\in\widetilde{\mathcal{B}}_{1}$. Then

[TABLE]

Proof.

Observe that we have

[TABLE]

recalling (7). We conclude due to Theorem 4.2. ∎

4.2. Return time to the original obstacle: Proof of Theorem 2.4

Recall that $\mathcal{I}_{k}(x)$ corresponds to the index of the atom $\bar{O}_{\mathcal{I}_{k}(x)}$ containing $\bar{T}^{k}x$ and that $S_{n}(x)$ corresponds to the label of the copy of $\bar{M}$ in $M$ containing $T^{k}(x,\mathbf{0})$. We also define $\mathcal{I}_{k}$ on $\bar{M}$ by canonical projection. We consider the set $B_{n}$ of $x\in\bar{M}$ such that the orbit $(T^{n}(x,\mathbf{0}))_{n\geq 0}$ won’t return to the initial atom $\bar{O}_{\mathcal{I}_{0}(x)}\times\{\mathbf{0}\}$ until time $n$:

[TABLE]

Analogously we define $B^{\prime}_{n}$ the set of points $x\in\bar{M}$ for which the atom visited at time $n$ has not been visited before:

[TABLE]

We set $B_{n}(a):=\bar{O}_{a}\cap B_{n}$ and $B^{\prime}_{n}(a):=\bar{T}^{-n}(\bar{O}_{a})\cap B_{n}$. We prove the following result on the probability of these sets.

Proposition 4.4**.**

*Assume Assumption 4.1.

If $\mathbf{1}_{\bar{O}_{a}}\in\widetilde{\mathcal{B}}_{1}$ and if $f\mapsto\mathbb{E}_{\bar{\mu}}[f\mathbf{1}_{B_{k}(a)}]$ are uniformly bounded (uniformly in $k$) in $\widetilde{\mathcal{B}}^{\prime}_{2}$, then

[TABLE]

If $f\mapsto\mathbb{E}_{\bar{\mu}}[f\mathbf{1}_{\bar{O}_{a}}]$ is in $\widetilde{\mathcal{B}}_{2}^{\prime}$ and if $P^{k}\mathbf{1}_{B^{\prime}_{k}(a)}$ are uniformly bounded (uniformly in $k$) in $\widetilde{\mathcal{B}}_{1}$, then

[TABLE]

Proof.

As in [28], we follow the idea of the proof of Dvoretzy and Erdös and adapt it to our context. Considering the last visit time to $\bar{O}_{a}\times\{\mathbf{0}\}$ of $(T^{k}(x,\mathbf{0}))$ until time $n$, we write

[TABLE]

and, analogously,

[TABLE]

considering the first visit time to $\bar{O}_{a}\times\{S_{n}\}$ before time $n$. Moreover, due to Corollary 4.3 and to our assumptions on $\bar{O}_{a}$ and on $B_{n}(a)$, there exists $C">0$ such that

[TABLE]

and, since $\bar{\mu}\left((\bar{T}^{-n}\bar{O}_{a})\cap\{S_{n}-S_{n-k}=\mathbf{0}\}\cap B^{\prime}_{n-k}(a)\right)=\mathbb{E}_{\bar{\mu}}\left[\mathbf{1}_{\bar{O}_{a}}P^{k}\left(\mathbf{1}_{\{S_{k}=\mathbf{0}\}}P^{n-k}\mathbf{1}_{B^{\prime}_{n-k(a)}}\right)\right]$. Due to Theorem 4.2, we also have

[TABLE]

We will prove (24) using (26) and (28). The proof of (25) using (27) and (29) follows the same scheme, and we omit it.

[TABLE]

with $m_{n}=\lceil(\log n)^{2}\rceil$, which leads to

[TABLE]

Moreover

[TABLE]

where we used the facts that $\bar{\mu}(B_{\lceil n\log n\rceil-k}(a))\leq\bar{\mu}(B_{n}(a))=O((\log n)^{-1})$ for every $k\leq\lceil n\log n\rceil-n$ and that $\bar{\mu}(B_{m}(a))\leq 1$ for $k>\lceil n\log n\rceil-n$. This leads us to

[TABLE]

by taking $m^{\prime}_{n}:=\lfloor(\log n)^{\frac{2}{3}}\rfloor$ and so

[TABLE]

The proposition follows from (30) and (31). ∎

Lemma 4.5**.**

There exists $K_{1}>0$ such that, for every positive integer $\ell$, for every $(\omega_{1},...,\omega_{\ell})\in E^{\ell}$, for every uniformly bounded function $g:\bar{M}_{0}\rightarrow\mathbb{R}$ which is uniformly $p$-Hölder continuous on connected components of $\bar{M}_{0}\setminus\left(\cup_{k=1}^{\ell}\bar{T}_{\omega_{1}}^{-1}\circ\cdots\circ\bar{T}_{\omega_{k}}^{-1}(\mathcal{S}_{0,H})\right)$, and for all $f\in\mathcal{B}_{w}$,

[TABLE]

Moreover, for every $f\in\mathcal{B}$,

[TABLE]

where $\mathcal{C}_{\omega_{1},...,\omega_{\ell}}$ is the set of connected components of $\bar{M}_{0}\setminus\left(\cup_{k=1}^{\ell}\bar{T}_{\omega_{1}}^{-1}\circ\cdots\circ\bar{T}_{\omega_{k}}^{-1}(\mathcal{S}_{0,H})\right)$ and where $C_{g_{|C}}^{(p)}$ is the Hölder constant of $g$ restricted to $C$.

The proof of Lemma 4.5 can be found in Appendix A.

Remark 4.6**.**

The purpose of Lemma 4.5 is to show that $K_{1}$ can be chosen independently of $\ell$. If one wishes similar bounds on piecewise Hölder continuous functions on $M_{0}$ with respect to a fixed partition, then Remark 3.1 and Lemma 3.3 provide such estimates under general conditions on the boundaries of partition elements.

Indeed, we will apply the lemma to the function $g=\mathbf{1}_{B_{n}(a)}$, where $B_{n}(a)$ is defined in Section 2.2 (see also Section 4.2).

Next we are ready to prove the main Theorem 2.4.

Proof of Theorem 2.4.

Assumption 4.1 follows from Theorem 3.17. The other assumptions of Proposition 4.4 follow from Lemma 4.5 since $\mathbf{1}_{B_{n}(a)}$ satisfies the assumptions on $g$ in that lemma (uniformly in $n$). ∎

4.3. Number of self-intersections: Proof of Theorem 2.5

We consider the number of self-intersections $\mathcal{V}_{n}$ of the process $(\mathcal{I}_{k},S_{k})_{k}$ defined by

[TABLE]

Theorem 4.7**.**

Assume Assumptions 4.1 with $\widetilde{\mathcal{B}}_{2}=\widetilde{\mathcal{B}}_{1}$. Assume moreover:

• (A5)

the operator $f\mapsto f\mathbf{1}_{\bar{O}_{a}}$ is a linear operator on $\widetilde{\mathcal{B}}_{1}$ for every $a\in\{1,\ldots,I\}$.

Then $(\mathcal{V}_{n}/(n\log n))_{n}$ converges $\bar{\mu}$-almost surely to $\frac{1}{\pi\sqrt{\det\Sigma^{2}}}\sum_{a=1}^{I}\bar{\mu}(\mathcal{I}_{0}=a)^{2}$.

The proof of Theorem 2.5 will follow from the following lemmas. Recalling (34), let us write $E_{k,\ell}:=\{S_{k}=S_{\ell},\mathcal{I}_{k}=\mathcal{I}_{\ell}\}$ and $E_{\ell}:=E_{0,\ell}$.

Lemma 4.8**.**

For $\ell>k$, we have

[TABLE]

and so

[TABLE]

with $c_{1}:=\frac{1}{2\pi\sqrt{\det\Sigma^{2}}}\sum_{a=1}^{I}\bar{\mu}(\mathcal{I}_{0}=a)^{2}$.

Proof.

Since $\bar{\mu}$ is $\bar{T}$-invariant, for $k<\ell$, recalling (18) we have

[TABLE]

due to Theorem 4.2 since $\mathbf{1}_{\bar{O}_{a}}\in\widetilde{\mathcal{B}}_{1}$ and since $\mathbb{E}_{\bar{\mu}}[\mathbf{1}_{\bar{O}_{a}}\cdot]\in\widetilde{\mathcal{B}}^{\prime}_{1}$. Hence

[TABLE]

and

[TABLE]

∎

Lemma 4.9**.**

There exists $C_{1}>0$ such that for all non-negative integers $n,m,k$, for all $i,j,i^{\prime},j^{\prime}\in\{1,...,I\}$, and for all $N_{1},N_{2}\in\mathbb{Z}^{2}$, we have

[TABLE]

In particular

[TABLE]

Proof.

The covariance we are interested in can be rewritten

[TABLE]

[TABLE]

Moreover, using several times $P^{m}(f\,g\circ\bar{T}^{m})=g\,P^{m}(f)$ and the definition of $\mathcal{H}_{\ell,n}$, we obtain that this quantity is equal to

[TABLE]

and so is bounded by

[TABLE]

due to (21) and assumption (A5) of Theorem 4.7, together with (A1) of Assumptions 4.1 applied to $u=0$. Here $a_{i}:=\|\mathbf{1}_{\bar{O}_{i}}\times\cdot\|_{L(\widetilde{\mathcal{B}}_{1},\widetilde{\mathcal{B}}_{1})}$.

This gives the first estimate of the lemma. To get the second one from the first one, we just observe that

[TABLE]

∎

We will use the notation $A_{n}\sim B_{n}$ for two positive quantities whenever $\lim_{n\to\infty}\frac{A_{n}}{B_{n}}=1$.

Lemma 4.10**.**

We have $Var_{\bar{\mu}}(\mathcal{V}_{n})\sim cn^{2}\,,$ with

[TABLE]

[TABLE]

The proof of Lemma 4.10 is rather technical and involved, so we move it to the appendix B.

Proof of Theorem 4.7.

Set $n_{k}:=\exp(\sqrt{k}\log k)$. For every $\varepsilon>0$, due to the Bienaymé-Chebychev inequality and using Lemmas 4.8 and 4.10,

[TABLE]

Hence $(\mathcal{V}_{n_{k}}/\mathbb{E}_{\bar{\mu}}[\mathcal{V}_{n_{k}}])_{k}$ converges $\bar{\mu}$-almost surely to 1. Due to Lemma 4.8, $(\mathcal{V}_{n_{k}}/(n_{k}\log n_{k}))_{k}$ converges almost surely to $2c_{1}$. Since $n_{k}\log n_{k}\sim n_{k+1}\log n_{k+1}$ and since $(\mathcal{V}_{n})_{n}$ is increasing, if $n\in\{n_{k},...,n_{k+1}\}$, then

[TABLE]

and so $(\mathcal{\mathcal{V}}_{n}/(n\log n))_{n}$ converges $\bar{\mu}$-almost surely to $2c_{1}$. ∎

Proof of Theorem 2.5.

Due to Remark 3.5, Theorem 3.17 and to Lemma 4.5, the assumptions of Theorem 4.7 are satisfied. Therefore $(\mathcal{V}_{n}/(n\log n))_{n}$ converges $\bar{\mu}$-almost surely to

[TABLE]

∎

4.4. Random scenery: Proof of Theorem 2.6

Assume that to each atom $\bar{O}_{i}\times\{\ell\}$ is associated a random variable $\xi_{i,\ell}$, independent and identically distributed across $i\in[1,\ldots I]$ and $\ell\in\mathbb{Z}^{2}$, centered with variance $\sigma^{2}_{\xi}$ and defined on a common probability space $(\Omega,\mathcal{F},\mathbb{P})$. We define the random variable (defined on $\bar{M}\times\Omega$):

[TABLE]

We also define a linearly interpolated version of $\mathcal{Z}_{n}$ by,

[TABLE]

Theorem 4.11** (Annealed and $\xi$-quenched CLT for $\mathcal{Z}$).**

Assume Assumption 4.1 and that,

• i)

for every $a\in\{1,...,I\}$, $f\mapsto\mathbf{1}_{\bar{O}_{a}}f$ is a continuous linear operator on $\widetilde{\mathcal{B}}_{1}$;

• ii)

and $\sup_{k\in\mathbb{N}}\|P^{k}\mathbf{1}_{B^{\prime}_{k}(a)}\|_{\widetilde{\mathcal{B}}_{1}}<\infty$ (recalling (23)).

Then, $(\widetilde{\mathcal{Z}}_{n}:=(\widetilde{\mathcal{Z}}_{n}(t)/\sqrt{n\log n})_{t>0})_{n}$ converges in distribution, with respect to $\bar{\mu}\otimes\mathbb{P}$ (and to the uniform norm on $[0,T]$ for every $T>0$), to a Brownian motion $B=(B_{t})_{t\geq 0}$ such that $\mathbb{E}[B_{1}^{2}]=\frac{\sigma^{2}_{\xi}}{\pi\sqrt{\det\Sigma^{2}}}\sum_{a=1}^{I}\bar{\mu}(\mathcal{I}_{0}=a)^{2}$.

If, moreover, there exists $\chi>0$ such that $\mathbb{E}[|\xi_{(1,0)}|^{2}(\log^{+}|\xi_{(1,0)}|)^{\chi}|)]<\infty$, then, for $\mathbb{P}$-a.e. realization of $(\xi_{i,\ell})_{i,\ell}$, $(\widetilde{\mathcal{Z}}_{n})_{n}$ converges also in distribution, with respect to $\bar{\mu}$, to the same Brownian motion $B$.

Proof of Theorem 2.6.

Using Theorem 4.11, we prove Theorem 2.6. Assumptions 5.1 hold in the setting of Theorem 2.6 due to Theorem 3.17. Moreover, assumption (i) of Theorem 4.11 follows from Remark 3.5, while assumption (ii) follows from Lemma 4.5. With the hypotheses of Theorem 4.11 verified, Theorem 2.6 follows using the same calculation as in (35). ∎

We proceed to prove Theorem 4.11.

For the annealed central limit theorem, we mostly follow the proof by Bolthausen for random walks in random scenery in dimension 2 [5]. In comparison with [29], the fact that the almost sure convergence of $\mathcal{V}_{n}$ has been proved greatly simplifies the proof.

Lemma 4.12**.**

Fix $\vartheta>0$. For $\bar{\mu}$-almost every $x\in\bar{M}$, $\sup_{\ell}\sum_{k=1}^{n}\mathbf{1}_{\{S_{k}=\ell\}}=o(n^{\vartheta})$.

Proof.

For every $\ell\in\mathbb{Z}^{2}$ and every $N\in\mathbb{N}^{*}$,

[TABLE]

due to Lemma 4.2. Moreover, due to Theorem 2.1, there exists $c^{\prime}>0$ such that $\mathbb{E}_{\bar{\mu}}[|S_{n}|^{2}]\sim c^{\prime}n$ and so due to a result by Billingsley (see [4] and [35])

[TABLE]

and so due to the Markov inequality, for every $s>0$, $\bar{\mu}\left(\max_{k=1,...,n}|S_{k}|>n^{1+s}\right)\leq\frac{\mathbb{E}_{\bar{\mu}}\left[\max_{k=1,...,n}|S_{k}|^{2}\right]}{n^{2+2s}}=O(n^{-1-s})\,.$ Now fix $\vartheta>0$. Then,

[TABLE]

where we used the inequality $\mathbb{E}[X>n^{\vartheta}]\leq\mathbb{E}[X^{N}]n^{-\vartheta N}$ for any $N\in\mathbb{N}^{*}$ combined with (36). Now choosing $N>(3+3\vartheta)/\vartheta$, we conclude the proof of the lemma by the Borel-Cantelli lemma. ∎

Recall that, for $x\in\bar{M}$, the random variable $\mathcal{Z}_{n}(x)$ can be rewritten: $\mathcal{Z}_{n}(x)=\sum_{k=1}^{n}\xi_{\mathcal{I}_{k},S_{k}}=\sum_{i=1}^{I}\sum_{\ell\in\mathbb{Z}^{2}}\xi_{i,\ell}\mathcal{N}_{n}(i,\ell)(x),$ where $\mathcal{N}_{n}(i,\ell)(x):=\sum_{k=1}^{n}\mathbf{1}_{\{S_{k}=\ell,\mathcal{I}_{k}=i\}}(x)$ is the number of visits to the obstacle of index $(i,\ell)$ up to time $n$ and where $(\xi_{i,\ell})_{i,\ell}$ is a sequence of i.i.d. centered square integrable random variables defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$.

Note that the variance of $\mathcal{Z}_{n}(x)$ (with respect to $\mathbb{P}$) is $\sigma^{2}_{\xi}\mathcal{V}_{n}(x)$, where $\sigma^{2}_{\xi}:=\mathbb{E}[\xi_{(1,\mathbf{0})}^{2}]$.

Lemma 4.13** (Convergence of finite-dimensional distributions).**

For every $m\geq 1$, every $0<t_{1}<t_{2}<...<t_{m}$, $\bar{\mu}$-almost surely, $\left(\sum_{j=1}^{m}a_{j}\left(\mathcal{Z}_{\lfloor nt_{j}\rfloor}-\mathcal{Z}_{\lfloor nt_{j-1}\rfloor}\right)(x)/\sqrt{n\log n}\right)_{n}$ converges in distribution (with respect to $\bar{\mu}\otimes\mathbb{P}$) to a centered Gaussian random variable with variance $2c_{1}\sigma^{2}_{\xi}\sum_{j=1}^{m}a_{j}^{2}(t_{j}-t_{j-1})$.

Proof.

We fix $x\in\bar{M}$. The variance of $\sum_{j=1}^{m}a_{j}\left(\mathcal{Z}_{\lfloor nt_{j}\rfloor}-\mathcal{Z}_{\lfloor nt_{j-1}\rfloor}\right)(x)$ (with respect to $\mathbb{P}$) is equal to, recalling (34),

[TABLE]

for $\bar{\mu}$-a.e. $x\in\bar{M}$, due to the proof of Theorem 2.5 (since $(\mathcal{V}_{n}/(n\log n))_{n}$ $\bar{\mu}$-converges almost surely to $2c_{1}$, as well as any sequence of random variables with the same marginal distributions).

Note that, with respect to $\mathbb{P}$, $\sum_{j=1}^{m}a_{j}\left(\mathcal{Z}_{\lfloor nt_{j}\rfloor}-\mathcal{Z}_{\lfloor nt_{j-1}\rfloor}\right)(x)$ is a sum of independent centered random variables with variances

[TABLE]

Hence, due to Lemma 4.12 and to the Lindeberg Theorem, for $\bar{\mu}$-almost every $x\in\bar{M}$, the sequence of random variables

[TABLE]

converges in distribution (with respect to $\mathbb{P}$) to a standard Gaussian random variable. The conclusion then follows from (37). ∎

Lemma 4.14**.**

The sequence of random variables $\left(\widetilde{\mathcal{Z}}_{n}(t)/\sqrt{n\log n}\right)_{n}$ is tight (with respect to $\bar{\mu}\otimes\mathbb{P}$) in $C([0,T])$ for every $T>0$.

Proof.

Due to Theorem 2.5, it is enough to prove the tightness of $\left(\widetilde{\mathcal{Z}}_{n}(t)/\sqrt{\sigma^{2}_{\xi}\mathcal{V}_{n}}\right)_{n}$. Due to [4, Lemma p. 88], it is enough to prove that

[TABLE]

We modify the proof of tightness of Bolthausen in [5]. For completeness, we explain the adaptations to make. Following [5] (see also [29, bottom of page 824], using the fact that $(\mathcal{Z}_{n})_{n}$ has positively associated increments knowing $(S_{n})_{n}$, we obtain that, for any $\lambda>\sqrt{2}$,

[TABLE]

Now we simplify the conclusion of [5]. Since we know that $(\mathcal{Z}_{n}/\sqrt{\mathcal{V}_{n}})_{n}$ converges in distribution to a Gaussian random variable $Y$, so

[TABLE]

and $\mathbb{P}\left(|Y|>x\right)=O(e^{-c_{Y}x^{2}})$ for some $c_{Y}>0$, which proves (38) and so the tightness. ∎

Proof of Theorem 4.11.

The first result of Theorem 4.11 is a direct consequence of Lemmas 4.13 and 4.14.

Now let us prove the last point. For this, we use the general argument developed by Guillotin-Plantard, Dos Santos and Poisat in [18]. Indeed the proof of [18] only uses the following assumptions:

• •

$\Gamma$ is a denumerable set,

• •

$\widetilde{S}:=(\widetilde{S}_{n})_{n\geq 0}$ is a sequence of $\Gamma$-valued random variables,

• •

$\xi:=(\xi)_{y\in\Gamma}$ is a sequence of independent identically distributed real valued random variables, which are centered and such that $\mathbb{E}[|\xi_{y}|^{2}(\log^{+}|\xi_{y}|)^{\chi}|)]<\infty$ for some $\chi>0$,

• •

the sequences of random variables $\xi$ and $\widetilde{S}$ are independent,

• •

$\left(\frac{1}{\sqrt{n\log n}}(\sum_{k=0}^{\lfloor nt\rfloor-1}\xi_{\widetilde{S}_{k}}+(nt-\lfloor nt\rfloor)\xi_{\widetilde{S}_{\lfloor nt\rfloor}})\right)_{t\in[0,1]}$ converges in distribution in $C(0,T)$ to the Brownian motion $B$,

• •

$\sup_{y\in\Gamma}\mathbb{E}[\widetilde{N}_{n}(y)]=O(\log n)$ with $\widetilde{N}_{n}(y):=\#\{k=0,...,n-1\,:\,\widetilde{S}_{k}=y\}=\sum_{k=0}^{n-1}\mathbf{1}_{\{\widetilde{S}_{k}=y\}}$ being the local time of $\widetilde{S}$.

• •

$\sum_{y\in\Gamma}(\mathbb{E}[(\widetilde{N}_{n}(y))])^{2}=O(n)$, with the same notation.

• •

$\mathbb{P}(\widetilde{S}_{n}\not\in\{\widetilde{S}_{0},...,\widetilde{S}_{n-1}\})=O((\log n)^{-1})$.

We apply this to $\Gamma=\{1,...,I\}\times\mathbb{Z}^{2}$ and $\widetilde{S}_{n}=(\mathcal{I}_{n},S_{n})$. For the antepenultimate condition, observe that, due to Corollary 4.3,

[TABLE]

For the penultimate condition,

[TABLE]

considering an independent copy $\widetilde{S}^{\prime}=(\widetilde{S}^{\prime}_{n}=(\mathcal{I}^{\prime}_{n},S^{\prime}_{n}))_{n}$ of $\widetilde{S}$. Now, using again (19) combined with Assumption 4.1 with $\beta$ and $a>0$ as in the proof of Theorem 4.2, we obtain

[TABLE]

Therefore

[TABLE]

The last condition comes from the second part of Proposition 4.4. Note that in order to invoke Proposition 4.4, we need that the operator $f\mapsto\mathbb{E}_{\bar{\mu}}[f\,\mathbf{1}_{\bar{O}_{a}}]$ is continuous on $\widetilde{B}_{2}^{\prime}$. This follows from the fact that we have assumed (i) in the statement of the theorem, that $f\mapsto f\mathbf{1}_{\bar{O}_{a}}$ is a continuous operator on $\widetilde{\mathcal{B}}_{1}$, and that by Assumption 4.1, $\mathbb{E}_{\bar{\mu}}[\cdot]$ acts continuously on $\widetilde{\mathcal{B}}_{2}$. The second condition needed to conclude (25) from Proposition 4.4 is precisely assumption (ii) in the statement of the theorem. ∎

Appendix A Proof of Lemma 4.5

Here we prove the Lemma 4.5, which was used in Subsection 4.2, especially used in the proof of Theorem 2.4].

Let us prove that (32) holds true. By density, it suffices to perform the estimate for $f\in\mathcal{C}^{1}(\bar{M}_{0})$. In the proof below, we use the fact that the invariant measure $\bar{\mu}_{0}$ is absolutely continuous with respect to the Lebesgure measure.

Choose $\ell\geq 1$ and fix $\underline{\omega}_{\ell}:=(\omega_{1},\ldots,\omega_{\ell})$. Let $g$ be as in the statement of the lemma. For brevity, denote by $\bar{T}_{\underline{\omega}_{\ell}}^{\ell}=\bar{T}_{\omega_{\ell}}\circ\cdots\circ\bar{T}_{\omega_{1}}$ the composition of random maps and by $\mathcal{L}_{\underline{\omega}_{\ell}}^{\ell}$ its associated transfer operator. Also, set $H_{\ell}^{p}(g)=|g|_{\infty}+\sup_{C\in\mathcal{C}_{\omega_{1},...,\omega_{\ell}}}C_{g_{|C}}^{(p)}$. We must estimate

[TABLE]

To do this, we decompose $\bar{M}_{0}$ into a countable collection of local rectangles, each foliated by a smooth collection of stable curves on which we may apply our norms. This technique follows closely the decomposition used in [15, Lemma 3.4].

We partition each connected component of $\bar{M}_{0}\setminus(\cup_{|k|\geq k_{0}}\mathbb{H}_{k})$, into finitely many boxes $B_{j}$ whose boundary curves are elements of $\mathcal{W}^{s}$ and $\mathcal{W}^{u}$, as well as the horizontal boundaries of $\mathbb{H}_{\pm k_{0}}$. We construct the boxes $B_{j}$ so that each has diameter in $(\delta/2,\delta)$, for some $\delta>0$, and is foliated by a smooth foliation of stable curves $\{W_{\xi}\}_{\xi\in\Xi_{j}}$, such that each curve $W_{\xi}$ is stretched completely between the two unstable boundaries of $B_{j}$. Indeed, due to the continuity of the cones $C^{s}(x)$ from (H1), we can choose $\delta$ sufficiently small that the family $\{W_{\xi}\}_{\xi\in\Xi_{j}}$ is a family of parallel line segments.

We disintegrate the measure $\bar{\mu}_{0}$ on $B_{j}$ into a family of conditional probability measures $d\mu_{\xi}=c_{\xi}\cos\varphi\,dm_{W_{\xi}}$, $\xi\in\Xi_{j}$, where $c_{\xi}$ is a normalizing constant, and a factor measure $\lambda_{j}(\xi)$ on the index set $\Xi_{j}$. Since $\bar{\mu}_{0}$ is absolutely continuous with respect to Lebesgue measure on $\bar{M}_{0}$, we have $\lambda_{j}(\Xi_{j})=\bar{\mu}_{0}(B_{j})=\mathcal{O}(\delta^{2})$.

Similarly, on each homogeneity strip $\mathbb{H}_{t}$, $t\geq k_{0}$, we choose a smooth foliation of parallel line segments $\{W_{\xi}\}_{\xi\in\Xi_{t}}\subset\mathbb{H}_{t}$ which completely cross $\mathbb{H}_{t}$. Due to the uniform transversality of the stable cone with $\partial\mathbb{H}_{t}$, we may choose a single index set $\Xi_{t}$ for each homogeneity strip. We again disintegrate $\bar{\mu}_{0}$ into a family of conditional probability measures $d\mu_{\xi}=c_{\xi}\cos\varphi\,dm_{W_{\xi}}$, $\xi\in\Xi_{t}$, and a transverse measure $\lambda_{t}(\xi)$ on the index set $\Xi_{t}$. This implies that $\lambda_{t}(\Xi_{t})=\bar{\mu}_{0}(\mathbb{H}_{t})=\mathcal{O}(|t|^{-5})$ for each $|t|\geq k_{0}$.

Notice that on each homogeneity strip $\mathbb{H}_{k}$, the function $\cos\varphi$ satisfies,

[TABLE]

for some uniform constant $C>0$ (uniform in $k$).

We are ready to estimate the required integral. Let $\mathcal{G}_{\ell}(W_{\xi})$ denote the components of $(\bar{T}_{\underline{\omega}_{\ell}}^{\ell})^{-1}W_{\xi}$ , with long pieces subdivided to have length between $\delta_{0}/2$ and $\delta_{0}$, as in the proof of Lemma 3.14.

[TABLE]

Next we use the assumption that $g$ is Hölder continuous on connected componts of $\bar{M}_{0}\setminus(\cup_{k=1}^{\ell}\bar{T}_{\omega_{1}}^{-1}\circ\cdots\circ\bar{T}_{\omega_{k}}^{-1}(\mathcal{S}_{0,H}))$. Since elements of $\mathcal{G}_{\ell}(W_{\xi})$ are also subdivided according to these singularity sets, we have that $g$ is Hölder continuous on each $W_{\xi,i}\in\mathcal{G}_{\ell}(W_{\xi})$. Thus,

[TABLE]

where we used (39) in the last estimate, as well as the fact that the normalizing constant $c_{\xi}$ is proportional to $|W_{\xi}|^{-1}$. This implies that

[TABLE]

Now $\sum_{W_{\xi,i}\in\mathcal{G}_{\ell}(W_{\xi})}|J_{W_{\xi,i}}\bar{T}_{\underline{\omega}_{\ell}}^{\ell}|_{\mathcal{C}^{0}(W_{\xi,i})}$ is bounded by a uniform constant independent of $\xi$ and $\underline{\omega}_{\ell}$ by [15, Lemma 5.5(b)]. Moreover, $\int_{\Xi_{j}}|W_{\xi}|^{-1}d\lambda_{j}(\xi)\leq C\delta_{0}$ for some constant $C>0$ since we chose our foliation to be comprised of long cone-stable curves. We conclude that the first term to the right hand side of the last inequality is uniformly bounded by $C_{1}|f|_{w}H_{\ell}^{p}(g)$ since the sum over $j$ is finite.

For the second term on the right hand side of the last equality, we again use [15, Lemma 5.5(b)] as well as the fact that $|W_{\xi}|^{-1}=\mathcal{O}(t^{3})$ for $\xi\in\Xi_{t}$, while $\lambda_{t}(\Xi_{t})=\mathcal{O}(t^{-5})$. Thus

[TABLE]

We conclude that

[TABLE]

for some uniform constant $K_{1}$ depending on $\bar{\mathcal{F}}_{\vartheta_{0}}$, but not on $f$, $\ell$ or $\underline{\omega}_{\ell}$. This completes the proof of (32).

To prove (33), we follow the proof of Lemma 3.14. Note that for $f\in\mathcal{C}^{1}(\bar{M}_{0})$, $W\in\mathcal{W}^{s}$, and a test function $\psi$, we have

[TABLE]

where the sum is taken over $W_{i}\in\mathcal{G}_{\ell}(W)$, the components of $(\bar{T}^{\ell}_{\underline{\omega}_{\ell}})^{-1}W$, subdivided as before. This is the same type of expression as in [15, eq. (5.24)] or [15, eq. (4.4)], but now the test function is

[TABLE]

rather than simply $\psi\circ\bar{T}^{\ell}_{\underline{\omega}_{\ell}}\,J_{W_{i}}\bar{T}^{\ell}_{\underline{\omega}_{\ell}}$. Since $S_{\ell}$ is constant on each $W_{i}\in\mathcal{G}_{\ell}(W)$, and we have assumed that $g$ is (uniformly in $\ell$) Hölder continuous on each $W_{i}\in\mathcal{G}_{\ell}(W)$, the proof of the Lasota-Yorke inequalities follows as in the proof of [15, Proposition 5.6]. The bound (33) then follows as in the proof of Lemma 3.14.

Remark A.1**.**

As a consequence of this lemma, if $g:\bar{M}\rightarrow\mathbb{R}$ is a bounded measurable function such that, for every $\underline{\omega}=(\omega_{k})_{k\geq 0}\in E^{\mathbb{N}}$, there exists positive integer $\ell_{\underline{\omega}}$ such that $g(\cdot,\underline{\omega})$ is $p$-Hölder on every connected component (uniformly on $\underline{\omega}$) of $\bar{M}_{0}\setminus\left(\cup_{k=0}^{\ell_{\underline{\omega}}-1}\bar{T}_{\omega_{0}}^{-1}\circ\cdots\circ\bar{T}_{\omega_{\ell(\underline{\omega})-1}}^{-1}(\mathcal{S}_{0,H})\right)$. Then, for every $f\in\widetilde{\mathcal{B}}_{w}$, we have

[TABLE]

*with the same notations as in the previous lemma. Therefore, $\mathbb{E}_{\bar{\mu}}[g\cdot]$ is in $\widetilde{\mathcal{\mathcal{B}}}_{w}^{\prime}$. *

Appendix B Proof of Lemma 4.10.

Note that $\mathcal{V}_{n}=n+2\sum_{1\leq k<\ell\leq n}\mathbf{1}_{\{S_{\ell}=S_{k},\mathcal{I}_{\ell}=\mathcal{I}_{k}\}}$. Hence

[TABLE]

with $D_{k_{1},\ell_{1},k_{2},\ell_{2}}:=\bar{\mu}(E_{k_{1},\ell_{1}}\cap E_{k_{2},\ell_{2}})-\bar{\mu}(E_{k_{1},\ell_{1}})\bar{\mu}(E_{k_{2},\ell_{2}})$. It follows that

[TABLE]

with

[TABLE]

[TABLE]

with

[TABLE]

[TABLE]

We will start with the two easiest estimates: the estimates of the error terms $A_{1}$ and $A_{4}$. The method we will use to estimate the main terms $A_{2}$ and $A_{3}$ differs from [30].

Due to Lemma 4.9,

[TABLE]

Let us now prove that $A_{4}=o(n^{2})$ by writing

[TABLE]

for some $K^{\prime}_{0}>0$ due to Theorem 4.2, since $\mathbb{E}_{\bar{\mu}}[\cdot]$ is a continuous linear operator on $\widetilde{\mathcal{B}}_{1}$ and since $\mathbf{1}\in\widetilde{\mathcal{B}}_{1}$. This leads to $\sum_{(k_{1},k_{2},\ell_{1},\ell_{2})\in E_{n}}\left|D_{k_{1},\ell_{1},k_{2},\ell_{2}}\right|=O(n(\log n)^{2})$. Analogously, we obtain $\sum_{(k_{1},k_{2},\ell_{1},\ell_{2})\in F_{n}}\left|D_{k_{1},\ell_{1},k_{2},\ell_{2}}\right|=O(n(\log n)^{2})$. Hence $A_{4}=o(n^{2})$.

For $A_{2}$, we study separately the terms $\bar{\mu}(E_{k_{1},\ell_{1}}\cap E_{k_{2},\ell_{2}})$ and the terms $\bar{\mu}(E_{k_{1},\ell_{1}})\bar{\mu}(E_{k_{2},\ell_{2}})$. First by Lemma 4.8,

[TABLE]

where we used the fact that

[TABLE]

Therefore, due to the Lebesgue dominated convergence theorem, we obtain

[TABLE]

The rest of the estimate of $A_{2}$ is new (it is different from [30]). Fix for the moment $1\leq k_{1}<k_{2}<\ell_{1}<\ell_{2}\leq n$. Note that

[TABLE]

Using now (19) as for (20), we observe that $\mathbf{1}_{\{S_{k_{2}}-S_{k_{1}}=-(S_{\ell_{1}}-S_{k_{2}})=S_{\ell_{2}}-S_{\ell_{1}}\}}$ is equal to the following quantity

[TABLE]

which is also equal to

[TABLE]

Now using the $P$-invariance and $\bar{T}$-invariance of $\bar{\mu}$ and several times the formula $P^{m}(f.g\circ\bar{T}^{m})=gP^{m}(f)$, we obtain

[TABLE]

Due to our spectral assumptions, we observe that

[TABLE]

up to defining $\lambda_{u}=e^{-\frac{1}{2}\Sigma^{2}u\cdot u}$ for $u$ outside $[-\beta,\beta]$ and so, proceding as in the proof of Theorem 4.2, we obtain that, for every $n\geq 2$ and every $u,v\in[-\pi,\pi]^{2}$,

[TABLE]

and $|\lambda_{u}^{n}|\leq e^{-2a|u|^{2}}$ for some $a>0$ (such that $e^{-2a|\pi|^{2}}>\alpha^{n}$, $\max(\lambda_{u}^{n-1},e^{-\frac{n-1}{2}\Sigma^{2}u\cdot u})\leq e^{-2an|u|^{2}}$) since $n|u|^{2}e^{-2na|u|^{2}}=O(e^{-na|u|^{2}})$. Therefore, we obtain

[TABLE]

where we have set

[TABLE]

with A^{2}_{Q}:=\left(\begin{array}[]{cccc}\ell_{1}-k_{1}&0&\ell_{1}-k_{2}&0\\ 0&\ell_{1}-k_{1}&0&\ell_{1}-k_{2}\\ \ell_{1}-k_{2}&0&\ell_{2}-k_{2}&0\\ 0&\ell_{1}-k_{2}&0&\ell_{2}-k_{2}\end{array}\right) which is symmetric with determinant

[TABLE]

Due to the form of $A^{2}_{Q}$, we observe that $A^{2}_{Q}$ has eigenvectors of the forms $(*,0,*,0)$ and $(0,*,0,*)$, that it has two double eigenvalues of sum (without multiplicity) $\ell_{1}-k_{1}+\ell_{2}-k_{2}$ and of product (without multiplicity) $\sqrt{\det A_{Q}^{2}}$. Therefore its dominating eigenvalue is smaller than the sum and so is less than $4\max(k_{2}-k_{1},\ell_{1}-k_{2},\ell_{2}-\ell_{1})$ and so (using the fact that the product of the two eigenvalues is larger than the maximum times the median of these three values) the smallest eigenvalue of $A^{2}_{Q}$ cannot be smaller than a quarter of the median of $k_{2}-k_{1},\ell_{1}-k_{2},\ell_{2}-\ell_{1}$, that we denote by $med(k_{2}-k_{1},\ell_{1}-k_{2},\ell_{2}-\ell_{1})$. So

[TABLE]

for some $a_{1}>0$. Moreover

[TABLE]

Therefore

[TABLE]

But using (44),

[TABLE]

due to the dominated convergence theorem. Therefore

[TABLE]

Analogously

[TABLE]

Equations (45), (46) and (47) lead to

[TABLE]

Combining this with (42), we conclude that

[TABLE]

The study of $A_{3}$ is the most delicate. We can observe that both sums $\sum_{1\leq k_{1}<k_{2}<\ell_{2}<\ell_{1}\leq n}\bar{\mu}(E_{k_{1},\ell_{1}}\cap E_{k_{2},\ell_{2}})$ and $\sum_{1\leq k_{1}<k_{2}<\ell_{2}<\ell_{1}\leq n}\bar{\mu}(E_{k_{1},\ell_{1}})\bar{\mu}(E_{k_{2},\ell_{2}})$ are in $O(n^{2}\log n)$. However, we will see that their difference is in $n^{2}$. Once again our proof differs from the one in [30] and is based on the same idea as the one used to prove $A_{2}$. We set $E_{k,\ell}(b):=E_{k,\ell}\cap\{\mathcal{I}_{k}=b\}$. Due to the first part of Lemma 4.8,

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

Now, as we did for (43) (and using Theorem 4.2), we get that

[TABLE]

Therefore

[TABLE]

We will now prove that the term in $O$ in this last formula is negligable. Indeed its sum over $\{1\leq k_{1}\leq k_{2}\leq\ell_{2}\leq\ell_{1}\leq n\}$ is in $O$ of the following quantity:

[TABLE]

This combined with (50) and (51) leads to

[TABLE]

i.e.

[TABLE]

since

[TABLE]

Therefore, due to the Lebesgue dominated convergence theorem,

[TABLE]

To conclude the proof of the lemma, we use the estimate for $A_{3}$ together with (40) and (48) to obtain,

[TABLE]

This finished the proof.

Appendix C Spectrum of $\mathcal{P}_{u}$

In this appendix, we are interested in the spectrum of the family of operators $\mathcal{P}_{u}$. We start by stating a result for the unperturbed operators $\mathcal{L}_{u,0}$.

Lemma C.1**.**

Let $u\in\mathbb{R}^{2}$, $h\in\mathcal{B}$ and $\lambda\in\mathbb{C}$ be such that $\mathcal{L}_{u,0}h=\lambda h$ in ${\mathcal{B}}$ and $|\lambda|\geq 1$. Then either $h\equiv 0$ or $u\in 2\pi\mathbb{Z}^{2}$, $\lambda=1$ and $h$ is $\bar{\mu}_{0}$-almost surely constant.

Proof.

Recall that for $\psi\in\mathcal{C}^{p}(\bar{M}_{0})$, we have $\psi\circ\bar{T}_{0}^{n}\in\mathcal{C}^{p}(\bar{T}^{-n}\mathcal{W}^{s})$. Note that

[TABLE]

Thus for $n\geq 1$,

[TABLE]

where $S_{n}\Phi_{0}=\Phi_{0}+\Phi_{0}\circ\bar{T}_{0}+\cdots+\Phi_{0}\circ\bar{T}_{0}^{n-1}$ denotes the partial sum. By [15, Lemma 3.4], using the invariance of $h$,

[TABLE]

where $C^{(p)}_{\bar{T}_{0}^{-n}\mathcal{W}^{s}}(\cdot)$ denotes the Hölder constant of exponent $p$ measured along elements of $\bar{T}_{0}^{-n}\mathcal{W}^{s}$. Since $|e^{iu\cdot S_{n}\Phi_{0}}|=1$ and $S_{n}\Phi_{0}$ is constant on each element of $\bar{T}_{0}^{-n}\mathcal{W}^{s}$, we have

[TABLE]

Using this estimate in (52) and taking the limit as $n\to\infty$ yields $|h(\psi)|=0$ if $|\lambda|>1$ and $|h(\psi)|\leq C|h|_{w}|\psi|_{\infty}$ for all $\psi\in\mathcal{C}^{p}(\mathcal{W}^{s})$ if $|\lambda|=1$. From this we conclude that the spectrum of $\mathcal{L}_{u,0}$ is always contained in the unit disk. Furthermore, when $|\lambda|=1$, then $h$ is a signed measure. For the remainder of the proof, we assume $|\lambda|=1$.

Let $\mathbb{V}_{u,0}$ be the eigenspace of $\mathcal{L}_{u,0}$ corresponding to eigenvalue $\lambda_{u,0}$, and $\Pi_{u,0}$ the eigenprojection operator. Since we are assuming $\mathbb{V}_{u,0}$ is non-empty, Lemma 3.14 implies that $\mathcal{L}_{u,0}$ is quasi-compact with essential spectral radius bounded by $\tau<1$. Moreover, Lemma 3.14 implies that $\|\mathcal{L}_{u,0}^{n}\|_{L(\mathcal{B},\mathcal{B})}$ remains bounded for all $n\geq 0$, so using [14, Lemma 5.1], we conclude that $\mathcal{L}_{u,0}$ has no Jordan blocks corresponding to its peripheral spectrum.

Using these facts, $\Pi_{u,0}$ has the representation

[TABLE]

In addition, for $f\in\mathcal{C}^{1}(\bar{M}_{0})$, $\psi\in\mathcal{C}^{p}(\mathcal{W}^{s})$,

[TABLE]

Since $\Pi_{u,0}\mathcal{C}^{1}(\bar{M}_{0})$ is dense in the finite dimensional space $\Pi_{u,0}\mathcal{B}$, therefore $\Pi_{u,0}\mathcal{C}^{1}(\bar{M}_{0})=\Pi_{u,0}\mathcal{B}=\mathbb{V}_{u,0}$. So for $h\in\mathbb{V}_{u,0}$, there exists $f\in\mathcal{C}^{1}(\bar{M}_{0})$ such that $\Pi_{u,0}f=h$. Now for each $\psi\in\mathcal{C}^{p}(\bar{M}_{0})$,

[TABLE]

Thus $h$ is absolutely continuous with respect to $\bar{\mu}_{0}$. For simplicity, we identify $h$ and its density with respect to $\bar{\mu}_{0}$; then $h\in L^{\infty}(\bar{M}_{0},\bar{\mu}_{0})$. Now for any $\psi\in\mathcal{C}^{p}(\mathcal{W}^{s})$, we have

[TABLE]

Accordingly, $\lambda\,h=(e^{iu\cdot\Phi_{0}}h)\circ\bar{T}_{0}^{-1}$, $\bar{\mu}_{0}$-a.e. Or equivalently, we have $\lambda\,h\circ\bar{T}_{0}=e^{iu\cdot\Phi_{0}}h$. Hence $\lambda^{n}\,h\circ\bar{T}_{0}^{n}=e^{iu\cdot S_{n}\Phi_{0}}h$.

Let $G_{\lambda}$ be the closed multiplicative group generated by $\lambda$ and let $m_{\lambda}$ be the normalized Haar measure on $G_{\lambda}$. ($G_{\lambda}$ is finite if $\lambda$ is a root of unity; it is $\{z\in\mathbb{C}\,:\,|z|=1\}$ otherwise.) The dynamical system $(G_{\lambda},m_{\lambda},T_{\lambda})$ is ergodic, where $T_{\lambda}$ denotes multiplication by $\lambda$ in $G_{\lambda}$. Due to [27], the dynamical system $(M_{0}\times G_{\lambda},\mu_{0}\otimes m_{\lambda},T_{0}\times T_{\lambda})$ in infinite measure is conservative and ergodic. But the function $H:M_{0}\times G_{\lambda}\rightarrow\mathbb{C}$ defined as follows is $(T_{0}\times T_{\lambda})$-invariant:

[TABLE]

Indeed, for $\mu_{0}\otimes m_{\lambda}$-a.e. $(\bar{x}+\ell,y)\in M_{0}\times G_{\lambda}$,

[TABLE]

due to our assumption on $h$. We conclude that $H$ is a.e. equal to a constant, which implies that $u\in 2\pi\mathbb{Z}^{2}$, $\lambda=1$, and $h$ is $\bar{\mu}_{0}$-a.s. constant. ∎

Proposition C.2**.**

Given $\beta>0$, there exists $C>1$ and $\alpha\in(0,1)$ such that

[TABLE]

Proof.

Fix $\beta>0$. Due to [1, Lemma 4.3], Lemma C.1, and the continuity in $u$ provided by [16, Lemma 5.4] (see also Lemma 3.16 applied to $\mathcal{L}_{u,0}$ rather than $P_{u}$), we know that there exists $C>1$ and $\alpha\in(0,1)$ such that

[TABLE]

Therefore, for every $f\in\widetilde{\mathcal{B}}$, we have

[TABLE]

where we used Lemma 3.7 to obtain the second line. Analogously,

[TABLE]

We conclude by putting these two estimates together. ∎

Acknowledgements.

This work was begun at the AIM workshop Stochstic Methods for Non-Equilibrium Dynamical Systems, in June 2015. Part of this work was carried out during visits by the authors to ESI, Vienna in 2016, to CIRM, Luminy in 2017 and 2018, and to BIRS, Canada in 2018, and by a visit of FP to the University of Massachusetts at Amherst in 2018. MD was supported in part by NSF Grant DMS 1800321.