D Dragičević
email: ddragicevic@math.uniri.hr.

G Froyland
email: g.froyland@unsw.edu.au.

C González-Tokman

S Vaienti
email: vaienti@cpt.univ-mrs.fr.

A SPECTRAL APPROACH FOR QUENCHED LIMIT THEOREMS FOR RANDOM HYPERBOLIC DYNAMICAL SYSTEMS

 to quenched random piecewise hyperbolic dynamics including some classes of billiards.

Introduction

In our previous paper [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] we extended the Nagaev-Guivarc'h spectral method to obtain limit theorems, such as the Central Limit Theorem (CLT), the Large Deviation Principle (LDP) and the Local Central Limit Theorem (LCLT), for random dynamical systems governed by a cocycle of maps

T (n) ω := T σ n-1 ω • • • • • T σω • T ω ,
assuming uniform-in-ω eventual expansivity conditions on the maps T ω . The random driving was a general ergodic, invertible transformation σ : Ω on a probability space (Ω, P), and the real observable g was defined on the product space Ω × X → R.

Before introducing our new results, we briefly recap the essence of the Nagaev-Guivarc'h spectral method in the deterministic setting, where one deals with a single map T , deferring to the original articles by Nagaev [START_REF] Nagaev | Some limit theorems for stationary markov chains[END_REF][START_REF] Nagaev | More exact statement of limit theorems for homogeneous markov chains[END_REF] and Guivarc'h [START_REF] Rousseau-Egele | Un théoreme de la limite locale pour une classe de transformations dilatantes et monotones par morceaux[END_REF][START_REF] Guivarc | Théorèmes limites pour une classe de chaînes de markov et applications aux difféomorphismes d'anosov[END_REF] and to the excellent survey [START_REF] Gouëzel | Limit theorems in dynamical systems using the spectral method[END_REF] for more details. The spectral method uses the transfer operator L : B acting on a Banach space B, and in particular, the twisted transfer operator L θ f := L(e θg f ), for f and g ∈ B. In the situation where L θ is quasi-compact for θ near zero, regularity of the leading eigenvalues and eigenprojectors have been used to prove limit theorems [START_REF] Keller | Spectral theory, zeta functions and the distribution of periodic points for collet-eckmann maps[END_REF][START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF][START_REF] Rey-Bellet | Large deviations in non-uniformly hyperbolic dynamical systems[END_REF][START_REF] Rousseau-Egele | Un théoreme de la limite locale pour une classe de transformations dilatantes et monotones par morceaux[END_REF][START_REF] Keller | Spectral theory, zeta functions and the distribution of periodic points for collet-eckmann maps[END_REF][START_REF] Broise | Transformations dilatantes de l'intervalle et théorèmes limites[END_REF][START_REF] Rousseau-Egele | Un théoreme de la limite locale pour une classe de transformations dilatantes et monotones par morceaux[END_REF][START_REF] Morita | Local limit theorem and distribution of periodic orbits of lasota-yorke transformations with infinite markov partition[END_REF][START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF][START_REF] Szász | Local limit theorem for the lorentz process and its recurrence in the plane[END_REF][START_REF] Gouëzel | Berry-esseen theorem and local limit theorem for non uniformly expanding maps[END_REF] and more, namely Berry-Esseen theorems [START_REF] Guivarc | Théorèmes limites pour une classe de chaînes de markov et applications aux difféomorphismes d'anosov[END_REF][START_REF] Gouëzel | Berry-esseen theorem and local limit theorem for non uniformly expanding maps[END_REF] and almost-sure invariance principles [START_REF] Gouëzel | Almost sure invariance principle for dynamical systems by spectral methods[END_REF]. The key equality was E(e θSng f ) = E((L θ ) n f ), where S n g denotes the Birhkoff sum of the observable g and the expectation is taken with respect to the unique eigenmeasure m of the adjoint of L. Since the map θ → L θ is holomorphic, classical perturbation theory allows one to obtain E(e θSng ) = c(θ)λ(θ) n + d n (θ), where λ(θ) is the leading eigenvalue of L θ , with c and λ analytic in θ, and sup θ |d n (θ)| → 0. We can therefore easily compute the characteristic function and the log generating function of the process g •T n with respect to the invariant probability measure of T , which has density dh/dm where h is the eigenfunction of L corresponding to the simple eigenvalue 1.

In the quenched random setting we must replace the n-th power of the twisted operator with the twisted transfer operator cocycle L θ,(n) ω

:= L θ σ n-1 ω • • • • • L θ σω • L θ ω .
By using the multiplicative ergodic theorem adapted to the study of such cocycles and generalizing a theorem of Hennion and Hérve [START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF] to the random setting, we were able in our previous paper [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] to show that the cocycle L θ,(n) ω is quasi-compact for θ near to 0. We therefore thus obtained that for such values of θ and for P-a.e. ω ∈ Ω, the top Lyapunov exponent Λ(θ) (analogous to the logarithm of λ(θ) in the deterministic setting) of the cocycle is analytic and given by

lim n→∞ 1 n log |E µω (e θSng(ω,•) )| = Λ(θ),
where µ ω is the equivariant probability measure on the ω-fiber (see below). This result together with the exponential decay of the norm of the elements in the complement of the top Oseledets space, which handled the error corresponding to quantity d n above, allowed us to achieve the desired limit theorems.

In the present paper we move from cocycles of piecewise expanding maps to cocycles of hyperbolic maps (both smooth and piecewise smooth), including some classes of billiards. To our knowledge, this is the first time that this setting has been investigated with multiplicative ergodic theory tools. One of the primary differences with [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] is the use of anisotropic Banach spaces here in place of the space of functions of bounded variation in [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF]. Specifically, in the smooth hyperbolic setting and in any dimension, we use the functional analytic setup of Gouëzel and Liverani [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF], and in the piecewise hyperbolic case in dimension two we use the spaces from Demers and Liverani [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF] (as well as Demers and Zhang [START_REF] Demers | Spectral analysis of the transfer operator for the lorentz gas[END_REF][START_REF] Demers | A functional analytic approach to perturbations of the lorentz gas[END_REF]). This increased technicality in the underlying spaces necessitates a certain amount of checking of relevant conditions, however, we wish to highlight the fact that a wholesale change of the theory of [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] is not required, which demonstrates the power and flexibility of our approach. The use of transfer operators in the study of statistical properties and limit theorems for hyperbolic dynamical systems has flourished in the last years, and [START_REF] Baladi | The quest for the ultimate anisotropic Banach space[END_REF] presents a thorough discussion of the various spaces that have been used in the literature. Our intention in this work has not been to find the most general version of the results, but rather to illustrate the applicability of the methods. In fact, we expect the methods presented here to remain applicable in some (or all) of these functional analytic scenarios.

We first consider cocycles

T (n) ω
where the family of maps {T ω } ω∈Ω are selected from a C r+1 -neighbourhood of a topologically transitive Anosov map T of class C r+1 (in Section 9 we consider piecewise hyperbolic maps also describing periodic Lorentz gas). The random driving σ : Ω is a general (ergodic, invertible) automorphism preserving a probability measure P. If d C r+1 (T ω , T ) < ∆ for P-a.e. ω ∈ Ω and ∆ is sufficiently small, the random dynamical system generated by the cocycle T (n) ω supports a measure µ, invariant under the skew product τ (ω, x) = (σω, T ω x). We obtain this measure by explicitly constructing the family µ ω along the marginal P, namely µ = Ω µ ω dP(ω), and satisfying the usual equivariance condition µ ω • T -1 ω = µ σω . Our observable g satisfies g(ω, •) ∈ C r for P-a.e. ω, ess sup ω∈Ω g(ω, •) C r < ∞, and is fiberwise centred: X g(ω, x) dµ ω (x) = 0 for P-a.e. ω. Our limit theorems concern random Birkhoff sums S n g(ω, x) := n-1 i=0 g(τ i (ω, x)) = n-1 i=0 g(σ i , T (i) ω x), (ω, x) ∈ Ω × X, n ∈ N.

Our main theorems are:

Theorem A (Quenched large deviations theorem). In the above setting, there exists ǫ 0 > 0 and a non-random function c : (-ǫ 0 , ǫ 0 ) → R which is nonnegative, continuous, strictly convex, vanishing only at 0 and such that lim n→∞ 1 n log µ ω (S n g(ω, •) > nǫ) = -c(ǫ), for 0 < ǫ < ǫ 0 and P-a.e. ω ∈ Ω.

We now define the non-random quantity

Σ 2 := Ω×X g(ω, x) 2 dµ(ω, x) + 2 ∞ n=1 Ω×X g(ω, x)g(τ n (ω, x)) dµ(ω, x). (2)
It is clear that Σ 2 ≥ 0.

Theorem B (Quenched central limit theorem). In the above setting, assume that the non-random variance Σ 2 , defined in (2) satisfies Σ 2 > 0. Then, for every bounded and continuous function φ : R → R and P-a.e. ω ∈ Ω, we have

lim n→∞ φ S n g(ω, x) √ n dµ ω (x) = φ dN (0, Σ 2 ).
(The discussion in §6.3 deals with the degenerate case Σ 2 = 0).

One of the main achievements of our previous paper was the proof of the local central limit theorem (LCLT) in the non-arithmetic and arithmetic cases. Our basic assumption, which for convenience we simply call (L), expresses the exponential decay of the strong norm of the twisted operator when the parameter θ = it has t = 0. Moreover we showed under additional assumptions that we will recall in section 8, that hypothesis (L) was equivalent to a co-boundary condition which is better known as the aperiodicity condition.

In the present paper we prove the LCLT in the non-arithmetic case by assuming (L). Recently Hafouta and Kifer [START_REF] Hafouta | Nonconventional limit theorems and random dynamics[END_REF] proposed a new set of assumptions which allow us to check condition (L). We will see that some of these assumptions can be verified easily for our systems, provided we restrict the class of the driving maps.

Theorem C (Quenched local central limit theorem). In the above setting suppose condition (L) holds. Then, for P-a.e. ω ∈ Ω and every bounded interval J ⊂ R, we have

lim n→∞ sup s∈R Σ √ nµ ω (s + S n g(ω, •) ∈ J) - 1 √ 2π e -s 2 2nΣ 2 |J| = 0.
In Section 9 we consider random cocycles of piecewise hyperbolic maps of the type considered in [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF] on two-dimensional compact Riemannian manifolds, and billiard maps associated to periodic Lorentz gas [START_REF] Demers | Spectral analysis of the transfer operator for the lorentz gas[END_REF][START_REF] Demers | A functional analytic approach to perturbations of the lorentz gas[END_REF]. As we will explain later on and in order to apply the multiplicative ergodic theorem, we have now less choice for the random distribution of the maps, but for instance we can deal with countably many maps. All the preceding theorems A, B and C still hold.

Apart from [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] there are some quenched limit theorems (LDP and CLT) that have been obtained using different methods. Kifer derives a large deviation principle [START_REF] Kifer | Large deviations in dynamical systems and stochastic processes[END_REF][START_REF] Kifer | Equilibrium states for random expanding transformations[END_REF][START_REF] Kifer | Limit theorems for random transformations and processes in random environments[END_REF] for occupational measures using theory of equilibrium states, and a central limit theorem via martingale methods; in both cases, he treats random subshifts of finite type and random smooth expanding maps. Recently, Hafouta and Kifer [START_REF] Hafouta | Nonconventional limit theorems and random dynamics[END_REF] proved limit theorems for these systems in the more general "nonconventional setting". They used (complex) cone techniques, where the cones were defined in the functional space upon which the transfer operator acts. We emphasize that they don't consider the case of hyperbolic dynamics studied in the present paper. In fact, is not clear if their cone techniques can be adapted to the present setting. Bakhtin [START_REF] Bakhtin | Random processes generated by a hyperbolic sequence of mappings. i[END_REF] is probably the closest to our work; he proves a central limit theorem and large deviation estimates for mixing sequences of smooth hyperbolic maps with common expanding and contracting directions, under a variance growth condition on the Birkhoff sums. He also used cones, but living on the tangent space of the manifold. In comparison to Bakhtin, we can additionally treat the case of random piecewise hyperbolic maps (with singularities and including billiards), and moreover we exhibit explicitly the rate function which produces asymptotic large deviation bounds; the local CLT is also new in this setting.

Preliminaries

Let X be a d-dimensional C ∞ compact connected Riemannian manifold and let T be a topologically transitive Anosov map of class C r+1 , where r > 2. We follow the setup of [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]. Replacing the Riemannian metric by an adapted metric [START_REF] Mather | Characterization of Anosov diffeomorphisms[END_REF], we use hyperbolicity constants 0 < ν < 1 < λ, where λ is less than the minimal expansion along the unstable directions, ν is greater than the minimal contraction along the stable directions, and the angles between the stable and unstable spaces (of dimensions d s , d u , respectively) are close to π/2. A collection of C ∞ coordinate charts ψ i : (-r i , r i ) d → X, i = 1, . . . , N are defined so that N i=1 ψ i ((-r i /2, r i /2) d ) cover X, with the r i small enough that Dψ i (0)

• (R ds × {0}) = E s (ψ i (0)), |ψ i | C r+1 , |ψ -1 i | C r+1 ≤ 1 + κ,
and κ small enough in such a way that the stable cone at x in R d is compatibly mapped to the stable cone at ψ i (x) in X (see [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF] for details). Let G i (K) denote the set of graphs of C r+1 functions χ : (-r i , r i ) ds → (-r i , r i ) du with |χ| C r+1 ≤ K (and with |Dχ| ≤ c i so that the tangent space of the graph belongs to the stable cone in R d mentioned above). For large enough K, the coordinate map

ψ -1 j • T -1 • ψ i maps G i (K) into G j (K ′ )
for some K ′ < K. For A sufficiently large, (depending on κ and ν) and δ small enough that Aδ < min i r i /6, an admissible graph is a map χ : B(x, Aδ) → (-2r i /3, 2r i /3) du , B(x, Aδ) ⊂ (-2r i /3, 2r i /3) ds ; the collection of admissible graphs is denoted Ξ i .

For p ∈ N, p ≤ r, q ≥ 0 and h ∈ C r (X, C), ϕ ∈ C q (X, C) we define (using the notation in [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF])

h ∼ p,q := sup |α|=p 1≤i≤N sup χ : B(x,Aδ)→R du χ∈Ξ i sup ϕ∈C q 0 (B(x,δ),C) |ϕ| C q ≤1 B(x,δ) ∂ α (h • ψ i ) • (Id, χ) • ϕ . (3)
Finally, for p and q as above satisfying p + q < r, we set

h p,q := sup 0≤k≤p h ∼ k,q+k = sup p ′ ≤p,q ′ ≥q+p ′ h ∼ p ′ ,q ′ . (4)
The space B p,q is defined to be the completion of C r (X, C) with respect to the norm • p,q .

The following proposition will be useful when applying the multiplicative ergodic theorem.

Proposition 2.1. The space B p,q is separable.

Proof. The desired conclusion follows directly from [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]Remark 4.3] after we note that C ∞ (X, C) has a countable subset which is dense with respect to the C r norm.

We recall from [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]Section 4] that the elements of B p,q are distributions of order at most q. More precisely, there exists C > 0 such that any h ∈ B p,q induces a linear functional ϕ → h(ϕ) with the property that |h(ϕ)| ≤ C h p,q |ϕ| C q , for ϕ ∈ C q (X, C).

(

) 5
In particular, for h ∈ C r we have that

h(ϕ) = X hϕ, for ϕ ∈ C q (X, C). (6)
We say that h ∈ B p,q is nonnegative and write h ≥ 0 if h(ϕ) ≥ 0 for any ϕ ∈ C q (X, R) such that ϕ ≥ 0. Let L T : B p,q → B p,q be the transfer operator associated to T defined by

(L T h)(ϕ) = h(ϕ • T ), for h ∈ B p,q and ϕ ∈ C q (X, C). (7)
We recall that for h ∈ C r (X, C), L T is the function given by

L T h = h |det T | • T -1 . (8)
Take g ∈ C r (X, C) and h ∈ B p,q . Then, there exists a sequence (h n ) n ⊂ C r (X, C) that converges to h in B p,q . It follows that (gh n ) n ⊂ C r (X, C) is a Cauchy sequence in B p,q and therefore it converges to some element of B p,q which we denote by g • h. It is straightfoward to verify that the above construction does not depend on the particular choice of the sequence (h n ) n . Moreover, the action of g • h as a distribution is given by

(g • h)(ϕ) = h(gϕ), ϕ ∈ C q (X, C). (9)
We will need the following result.

Lemma 2.2. For h ∈ B p,q , g ∈ C r (X, C) one has L T (g

• T • h) = g • L T h.
Proof. Let ϕ ∈ C q (X, C). It follows from ( 7) and ( 9) that [L T (g

• T • h)](ϕ) = (g • T • h)(ϕ•T ) = h(g •T •ϕ•T ) = L T h(g •ϕ) = (g •L T h)(ϕ)
, which yields the desired result.

Building the cocycle L

In the sequel we will consider the case p = q = 1 and r > 2, but we will also require T to be C r+1 , to be in a suitable framework for perturbations. Using the fact that the unit ball in B 1,1 is relatively compact in B 0,2 [15, Lemma 2.1], it follows from [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]Theorem 2.3] that the associated transfer operator L T is quasicompact on B 1,1 , 1 is a simple eigenvalue and there are no other eigenvalues of modulus 1. This in particular implies (using the terminology as in [START_REF] Conze | Limit theorems for sequential expanding dynamical systems on [0, 1[END_REF]Definition 2.6]

) that L T is exact in {h ∈ B 1,1 : h(1) = 0}. Let M ǫ (T ) = {S : S is an Anosov map of class C r+1 satisfying d C r+1 (S, T ) < ǫ}.
We also recall (see [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]Lemmas 2.1. and 2.2] and the discussion at the beginning of §7 [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]) that there exist ǫ, A > 0 and c ∈ (0, 1) such that for any T ′ ∈ M ǫ (T ), one has

• L n T ′ h 0,2 ≤ A h 0,2 for each n ∈ N and h ∈ B 1,1 ; • L n T ′ h 1,1 ≤ Ac n h 1,1 + A h 0,2 for each n ∈ N and h ∈ B 1,1
. Consider the family of (bounded, linear) transfer operators, acting on the Banach space

(B 1,1 , • ), O ǫ (T, B 1,1 ) = {L S : B 1,1 → B 1,1 such that S ∈ M ǫ (T )}.
It follows from [4, Proposition 2.10] (applied to the case where

• = • 0,2 and |•| v = • 1,1
) that there exists 0 < ǫ 0 ≤ ǫ, D, λ > 0 such that for any L T 1 , . . . , L Tn ∈ O ǫ 0 (T, B 1,1 ), one has

L Tn • • • • • L T 2 • L T 1 h 1,1 ≤ De -λn h 1,1 for h ∈ B 1,1 satisfying h(1) = 0, (10)
where L T i denotes the transfer operator associated with T i . From now on, we replace ǫ 0 by ǫ so that (10) holds on O ǫ (T, B 1,1 ).

We now build the cocycle R := (Ω, F , P, σ, B 1,1 , L), simply referred to as L, as follows:

(1) Let (Ω, F , P) be a probability space, where Ω is a Borel subset of a separable, complete metric space and σ : Ω → Ω an ergodic, invertible P-preserving transformation. (2) Let T : Ω → M ǫ (T ) be a measurable map given by ω → T ω . By applying [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]Lemma 7.1] we find that there exists C > 0 such that for any S ∈ M ǫ (T ), sup

h 1,1 ≤1 (L S -L T )h 0,2 ≤ Cǫ.
3.1. Strong measurability of ω → L ω . In this section we demonstrate strong measurability of the map L : Ω → O ǫ (T, B 1,1 ) given by ω → L ω := L Tω ; this is required to establish the existence of measurable Oseledets spaces for the cocycle. To prove strong measurability of ω → L ω := L Tω , we will show that the map from M ǫ (T ) to B 1,1 defined by S → L S is strongly continuous. For this, let S ∈ M ǫ (T ) and h ∈ B 1,1 . We must show that L S h -L S h 1,1 → 0 as d C r+1 ( S, S) → 0. First, assume h ∈ C r . Then, we need to estimate differences of the form

B(x,δ) ∂ α (L S h • ψ i ) • (Id, χ) • ϕ - B(x,δ) ∂ α (L S h • ψ i ) • (Id, χ) • ϕ ,
where α, χ and ϕ vary as in the definition in (3), with p = q = 1. Arguing as in [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]Lemma 7.1], and employing the corresponding notation, we write

B(x,δ) ∂ α (L S h • ψ i ) • (Id, χ) • ϕ = |β|≤|α| l j=1 B(x j ,δ) ∂ β hj • (Id, χ j ) • F α,β,S,j • ρ j , (11)
where χ 1 , . . . , χ l are γ-admissible graphs whose corresponding γ-admissible leaves cover S -1 (W ), with W an admissible leaf corresponding to the graph of χ; hj = h • ψ i(j) ; {ρ j } j=1,...,l is a partition of unity subordinated to the γ-admissible leaves of χ j ; and F α,β,S,j are functions bounded in C q+|β| . A similar expression holds for B(x,δ) ∂ α (L S h • ψ i ) • (Id, χ) • ϕ, with F α,β,S,j replaced by F α,β, S,j and χ j replaced by χj , the graph corresponding to

ψ -1 i(j) • S-1 • S • ψ i(j) • (Id, χ j )(B(x j , γAδ)). Furthermore, if d C r+1 (S, S
) is small enough, each χj is a graph in Ξ i(j) , and |χ jχj | C 2 ( B(x j ,Aδ)) < Cd C r+1 (S, S). Also, F α,β,S,j C q+|β| , F α,β, S,j C q+|β| are uniformly bounded for S, S ∈ M ǫ (T ) and F α,β,S,j -F α,β, S,j C q+|β| → 0 as d C r+1 ( S, S) → 0, uniformly over ϕ as in [START_REF] Broise | Transformations dilatantes de l'intervalle et théorèmes limites[END_REF]. Hence, as d C r+1 ( S, S) → 0, we get

B(x j ,δ) ∂ β ( hj ) • (Id, χj ) • F α,β, S,j • ρ j -∂ β ( hj ) • (Id, χ j ) • F α,β,S,j • ρ j → 0,
uniformly over χ (and so χ j ) and ϕ as in [START_REF] Broise | Transformations dilatantes de l'intervalle et théorèmes limites[END_REF]. It then follows from [START_REF] González-Tokman | A semi-invertible operator Oseledets theorem[END_REF]

that L S h - L S h 1,1 → 0 as d C r+1 ( S, S) → 0, as claimed.
The result for general h ∈ B 1,1 follows from an approximation argument by C r functions, because if d C r+1 (S, S) is sufficiently small, then L S 1,1 ≤ 1 + L S 1,1 =: M. Indeed, let {h j } j∈N be a sequence of C r functions such that lim j→∞ h j = h in B 1,1 , and let ǫ > 0. Then, there exists n ∈ N such that h -

h n 1,1 < ǫ 3M . Hence, L S h -L S h 1,1 ≤ L S h -L S h n 1,1 + L S h n -L S h n 1,1 + L S h n -L S h 1,1 ≤ 2ǫ 3 + L S h n -L S h n 1,1 . Since h n ∈ C r , we have that lim sup d C r+1 ( S,S)→0 L S h -L S h 1,1 ≤ 2ǫ 3 .
Since the choice of ǫ > 0 is arbitrary, the result follows.

3.2. Quasi-compactness of the cocycle L and existence of Oseledets splitting. We may apply Kingman's subadditive ergodic theorem to form the following limits, which are constant for P-a.e. ω ∈ Ω:

Λ(R) := lim n→∞ 1 n log L (n) ω 1,1 , and
κ(R) := lim n→∞ 1 n log ic(L (n) ω ), where ic(A) := inf{r > 0 : A(B B 1,1
) can be covered with finitely many balls of radius r},

and B B 1,1 is the unit ball in B 1,1 . The cocycle R is called quasi-compact if Λ(R) > κ(R). For each ω ∈ Ω, n ∈ N, let L (n) ω := L σ n-1 ω • • • • • L σω • L ω . It follows readily from (10) that L (n) ω h 1,1 ≤ De -λn h 1,1 for any ω ∈ Ω, n ∈ N and h ∈ B 1,1 , h(1) = 0. (12)
After possibly decreasing ǫ we can also assume5 that there exist a ∈ (0, 1) and B, K > 0 such that for every ω ∈ Ω, n ∈ N and h ∈ B 1,1 ,

L (n) ω h 0,2 ≤ B h 0,2 , L (n) ω h 1,1 ≤ Ba n h 1,1 + B h 0,2 , (13)
which in particular implies that

L ω h 1,1 ≤ K h 1,1 , (14)
where K := Ba + B > 0. By [9, Lemma 2.1], the inequalities ( 13) and ( 14) imply that the cocycle R is quasi-compact. By separability of B 1,1 , and quasi-compactness and strong measurability of L, the multiplicative ergodic theorem (Theorem A, [START_REF] González-Tokman | A semi-invertible operator Oseledets theorem[END_REF]) yields the existence of a measurable Oseledets splitting

B 1,1 = l j=1 Y j (ω) ⊕ V (ω),
where each component of the splitting is equivariant under L ω . The Y j (ω) are finitedimensional and by 0 = λ 1 > λ 2 > . . . we denote the corresponding (finite or infinite) sequence of Lyapunov exponents.

3.3.

One-dimensionality of the top Oseledets space.

Proposition 3.1. There exists a unique family

(h 0 ω ) ω∈Ω ⊂ B 1,1 such that: (1) L ω h 0 ω = h 0 σω for P-a.e. ω ∈ Ω; (2) h 0 ω is nonnegative and h 0 ω (1) = 1 for P-a.e. ω ∈ Ω; (3) ω → h 0 ω is a measurable map from Ω to B 1,1 ; (4) ess sup ω∈Ω h 0 ω 1,1 < ∞. (15)
Proof. Let

Y = {v : Ω → B 1,1 : v is measurable and v ∞ := ess sup ω∈Ω v(ω) 1,1 < ∞}.
Then, Y = (Y, • ∞ ) is a Banach space. Furthermore, let Z be the subset of Y that consists of v ∈ Y with the property that v(ω) is nonnegative and v(ω)(1) = 1 for P-a.e. ω ∈ Ω. It is easy to verify that Z is a closed subset of Y . Indeed, assume that

(v n ) n∈Z is a sequence in Z converging to v ∈ Y . It follows from (5) that |v(ω)(ϕ) -v n (ω)(ϕ)| ≤ C v n (ω) -v(ω) 1,1 |ϕ| C 1 ≤ C v n -v ∞ |ϕ| C 1 ,
and thus v n (ω)(ϕ) → v(ω)(ϕ) for ϕ ∈ C 1 and P-a.e. ω ∈ Ω. Thus, v(ω)(ϕ) ≥ 0 for ϕ ≥ 0 and v(ω)(1) = 1 for P-a.e. ω ∈ Ω and we conclude that v ∈ Y . We define L : Z → Z by

(Lv)(ω) = L σ -1 ω v(σ -1 ω) ω ∈ Ω, v ∈ Z.
It follows from ( 7) and ( 14) that L is a well-defined and continuous map on Z. Using ( 12), one can easily verify (see [START_REF] Dragičević | Almost sure invariance principle for random piecewise expanding maps[END_REF]Proposition 1]) that there exists n 0 ∈ N such that L n 0 is a contraction on Z. Thus, L has a unique fixed point v ∈ Z. It is easy to verify that the family h 0 ω , ω ∈ Ω defined h 0 ω = v(ω), ω ∈ Ω satisfies the desired properties. Conversely, each family satisfying properties (1)-( 4) induces a fixed point of L which then must coincide with v. Proposition 3.2. Let (h 0 ω ) ω∈Ω be as in Proposition 3.1. Then h 0 ω is a probability measure on B 1,1 for P-a.e. ω ∈ Ω.

Proof. Using Lemma 2.2, we have that

h 0 ω (ϕ) = L (n) σ -n ω h 0 σ -n ω (ϕ) = h 0 σ -n ω (ϕ • T (n) σ -n ω ), for ω ∈ Ω and ϕ ∈ C 1 .
Hence, using the arguments as in [5, Lemma 5.3], and equations ( 5) and ( 15), we find that there exists a constant D > 0 such that

|h 0 ω (ϕ)| ≤ D|ϕ| ∞ for P-a.e. ω ∈ Ω and ϕ ∈ C 1 . Since C 1 is dense in C 0 ,
we conclude that for P-a.e. ω ∈ Ω, h 0 ω can be extended to a bounded linear functional on C 0 . By the Riesz representation theorem, h 0 ω is a signed measure. By invoking the nonnegativity of h 0 ω together with h 0 ω (1) = 1, we conclude that h 0 ω is a probability measure for P-a.e. ω ∈ Ω. Proposition 3.3. The top Oseledets space Y 1 (ω) of the cocycle L is one-dimensional, and spanned by h 0 ω . Proof. Proposition 3.1 and ( 13) imply that the top Lyapunov exponent of the cocycle L is equal to 0. Furthermore, it follows from (15) that

lim n→∞ 1 n log L (n) ω h 0 ω 1,1 = lim n→∞ 1 n log h 0 σ n ω 1,1 ≤ 0 for P-a.e. ω ∈ Ω.
On the other hand, it follows from ( 5) that 1 = h 0 ω (1) ≤ C h 0 ω 1,1 for P-a.e. ω ∈ Ω. Thus,

lim n→∞ 1 n log L (n) ω h 0 ω 1,1 = lim n→∞ 1 n log h 0 σ n ω 1,1 ≥ lim n→∞ 1 n log C -1 = 0 for P-a.e. ω ∈ Ω.
We conclude that h 0 ω ∈ Y 1 (ω) for P-a.e. ω ∈ Ω. We now claim that h 0 ω spans Y 1 (ω) for P-a.e. ω ∈ Ω. Indeed, assume that there exists

g ω / ∈ span{h 0 ω }, g ω ∈ Y 1 (ω) and choose α, β scalars (that depend on ω) such that |α| + |β| > 0 and (αh 0 ω + βg ω )(1) = 0. Then, it follows from (12) that lim n→∞ 1 n log L (n) ω (αh 0 ω + βg ω ) 1,1 ≤ -λ < 0.
On the other hand, since

αh 0 ω + βg ω ∈ Y 1 (ω) \ {0} we have lim n→∞ 1 n log L (n) ω (αh 0 ω + βg ω ) 1,1 = 0,
which yields a contradiction. We conclude that Y 1 (ω) = span{h 0 ω } and thus Y 1 (ω) is one-dimensional for P-a.e. ω ∈ Ω.

Quasi-compactness of the twisted cocycle L θ

We build a twisted cocycle L θ , by setting

L θ ω (h) = L ω (e θg(ω,•) • h), for ω ∈ Ω, θ ∈ C, and h ∈ B 1,1 . We will from now write e θg(ω,•) h instead of e θg(ω,•) • h. Our (centered) observable g will be a map g : Ω × X → R such that g(ω, •) ∈ C r for ω ∈ Ω, ess sup ω∈Ω g(ω, •) C r < ∞, (16)
and for P-a.e. ω ∈ Ω,

h 0 ω (g(ω, •)) = 0. (17)
This twisted cocycle gives us access to an ω-wise moment-generating function for Birkhoff sums of g. Lemma 4.1. For P-a.e. ω ∈ Ω, h ∈ B 1,1 and ϕ ∈ C 1 (X, C) one has

(L θ,(n) ω h)(ϕ) = h(e θSng(ω,•) (ϕ • T n ω )). (18)
Proof. One can follow the proof of Lemma 3.3 (part 2) [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF], using the definition of the untwisted transfer operator (7) and Lemma 2.2.

The following lemma is required as an auxiliary result in the proof of quasi-compactness of the twisted cocycle (Proposition 4.4).

Lemma 4.2. For θ 1 , θ 2 ∈ B C (0, 1) := {θ ∈ C : |θ| < 1}, we have that ess sup ω∈Ω e θ 1 g(σ -1 ω,•) -e θ 2 g(σ -1 ω,•) C 2 ≤ C|θ 1 -θ 2 |. Proof.
By applying the mean value theorem for the map g(z) = e zg(σ -1 ω,x) , where x ∈ X is fixed and using ( 16), we find that

ess sup ω∈Ω e θ 1 g(σ -1 ω,•) -e θ 2 g(σ -1 ω,•) C 0 ≤ C|θ 1 -θ 2 |. (19)
Furthermore, for j = 1, . . . , d 16) and ( 19) that

∂ j (e θ 1 g(σ -1 ω,•) -e θ 2 g(σ -1 ω,•) ) C 0 = e θ 1 g(σ -1 ω,•) θ 1 ∂ j g(σ -1 ω, •) -e θ 2 g(σ -1 ω,•) θ 2 ∂ j g(σ -1 ω, •) C 0 ≤ |θ 1 -θ 2 | • e θ 1 g(σ -1 ω,•) ∂ j g(σ -1 ω, •) C 0 + |θ 2 | • e θ 1 g(σ -1 ω,•) -e θ 2 g(σ -1 ω,•) C 0 • ∂ j g(σ -1 ω, •) C 0 . It now follows from (
ess sup ω∈Ω ∂ j (e θ 1 g(σ -1 ω,•) -e θ 2 g(σ -1 ω,•) ) C 0 ≤ C|θ 1 -θ 2 |.
One can now proceed and obtain the same estimates for the second derivatives of the map e θ 1 g(σ -1 ω,•)e θ 2 g(σ -1 ω,•) which implies the desired conclusion.

We need the following basic regularity result for the operators L θ ω . Proposition 4.3. There exists a continuous function K : C → (0, ∞) such that

L θ ω h 1,1 ≤ K(θ) h 1,1 , for h ∈ B 1,1 , θ ∈ C and P-a.e. ω ∈ Ω. (20)
Proof. We first note that it follows from ( 14) that

L θ ω h 1,1 = L ω (e θg(ω,•) h) 1,1 ≤ K e θg(ω,•) h 1,1 , for h ∈ B 1,1
, θ ∈ C and P-a.e. ω ∈ Ω. Hence, we need to estimate e θg(ω,•) h 1,1 . Note that by ( 4),

e θg(ω,•) h 1,1 = max{ e θg(ω,•) h ∼ 0,1 , e θg(ω,•) h ∼ 1,2 }. It follows easily from (3) that e θg(ω,•) h ∼ 0,1 ≤ max 1≤i≤N sup χ : B(x,Aδ)→R du χ∈Ξ i (e θg(ω,•) • ψ i ) • (Id, χ) C 1 • h ∼ 0,1 and e θg(ω,•) h ∼ 1,2 ≤ max 1≤i≤N sup χ : B(x,Aδ)→R du χ∈Ξ i (e θg(ω,•) • ψ i ) • (Id, χ) C 2 • h ∼ 1,2 + max 1≤j≤d 1≤i≤N sup χ : B(x,Aδ)→R du χ∈Ξ i ∂ j (e θg(ω,•) • ψ i )] • (Id, χ) C 1 • h ∼ 0,1 ,
which together with ( 16) implies the desired conclusion.

We can now state the main result of this section on quasi-compactness.

Proposition 4.4. For θ close to 0, the cocycle (L θ ω ) ω∈Ω is quasi-compact.

Proof. We follow closely [9, Lemma 3.13]. Observe [START_REF] Gouëzel | Almost sure invariance principle for dynamical systems by spectral methods[END_REF] and choose N ∈ N such that γ := Ba N < 1. Hence,

L θ,(N ) ω h 1,1 ≤ L (N ) ω h 1,1 + L θ,(N ) ω -L (N ) ω 1,1 • h 1,1 ≤ γ h 1,1 + B h 0,2 + L θ,(N ) ω -L (N ) ω 1,1 • h 1,1 .
On the other hand, we have that

L θ,(N ) ω -L (N ) ω = N -1 j=0 L θ,(j) σ N-j ω (L θ σ N-1-j ω -L σ N-1-j ω )L (N -1-j) ω .
It follows from ( 14) and ( 20) that

L (N -1-j) ω 1,1 ≤ K N -1-j and L θ,(j)
σ N-j ω 1,1 ≤ K(θ) j . Furthermore, using ( 14), we have that for any h ∈ B 1,1 and P-a.e. ω ∈ Ω,

(L θ ω -L ω )(h) 1,1 = L ω (e θg(ω,•) h -h) 1,1 ≤ K (e θg(ω,•) -1)h 1,1 . Moreover, (e θg(ω,•) -1)h 1,1 = max{ (e θg(ω,•) -1)h ∼ 0,1 , (e θg(ω,•) -1)h ∼ 1,2 }. Now Lemma 4.2 (applied for θ 1 = θ and θ 2 = 0) implies that there exists C > 0 such for θ ∈ B C (0, 1), (e θg(ω,•) -1)h 1,1 ≤ C|θ| h 1,1 for h ∈ B 1,1 .
We conclude that

L θ,(N ) ω -L (N ) ω 1,1 ≤ C|θ| N -1 j=0 K N -1-j K(θ) j ,
and therefore there exists γ ∈ (0, 1) such that for any θ sufficiently close to 0 and h ∈ B 1,1 ,

L θ,(N ) ω h 1,1 ≤ γ h 1,1 + B h 0,2 . (21)
Similarly, one can show that there exists B > 0 such that for any θ sufficiently close to 0 and h ∈ B 1,1 ,

L θ ω h 0,2 ≤ B h 0,2 . ( 22
)
The conclusion of the proposition follows from ( 21) and ( 22) by arguing as in the quasicompactness part of the proof of [9, Theorem 3.12].

Regularity of the top Oseledets space of the twisted cocycle

Let S ′ be the space of measurable maps V : Ω → B 1,1 with the property that

V ∞ := ess sup ω∈Ω V(ω) 1,1 < ∞.
Then, (S ′ , • ∞ ) is a Banach space. Furthermore, let S be the set of all V ∈ S ′ such that V(ω)(1) = 0 for P-a.e. ω ∈ Ω. Arguing as in the proof of Proposition 3.1, it is easy to verify that S is a closed subspace of S ′ . For V ∈ S ′ and ω ∈ Ω we will often write V ω instead of V(ω).

5.1.

Regularity of the cocycles.

Lemma 5.1.

(1) For P-a.e. ω ∈ Ω, the map θ → L θ ω is analytic in the norm topology of B 1,1 . (2) The map P : B C (0, 1) × S → S, given by P(θ, V) 1) is analytic in θ and bounded, linear in V. In particular, P 1 is C ∞ .

ω = L θ σ -1 ω (V σ -1 ω ) is analytic in θ and bounded, linear in V. In particular, P is C ∞ . (3) The map P 1 : B C (0, 1) × S → L ∞ (Ω), given by P 1 (θ, V) ω = (L θ σ -1 ω (V σ -1 ω ))(
Proof. We claim that for every h ∈ B 1,1 , the following holds:

L θ ω (h) = ∞ k=0 θ k k! L ω (g(ω, •) k h), in B 1,1 . (23)
To verify this, note that [15, Lemma 3.2] implies that

L ω (g(ω, •) k h) 1,1 ≤ C g(ω, •) k C 2 h 1,1 ≤ C g(ω, •) k C 2 h 1,1
, so by ( 16), the RHS of ( 23) is a well defined element of B 1,1 . The fact that it coincides with L θ ω (h) is straightforward to check, using linearity of L ω , the power series expansion of e θg(ω,•) , and testing against functions ϕ ∈ C 1 . This concludes the proof of Lemma 5.1 [START_REF] Bakhtin | Random processes generated by a hyperbolic sequence of mappings. i[END_REF].

Let us prove Lemma 5.1 [START_REF] Baladi | The quest for the ultimate anisotropic Banach space[END_REF]. For each k ≥ 0 and V ∈ S, let 16) and [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]Lemma 3.2]. We claim that

(g k • V)(ω, •) := g(ω, •) k V(ω, •). Then, g k • V ∈ S, because of (
P(θ, V) = ∞ k=0 θ k k! P(0, g k • V) in S. (24)
Indeed, [START_REF] Mather | Characterization of Anosov diffeomorphisms[END_REF] implies that

P(θ, V) ω = ∞ k=0 θ k k! P(0, g k • V) ω in B 1,1 . (25)
Furthermore, using once again [15, Lemma 3.2], in combination with the uniform over ω bounds ( 14) and ( 16), we have that there exists C > 0 such that for P-a.e. ω ∈ Ω,

P(θ, V) ω 1,1 ≤ ∞ k=0 θ k k! P(0, g k • V) ω 1,1 ≤ C ∞ k=0 θ k k! ess sup ω∈Ω g(ω, •) k C 2 V ∞ . ( 26
)
Hence, the series in [START_REF] Morita | Local limit theorem and distribution of periodic orbits of lasota-yorke transformations with infinite markov partition[END_REF] indeed converges in S and yields analyticity as required. The fact that V → P(θ, V), and also V → P(0, g k • V) is linear and bounded is straightforward to check. Hence, the C ∞ claim follows immediately. The proof of Lemma 5.1(3) is similar to that of Lemma 5.1(2). Indeed,

P 1 (θ, V) ω = ∞ k=0 θ k k! P(0, g k • V) ω , 1 , (27)
and ( 5) implies that | P(0,

g k •V) ω , 1 | ≤ C P(0, g k •V) ω 1,1
, which was bounded uniformly over ω in [START_REF] Nagaev | More exact statement of limit theorems for homogeneous markov chains[END_REF]. Hence, the series (27) converges to P 1 (θ, V) in L ∞ (Ω).

5.

2. An auxiliary function F and its regularity. For θ ∈ C and W ∈ S, set

F (θ, W)(ω) = L θ σ -1 ω (W(σ -1 ω) + h 0 σ -1 ω ) L θ σ -1 ω (W(σ -1 ω) + h 0 σ -1 ω )(1) -W(ω) -h 0 ω , ω ∈ Ω. (28)
We define two further auxiliary functions, which will be used in the sequel. Let G : C × S → S ′ and H : C × S → L ∞ (Ω) be given by

G(θ, W)(ω) := P(θ, W + h 0 )(ω) = L θ σ -1 ω (W σ -1 ω + h 0 σ -1 ω ), ω ∈ Ω, (29)
H(θ, W)(ω) := P 1 (θ, W + h 0 )(ω) = L θ σ -1 ω (W σ -1 ω + h 0 σ -1 ω )(1), ω ∈ Ω. (30
) It follows readily from [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF] and Lemma 5.1 that G and H are well defined, and in fact C ∞ functions. Direct calculations, analogous to those of [9, Appendix B], yield the following: Lemma 5.2. For ω ∈ Ω; θ, z ∈ C; W, H ∈ S, the following identities hold:

D 1 G(θ, W)(z) ω = zL σ -1 ω (g(σ -1 ω, •)e θg(σ -1 ω,•) (W σ -1 ω + h 0 σ -1 ω )), (31)
D 2 G(θ, W)(H) ω = L θ σ -1 ω (H σ -1 ω ), (32)
D 11 G(θ, W)(z 1 , z 2 ) ω = z 1 z 2 L σ -1 ω (g(σ -1 ω, •) 2 e θg(σ -1 ω,•) (W σ -1 ω + h 0 σ -1 ω )), (33)
D 12 G(θ, W)(z, H) ω = D 21 G(θ, W)(H, z) ω = zL σ -1 ω (g(σ -1 ω, •)e θg(σ -1 ω,•) H σ -1 ω ), (34)
D 22 G = 0. ( 35
)
Moreover, the expressions for the derivatives of H are equal to the corresponding expression for G applied to the constant function 1.

Lemma 5.3. There exist ǫ, R > 0 such that F : D → S is a well-defined map on

D := {θ ∈ C : |θ| < ǫ} × B S (0, R),
where B S (0, R) denotes the ball of radius R in S centered at 0.

Proof. Let G and H be defined as in [START_REF] Szász | Local limit theorem for the lorentz process and its recurrence in the plane[END_REF] and [START_REF] Thieullen | Fibrés dynamiques asymptotiquement compacts exposants de lyapounov. entropie. dimension[END_REF]. The function H is continuous on a neighborhood of (0, 0) in C × S and obviously H(0, 0)(ω) = h 0 ω (1) = 1 for P-a.e. ω ∈ Ω. Hence,

|H(θ, W)(ω)| ≥ 1 -|H(0, 0)(ω) -H(θ, W)(ω)| ≥ 1 -H(0, 0) -H(θ, W) L ∞ , for P-a.e. ω ∈ Ω. Continuity of H implies that H(0, 0) -H(θ, W) L ∞ ≤ 1
2 for all (θ, W) in a neighborhood of (0, 0) in C × S and hence, in such a neighborhood,

ess inf ω∈Ω |H(θ, W)(ω)| ≥ 1 2 .
This together with [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF] and a simple observation that F (θ, W)(1) = 0 immediately yields the desired conclusion.

Notice that map F defined by [START_REF] Rousseau-Egele | Un théoreme de la limite locale pour une classe de transformations dilatantes et monotones par morceaux[END_REF] satisfies

F (θ, W)(ω) = G(θ, W)(ω)/H(θ, W)(ω) - W(ω) -h 0 ω . The proof of Lemma 5.3 ensures that for (θ, W) in a neighbourhood D of (0, 0) ∈ C × S, ess inf ω∈Ω |H(θ, W)(ω)| ≥ 1
2 . Thus, the following result is a direct consequence of Lemma 5.2.

Proposition 5.4. The map F defined by (28) is of class C ∞ on the neighborhood D of (0, 0) ∈ C × S from Lemma 5.3. Moreover, for ω ∈ Ω, (θ, W) ∈ D and H ∈ S,

D 2 F (θ, W)(H) ω = 1 H(θ, W)(ω) L θ σ -1 ω H σ -1 ω - L θ σ -1 ω H σ -1 ω (1) [H(θ, W)(ω)] 2 G(θ, W) ω -H ω , D 1 F (θ, W) ω = 1 H(θ, W)(ω) L σ -1 ω (g(σ -1 ω, •)e θg(σ -1 ω,•) (W σ -1 ω + h 0 σ -1 ω )) - L σ -1 ω (g(σ -1 ω, •)e θg(σ -1 ω,•) (W σ -1 ω + h 0 σ -1 ω ))(1) [H(θ, W)(ω)] 2 L θ σ -1 ω (W σ -1 ω + h 0 σ -1 ω ),
where we have identified D 1 F (θ, W) with its value at 1. Proof. Note that F (0, 0) = 0. Furthermore, Proposition 5.4 implies that F is of class C ∞ on a neighborhood of (0, 0). In addition, Lemma 5.2 implies that

(D 2 F (0, 0)X )(ω) = L σ -1 ω X (σ -1 ω) -X (ω), ω ∈ Ω, X ∈ S.
Using [START_REF] Gouëzel | Berry-esseen theorem and local limit theorem for non uniformly expanding maps[END_REF] and proceeding as in [9, Lemma 3.5], one can show that D 2 F (0, 0) is invertible and that

(D 2 F (0, 0) -1 X )(ω) = - ∞ j=0 L (j) σ -j ω X (σ -j ω) ω ∈ Ω, X ∈ S. ( 37
)
The conclusion of the lemma now follows directly from the implicit function theorem.

Properties of Λ(θ)

Let 0 < ǫ < 1 be as in Lemma 5.3 and O(θ) be as in Lemma 5.5. Let

h θ ω := h 0 ω + O(θ)(ω) ∈ B 1,1 , ω ∈ Ω. (38)
We notice that h θ ω (1) = 1 and by Lemma 5.5, θ → h θ is continuously differentiable. Let us define Λ(θ) := log h θ ω (e θg(ω,•) ) dP(ω),

and

λ θ ω := h θ ω (e θg(ω,•) ) = L θ ω h θ ω (1). (40)
6.1. A differentiable lower bound for Λ(θ). Lemma 6.1 deals with differentiability properties of Λ(θ).

Lemma 6.1.

(1) For every θ ∈ B C (0, ǫ), Λ(θ) ≤ Λ(θ).

(2) Λ is differentiable on a neighborhood of 0, and

Λ′ (θ) = ℜ λ θ ω ((O(θ)(ω) + h 0 ω )(g(ω, •)e θg(ω,•) ) + O ′ (θ)(ω)(e θg(ω,•) )) |λ θ ω | 2 dP(ω) ,
where ℜ(z) denotes the real part of z and z the complex conjugate of z.

(3) For P-a.e. ω ∈ Ω, and θ in a neighborhood of 0, the map θ → Z ω (θ) := Z(θ, ω) is differentiable. Moreover,

Z ′ ω (θ) = ℜ λ θ ω ((O(θ)(ω) + h 0 ω )(g(ω, •)e θg(ω,•) ) + O ′ (θ)(ω)(e θg(ω,•) )) |λ θ ω | 2 . ( 4
) Λ′ (0) = 0.
Proof. The proof of part 1 is identical to the proof of Lemma 3.8 [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] replacing

• B with • 1,1 and L θ,(n) ω v θ ω 1 with |L θ,(n) ω h θ ω (1)
|. The proof of part 2 is identical to the proof of Lemma 3.9 [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF], using Lemma 5.5 in place of Lemma 3.5 [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] and replacing the final two equation blocks with: (1

|(O(θ)(ω) + h 0 ω )(g(ω, •)e θg(ω,•) )| ≤ C O(θ)(ω) + h 0 ω 1,1 • g(ω, •)e θg(ω,•) C 1 ≤ C O(θ) ∞ + C, and
|O ′ (θ)(ω)(e θg(ω,•) )| ≤ C O ′ (θ)(ω) 1,1 • e θg(ω,•) C 1 ≤ C O ′ (θ) ∞ . The proof
) dim Y θ 1 (ω) = 1. (2) Λ(θ) = Λ(θ).
In particular, Λ(θ) is differentiable near 0 and Λ ′ (0) = 0.

Convexity of Λ(θ)

. By Proposition 3.2, we can regard h 0 ω as Borel probability measure on X which we will denote by µ ω . The family (µ ω ) ω∈Ω induces a probability measure on Ω × X given by

µ(A × B) = A µ ω (B) dP(ω), for measurable sets A ⊂ Ω and B ⊂ X.
Then, µ is invariant for the skew-product transformation τ : Ω × X → Ω × X defined by

τ (ω, x) = (σω, T ω (x)), ω ∈ Ω, x ∈ X.
Obviously, the variance Σ 2 defined in (2) is nonnegative. From now on we shall assume that Σ 2 > 0. Otherwise we can invoke and adapt the coboundary case proved in Proposition 3 in our paper [START_REF] Dragičević | Almost sure invariance principle for random piecewise expanding maps[END_REF], which says that Σ 2 = 0 if and only if there exists r ∈ L 2 µ (Ω × X) such that g = rr • τ. Proposition 6.3. On a neighbourhood of 0,

(1) Λ is of class C 2 and Λ ′′ (0) = Σ 2 .

(2) Λ is strictly convex.

Proof. The proof of part 1 is identical to the proof of Lemma 3.15 [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] and part (ii) is a direct consequence of part 2.

Large deviation principle and central limit theorem

For θ ∈ C sufficiently close to 0, we have that dim

Y θ 1 (ω) = 1. Choose h θ ω ∈ Y θ 1 (ω) such that h θ ω (1) = 1.
We note that h θ ω is actually given by (38). Furthermore, let λ θ ω ∈ C be such that

L θ ω h θ ω = λ θ ω h θ σω . (41) Note that λ θ ω = h θ ω (e θg(ω,•) ), (42)
which coincides with (40). Next, let us fix φ θ ω ∈ Y * θ ω so that φ θ ω (h θ ω ) = 1. Furthermore, one can show (see [9, p. 30]) that Proof. Identical to the proof of Lemma 4.2 [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] with h ∈ B 1,1 in the present paper playing the role of f • dm in the proof of Lemma 4.2 [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF], and Lemma 18 replacing (43) [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF].

(L θ ω ) * φ θ σω = λ θ ω φ θ ω . (43
Lemma 7.3. For all complex θ in a neighborhood of 0, and P-a.e. ω ∈ Ω, we have that lim n→∞ 1 n log e θSng(ω,x) dµ ω (x) = Λ(θ).

Proof. We follow the proof of Lemma 4.3 [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF], observing that e θSng(ω,x) dµ ω (x) = h 0 ω (e θSng(ω,•) ),

and recalling the differentiability of the map θ → φ θ in Remark 7.1.

Proof of Theorem A. Following the proof of Theorem A [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF], by applying Proposition 6.3 and Lemma 7.3, together with the Gärtner-Ellis theorem ( [START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF] and Theorem 4.1 [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF]), we obtain the large deviation principle. 7.2. Central limit theorem. The proof of the following result is completely analogous to the proof of [9, Lemma 4.4].

Lemma 7.4. There exist C > 0, 0 < r < 1 such that for every θ ∈ C sufficiently close to 0, every n ∈ N and P-a.e. ω ∈ Ω, we have

L θ,(n) ω (h 0 ω -φ θ ω (h 0 ω )h θ ω )(1) ≤ Cr n . ( 44
)
Proof of Theorem B. The proof is identical to the proof of Theorem B [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF], with the same modifications as those listed in the proof of Lemma 7.2. Differentiability of θ → φ θ is used (see Remark 7.1) as well as Lemma 4.1 to obtain the coding of the Birkhoff sums via the twisted transfer operator. Lemma 4.5 [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] is proved is proved in an identical way.

Local central limit theorem

We begin by recalling the concept of P-continuity which we will also use in section 9.2.1. We say that our cocycle is P-continuous (a concept introduced in [START_REF] Thieullen | Fibrés dynamiques asymptotiquement compacts exposants de lyapounov. entropie. dimension[END_REF]) if the map ω → T ω has P-a.e. a countable range (besides being measurable). This implies that ω → L ω is continuous on each of countably many Borel subsets of Ω, whose union has full P measure. We refer to [START_REF] Froyland | A semi-invertible Oseledets theorem with applications to transfer operator cocycles[END_REF] for details.

In our earlier paper [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] we proved the local central limit theorem in the non-arithmetic case (we also separately treated the arithmetic case) under the condition that we called (L) in the introduction, namely

• (L) For P-a.e. ω ∈ Ω and for every compact interval J ⊂ R\{0} there exists C = C(ω) > 0 and ρ ∈ (0, 1) such that

||L it,(n) || B ≤ Cρ n , for t ∈ J and n ≥ 0. ( 45
)
Moreover under the assumption that the cocycle is P-continuous, we proved [9, Lemma 4.7.] that (L) is equivalent to the following aperiodicity condition • For every t ∈ R, either Λ(it) < 0 or the cocycle L it ω is quasicompact and the equation e itg(ω,x) L * ω ψ σω = γ it ω ψ ω , where γ it ω ∈ S 1 , L * ω denotes the adjoint of L ω and ψ ω ∈ B * , only has a measurable non-zero solution ψ := {ψ ω } ω∈Ω when t = 0. In this case γ 0 ω = 1 and ψ ω (f ) = f dm (up to a scalar multiplicative factor) for P-a.e. ω ∈ Ω. In our present Anosov setting the bound (45) will be replaced with the following:

L it,(n) ω 1,1 ≤ Cρ n , for t ∈ J and n ≥ 0. (46)
Still in the present setting, we can not prove at the moment the equivalence between (L) and the aperiodicity condition although several of the technical steps which formed the skeleton of our proof of [9, Lemma 4.7.] for expanding maps and functions of bounded variation can be transferred to Anosov maps and the anisotropic Banach spaces used in this work.

Proof of Theorem C. The proof assuming (L) follows now exactly as in the proof of Theorem C [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF], with the following minor modifications. We use Lemma 4.1 to obtain the coding of the Birkhoff sums through powers of the twisted transfer operator. The control of term (III) in the proof of Theorem C [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] uses Lemma 7.4 in place of Lemma 4.4 [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF]. The control of term (IV) in the proof of Theorem C [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] uses (L1) in place of the analogous condition (C5) in [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF].

As mentioned in the introduction, Hafouta and Kifer [START_REF] Hafouta | Nonconventional limit theorems and random dynamics[END_REF], Section 2.10, formulated a sort of classical aperiodicity condition in the random setting which allowed them to recover condition (L). We now state their assumptions and verify that some of them hold in our setting. This and additional hypothesis on the driving map σ will give us a new proof of the local central limit theorem. We first recall the Hafouta and Kifer assumptions adapted to our situations:

• HK A1: The probability measure P assigns positive measure to open sets, σ is a homeomorphism and there exist ω 0 ∈ Ω and m 0 ∈ N so that σ m 0 ω 0 = ω 0 . Moreover for each i ∈ {0, 1, . . . , m 0 -1}, there exists a neighborhood of σ i ω 0 on which the map ω → T ω is constant (note that on this neighborhood we also have that the map ω → L ω is constant).

• HK A2: For each compact interval J ⊂ R, the family of maps ω → L it ω , where t ∈ J, is equicontinuous at the points ω = σ i ω 0 , 0 ≤ i ≤ m 0 with respect to the operator norm, and there exists a constant B = B(J) ≥ 1, such that P-a.e.,

L it,(n) ω 1,1 ≤ B, (47)
for any n ∈ N and t ∈ J. • HK A3: For any compact interval J ⊂ R that does not contain the origin, there exists constants c = c(J) > 0 and b = b(J) ∈ (0, 1) such that

L s it 1,1 ≤ cb s , (48)
for any s ∈ N and t ∈ J, where the (deterministic) operator L it is defined as

L it := L it,(m 0 ) ω 0
. Under these three assumptions, it was proved in Lemma 2.10.4 [START_REF] Hafouta | Nonconventional limit theorems and random dynamics[END_REF] that condition (L) holds.

We now adapt the previous requirements to our setting. Assumption HK A1 is easily satisfied by requiring that σ is a homeomorphism that has at least one periodic point ω 0 and by building the cocycle in a way that ω → T ω is locally constant at all points that belong to the orbit of ω 0 . Of course, we also need to work with P that assigns positive measure to all open nonempty subsets of Ω.

Assumption HK A3 is equivalent to requiring the classical aperiodicity condition for deterministic systems (see [START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF]). Namely, it is sufficient to require that the spectral radius of L it,(m 0 ) ω 0 is strictly less than 1 for all t = 0. Then, taking J ⊂ R a compact interval, we can find c = c(J) > 0 and b = b(J) ∈ (0, 1) such that (48) holds by arguing as in [9, Proof of Lemma 4.7] (although the argument is simpler in our present setting since we deal with a deterministic situation).

Assumption HK A2 should instead be checked under suitable conditions since it relies on the properties of the maps, of the observable and of the functional spaces: we now show that it holds for our Anosov maps. This plus the necessary conditions in HK A1 and HK A3, will give us a proof of the LCLT in the non-arithmetic case. To verify HK A2 we first prove the equicontinuity property (under suitable conditions) in Lemma 8.1 and then in Lemma 8.3 we will develop a Lasota-Yorke inequality from which we can obtain the bound (47). The latter will be obtained from a Lasota-Yorke inequality for the twisted operator L it,(n) ω and with respect to the norm • 1,1 (strong) and • 0,2 (weak), and the fact that h 0,2 ≤ h 1,1 for any h ∈ B 1,1 , as we did in [START_REF] Gouëzel | Almost sure invariance principle for dynamical systems by spectral methods[END_REF] for the non-twisted operator.

Lemma 8.1. Let us suppose (Ω, F , P, σ) is an invertible and ergodic measure-preserving dynamical system verifying HK A1 and moreover for each i ∈ {0, 1, . . . , m 0 -1}, the observable g satisfies lim

ω→σ i ω 0 g(ω, •) -g(σ i ω 0 , •) C 2 = 0. ( 49
)
Furthermore, let J ⊂ R be a compact interval. Then, the family of maps {ω → L it ω : t ∈ J} is equicontinuous in all points ω that belong to the orbit of ω 0 . Remark 8.2. Observe that it is not enough to simply prescribe that (49) holds since we also need to make sure that this requirement is not spoiled when we center our observable (see [START_REF] Hafouta | Nonconventional limit theorems and random dynamics[END_REF]). It turns out that under suitable conditions we can make sure that (49) is preserved under centering. Let us first recall that (12) and (13) hold for every ω ∈ Ω. We assume that Ω is a metric space, σ : Ω → Ω is a homeomorphism and that ω → T ω is locally constant at points that belong to the orbit of ω 0 .

Let us now modify slightly Proposition 3.1 to ensure that under these additional assumptions we can say more about the top Oseledets space of our cocycle.

Set

Y = {v : Ω → B 1,1 : v measurable and v ∞ := sup ω∈Ω v(ω) 1,1 < ∞}.
Then, (Y, • ∞ ) is a Banach space. Let Y be a set of all v ∈ Y that are continuous at points σ i ω 0 , i = 0, 1, . . . , m 0 -1. We claim that Y is a closed subset of Y. Indeed, take a sequence

(v n ) n ⊂ Y such that v n → v in Y and fix i ∈ {0, 1, . . . , m 0 -1}. Then, we have that v(ω) -v(σ i ω 0 ) 1,1 ≤ v(ω) -v n (ω) 1,1 + v n (ω) -v n (σ i ω 0 ) 1,1 + v n (σ i ω 0 ) -v(σ i ω 0 ) 1,1 ≤ 2 v -v n ∞ + v n (ω) -v n (σ i ω 0 ) 1,1 .
Take ε > 0 and choose n such that

v -v n ∞ < ε 3 .
Since v n ∈ Y , we have that

v n (ω) -v n (σ i ω 0 ) 1,1 < ε 3 ,
whenever ω is sufficiently close to σ i ω 0 . Hence,

v(ω) -v(σ i ω 0 ) 1,1 < ε, whenever ω is sufficiently close to σ i ω 0 . Therefore, v ∈ Y and Y is closed. Set Z := {v ∈ Y ; v(ω) ≥ 0 and v(ω)(1) = 1 for ω ∈ Ω}.
Then, Z is a closed subset of Y (see the argument in the proof of Proposition 3.1) and hence it is a Banach space. We consider L : Z → Z defined by

(Lv)(ω) = L σ -1 ω v(σ -1 ω), ω ∈ Ω, v ∈ Z.
In order to show that L is well-defined, we only need to note that

ω → L σ -1 ω v(σ -1 ω)
is continuous at σ i ω 0 , i ∈ {0, . . . , m 0 -1}. However, this follows from the fact that v ∈ Z (and thus v ∈ Y ) and our assumption that ω → T ω (and thus also ω → L ω ) is locally constant along the orbit of ω 0 . It follows from ( 12) that L has the unique fixed point h 0 ∈ Z. This easily implies that ω → h 0 ω (g(ω, •)), h 0 ω := h 0 (ω) is continuous at σ i ω 0 , i = 0, . . . , m 0 -1 and therefore (49) will remain valid even after centering.

Proof of Lemma 8.1. We will prove the desired equicontinuity property in ω 0 . The argument for all other points in the orbit of ω 0 is completely analogous. Observe that for all ω ∈ Ω sufficiently close to ω 0 , we have that L ω = L ω 0 . Therefore, for all ω close to ω 0 , we have that (L it ω -L it ω 0 )(h) = L ω 0 ((e itg(ω,•)e itg(ω 0 ,•) )h), and thus

(L it ω -L it ω 0 )(h) 1,1 ≤ L ω 0 1,1 • (e itg(ω,•) -e itg(ω 0 ,•) )h 1,1 ,
for each h ∈ B 1,1 . Observe that (e itg(ω,•)e itg(ω 0 ,•) )h 1,1 = = max{ (e itg(ω,•)e itg(ω 0 ,•) )h ∼ 0,1 , (e itg(ω,•)e itg(ω 0 ,•) )h ∼ 1,2 }. As in the proofs of Propositions 4.3 and 4.4, we need to estimate e itg(ω,•)e itg(ω 0 ,•) C 2 . Take x ∈ X. By applying the mean-value theorem for the map z → e itz , we see that

|e itg(ω,x) -e itg(ω 0 ,x) | ≤ |t| • |g(ω, x) -g(ω 0 , x)|.
Thus,

e itg(ω,•) -e itg(ω 0 ,•) C 0 ≤ |t| • g(ω, •) -g(ω 0 , •) C 0 , which implies that e itg(ω,•) -e itg(ω 0 ,•) C 0 ≤ max{|t| : t ∈ J} g(ω, •) -g(ω 0 , •) C 2 . ( 50
)
Moreover, we have that

∂ j (e itg(ω,•) -e itg(ω 0 ,•) ) = ite itg(ω,•) ∂ j (g(ω, •)) -ite itg(ω 0 ,•) ∂ j (g(ω 0 , •)) = ite itg(ω,•) ∂ j (g(ω, •)) -ite itg(ω 0 ,•) ∂ j (g(ω, •)) + ite itg(ω 0 ,•) ∂ j (g(ω, •)) -ite itg(ω 0 ,•) ∂ j (g(ω 0 , •)),
for each j ∈ {1, . . . , d}. By (50), we have that

∂ j (e itg(ω,•) -e itg(ω 0 ,•) ) C 0 ≤ max{|t| 2 : t ∈ J} g(ω, •) -g(ω 0 , •) C 2 • g(ω, •) C 2 + max{|t| : t ∈ J} g(ω, •) -g(ω 0 , •) C 2 ,
for every j ∈ {1, . . . , d}. Thus,

max 1≤j≤d ∂ j (e itg(ω,•) -e itg(ω 0 ,•) ) C 0 ≤ C g(ω, •) -g(ω 0 , •) C 2 , (51)
for some C > 0 which is independent on t and ω. Finally, for each k, j ∈ {1, . . . , d}, we have that

∂ k ∂ j (e itg(ω,•) -e itg(ω 0 ,•) ) = -t 2 e itg(ω,•) ∂ k (g(ω, •))∂ j (g(ω, •)) + ite itg(ω,•) ∂ k ∂ j (g(ω, •)) + t 2 e itg(ω 0 ,•) ∂ k (g(ω 0 , •))∂ j (g(ω 0 , •)) -ite itg(ω 0 ,•) ∂ k ∂ j (g(ω 0 , •)).
Observe that

ite itg(ω,•) ∂ k ∂ j (g(ω, •)) -ite itg(ω 0 ,•) ∂ k ∂ j (g(ω 0 , •)) = ite itg(ω,•) ∂ k ∂ j (g(ω, •)) -ite itg(ω,•) ∂ k ∂ j (g(ω 0 , •)) + ite itg(ω,•) ∂ k ∂ j (g(ω 0 , •)) -ite itg(ω 0 ,•) ∂ k ∂ j (g(ω 0 , •)).
Thus (using (50)),

ite itg(ω,•) ∂ k ∂ j (g(ω, •)) -ite itg(ω 0 ,•) ∂ k ∂ j (g(ω 0 , •)) C 0 ≤ max{|t| : t ∈ J} g(ω, •) -g(ω 0 , •) C 2 + max{|t| 2 : t ∈ J} g(ω, •) -g(ω 0 , •) C 2 g(ω 0 , •) C 2 .
On the other hand,

-t 2 e itg(ω,•) ∂ k (g(ω, •))∂ j (g(ω, •)) + t 2 e itg(ω 0 ,•) ∂ k (g(ω 0 , •))∂ j (g(ω 0 , •)) = -t 2 e itg(ω,•) ∂ k (g(ω, •))∂ j (g(ω, •)) + t 2 e itg(ω 0 ,•) ∂ k (g(ω, •))∂ j (g(ω, •)) -t 2 e itg(ω 0 ,•) ∂ k (g(ω, •))∂ j (g(ω, •)) + t 2 e itg(ω 0 ,•) ∂ k (g(ω 0 , •))∂ j (g(ω, •)) -t 2 e itg(ω 0 ,•) ∂ k (g(ω 0 , •))∂ j (g(ω, •)) + t 2 e itg(ω 0 ,•) ∂ k (g(ω 0 , •))∂ j (g(ω 0 , •))
Hence, (50) implies that

-t 2 e itg(ω,•) ∂ k (g(ω, •))∂ j (g(ω, •)) + t 2 e itg(ω 0 ,•) ∂ k (g(ω 0 , •))∂ j (g(ω 0 , •)) C 0 ≤ max{|t| 3 : t ∈ J} g(ω, •) -g(ω 0 , •) C 2 • g(ω, •) 2 C 2 + t 2 g(ω, •) -g(ω 0 , •) C 2 • g(ω, •) C 2 + t 2 g(ω, •) -g(ω 0 , •) C 2 • g(ω 0 , •) C 2 .
We conclude that (by increasing C) we have that

sup 1≤k,j≤d -t 2 e itg(ω,•) ∂ k (g(ω, •))∂ j (g(ω, •)) + t 2 e itg(ω 0 ,•) ∂ k (g(ω 0 , •))∂ j (g(ω 0 , •)) C 0 ≤ C g(ω, •) -g(ω 0 , •) C 2 .
Thus,

e itg(ω,•) -e itg(ω 0 ,•) C 2 ≤ C g(ω, •) -g(ω 0 , •) C 2 , and L it ω -L it ω 0 ≤ C g(ω, •) -g(ω 0 , •) C 2 ,
for t ∈ J and ω in a neighborhood of ω 0 . The conclusion of the lemma follows directly from (49).

As we already announced, we now prove a Lasota-Yorke inequality for the twisted operator.

Lemma 8.3. For each t ∈ R, there exist A t , B t > 0, 0 < γ t < 1 such that for every n ≥ 0, h ∈ B 1,1 and P-a.e. ω ∈ Ω,

L it,(n) ω h 1,1 ≤ A t γ n t h 1,1 + B t h 0,2 . Moreover, for each J ⊂ R compact interval, we have that sup t∈J max{A t , B t } < ∞.
Proof. (Sketch). The proof follows verbatim the proof of Lemma 6.3 in [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF], with two differences. First, we work with composition of maps, but if they are close enough we can easily adapt the deterministic arguments (we recall that this was explicitly emphasized in [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]Section 7], in particular allowing for a random version of [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]Lemma 3.3] to be applied). Second, since we use the twisted operator instead of the usual one, in the various estimates in the proof of Lemma 6.3 [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF] we find the extra multiplicative factor e itSng(ω,•) . The proof of Lemma 6.3 [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF] is done by induction on the index p and the first step is to get a weak version of the Lasota-Yorke, namely for each t ∈ R, there exists C t ≥ 0 such that for every n ≥ 0 and P-a.e. ω ∈ Ω,

L it,(n) ω 0,q ≤ C t . (52)
We now prove (52) (with q = 1) to show how to handle the additional multiplicative factor. We use the notation as in [15, p.202]. Recall that

L (n) ω (h)(x) = h((T (n) ω ) -1 (x)) |det DT (n) ω ((T (n) ω ) -1 (x))|
, and therefore

L it,(n) ω (h)(x) = e itSng(ω,(T (n) ω ) -1 (x)) h((T (n) ω ) -1 (x)) |det DT (n) ω ((T (n) ω ) -1 (x))| , for each h ∈ C r and x ∈ X. Thus, W L it,(n) ω h • ϕ = (T (n) ω ) -1 (W ) hn • ϕ • T (n) ω • J W T (n) ω ,
where

J W T (n) ω is the Jacobian of T (n) ω : (T (n) ω ) -1 (W ) → W and hn := he itSng(ω,•) |det DT (n) ω | . Let ϕ j = ϕ • T (n) ω
• ρ j , where ρ 1 , . . . , ρ ℓ is a partition of unity on (T

ω ) -1 W , as provided by (the random analogue of) Lemma 3.3 [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF] (using γ = 1), and W 1 , . . . , W ℓ the corresponding admissible leaves such that (T

(n) ω ) -1 (W ) ⊂ ∪ ℓ j=1 W j .
Hence, [15, (6.2)] becomes

W j hn • ϕ j • J W T (n) ω ≤ C h 0,1 |det DT (n) ω | -1 • e itSng(ω,•) • ϕ j • J W T (n) ω C 1 (W j )
.

Note that

|det DT (n) ω | -1 •e itSng(ω,•) •ϕ j •J W T (n) ω C 1 (W j ) ≤ |det DT (n) ω | -1 •J W T (n) ω C 1 (W j ) •|ϕ j | C 1 (W j ) •|e itSng(ω,•) | C 1 (W j ) .
It follows from [15, Lemma 6.2] that

j≤ℓ ||det DT (n) ω | -1 • J W T (n) ω | C 1 (W j ) ≤ C.
In addition, from the argument at the bottom of [15, p. 203], it follows that

|ϕ j | C 1 (W j ) ≤ |ϕ • T (n) ω | C 1 (W j ) • |ρ j | C 1 (W j ) ≤ C.
Hence, in order to complete the proof of the weak Lasota-Yorke inequality, it is sufficient to show that

|e itSng(ω,•) | C 1 (W j ) ≤ C. (53) Note that |e itSng(ω,•) | C 0 (W j ) = 1 and |∂ α (e itSng(ω,•) )| C 0 (W j ) = |t| • |∂ α (S n g(ω, •))| C 0 (W j ) ≤ |t| n-1 i=0 |∂ α (g(σ i ω, T (i) ω (•)))| C 0 (W j ) .
In order to bound |∂ α (g(σ i ω, T

ω (•)))| C 0 (W j ) (i)
, we proceed as in [5, (4.3)]. For each i and x, y ∈ W j , we have

|g(σ i ω, T (i) ω x) -g(σ i ω, T (i) ω y)| d(x, y) = |g(σ i ω, T (i) ω x) -g(σ i ω, T (i) ω y)| d(T (i) ω x, T (i) ω y) • d(T (i) ω x, T (i)
ω y) d(x, y)

≤ Cν i ess sup ω∈Ω g(ω, •) C 1 ,
and thus

|∂ α (g(σ i ω, T (i) ω (•)))| C 0 (W j ) ≤ Cν i ess sup ω∈Ω g(ω, •) C 1 .
In view of ( 16), (53) holds. Now one can repeat arguments in [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF] to obtain the weak Lasota-Yorke inequality for the twisted cocycle, (52). The proof of the strong Lasota-Yorke inequality can be obtained in a similar manner.

Using Lemmas 8.1 and 8.3 we get: Theorem 8.4. Suppose that conditions HK A1 and HK A3 hold and the observable g satisfies (49). Then the same conclusions as those of Theorem C hold.

Piecewise hyperbolic dynamics

In this section, we apply the previous theory to obtain statistical laws for the random compositions T

(n) ω = T σ n-1 ω • • • • • T σω • T ω of
piecewise uniformly hyperbolic maps T ω of the type studied in [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]. The class of maps T ω considered contains piecewise toral automorphisms and piecewise hyperbolic maps with bounded derivatives; see Remark 2.2 [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]. 9.1. Preliminaries. We follow the construction of [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]. Let X be a two-dimensional compact Riemannian manifold, possibly with boundary and not necessarily connected and let T : X → X be a piecewise hyperbolic map in the sense of [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]. That is, the domain X is broken into a finite number of pairwise disjoint open regions {X + i } with piecewise C 1 boundary curves of finite length, such that i X + i = X. The image of each X + i under T is denoted X - i = T (X + i ); we assume that X - i = X. The sets S ± := X \ i X ± i are the "singularity sets" for T and T -1 , respectively. Assume that T is a C 2 diffeomorphism from the complement of S + to the complement of S -, and that for each i, there is a C 2 extension of T to X + i . On each X i , the map T is uniformly hyperbolic: there are two continuous, strictly DT -invariant families of cones C s and C u defined on X \ (S + ∪ ∂X) satisfying

λ := inf x∈X\S + inf v∈C u DT v v > 1, µ := inf x∈X\S + inf v∈C s DT v v < 1, µ -1 + := inf x∈X\S -inf v∈C s DT -1 v v > 1.
Assume that vectors tangent to the singularity curves in S -are bounded away from C s . The singularity curves and their images and preimages should not intersect at too many points. Denote by S - n (resp. S + n ) the set of singularity curves for T -n (resp. T n ), and let M(n) denote the maximum number of singularity curves that meet at a single point. Assume that there is an α 0 and an integer n 0 > 0 such that λµ α 0 > 1 and (λµ α 0 ) n 0 > M(n 0 ); this condition is satisfied if M(n) has polynomial growth, for example.

For each n ∈ N, let K n be the set of connected components of X \ S + n , and let C 1 (K, R) be the set of functions ϕ ∈ C 1 ( K, R) with C 1 extension in a neighbourhood of K. Let (C 1

S + n ) ′ := {ϕ ∈ L ∞ (X) : ϕ ∈ C 1 (K, R) ∀K ∈ K n }. If h ∈ (C 1 S + n ) ′ is an element of the dual of C 1 S + n , then L : (C 1 S + n ) ′ → (C 1 S + n-1
) ′ acts on h by

Lh(ϕ) = h(ϕ • T ) ∀ϕ ∈ C 1 S + n-1 .
In order to obtain useful spectral information from L, its action is restricted to a Banach space B, analogous to the space B p,q = B 1,1 in Section 2. We now briefly outline the construction of the norms on B and an associated "weak" space B w ; see [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF] for details. The norms are defined using "admissible leaves" W in a set of admissible leaves Σ. These leaves are smooth curves in approximately the stable direction, and are analogues of the ψ i • (Id, χ) defined in Section 2. Since we are going to recall several times estimates in [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF], we intend to comply with the notation there. In particular the functions χ defined on the charts will now become F and the image of the graph of F , namely the admissible leaves, will be denoted with G F . For α, β, q < 1 such that 0 < β ≤ α ≤ 1q ≤ α 0 let C α (W, C) denote the set of continuous complex-valued functions on W with Hölder exponent α and define the norm

|ϕ| W,α,q := |W | α • |ϕ| C q (W,C) , (54)
where |W | denotes unnormalised induced Riemannian volume of W . For h ∈ C 1 (X, C) we define the weak norm of h by

|h| w = sup W ∈Σ sup ϕ∈C 1 (W,C) |ϕ| C 1 (W,C) ≤1 W hϕ dm
and the strong norm by

h = h s + b h u ,
where the strong stable norm is

h s = sup W ∈Σ sup ϕ∈C 1 (W,C) |ϕ| W,α,q ≤1 W hϕ dm (55)
and the strong unstable norm is

h u = sup ǫ≤ǫ 0 sup W 1 ,W 2 ∈Σ d Σ (W 1 ,W 2 )≤ǫ sup |ϕ i | C 1 (W i ,C) ≤1 dq(ϕ 1 ,ϕ 2 )≤ǫ 1 ǫ β W 1 hϕ 1 dm - W 2 hϕ 2 dm , (56)
where d Σ and d q are defined precisely in §3.1 [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]. In comparison to the setting in Section 2, the norm | • | w plays the role of • p-1,q+1 = • 0,2 , and the norm • plays the role of • p,q = • 1,1 .

Let B be the completion of C 1 (X, C) with respect to the norm • . Similarly, we define B w to the completion of C 1 (X, C) with respect to the norm |•| w .

We recall that the elements of B are distributions. More precisely, there exists C > 0 such that any h ∈ B induces a linear functional ϕ → h(ϕ) with the property that

|h(ϕ)| ≤ C|h| w |ϕ| C 1 , for ϕ ∈ C 1 (X, C), (57)
see [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]Remark 3.4] for details. In particular, for h ∈ C 1 (X, C) we have that (see [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]Remark 2.5])

h(ϕ) = X hϕ, for ϕ ∈ C 1 (X, C). (58)
We say that h ∈ B is nonnegative and write h ≥ 0 if h(ϕ) ≥ 0 for any ϕ ∈ C 1 (X, R) such that ϕ ≥ 0. Finally, we recall (see [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]Section 2.1]) that for h ∈ L 1 (X, C),

Lh = h |det DT | • T -1 . ( 59
)
Proposition 9.1. We have that

(Lh)(ϕ) = h(ϕ • T ), for h ∈ B and ϕ ∈ C 1 (X, C).
Proof. For h ∈ C 1 (X, C) the desired conclusion can be easily obtained from (58) and (59) by using a change of variables. This immediately implies that the conclusion holds for any h ∈ B. 9.2. Building the cocycle. This section follows the material in Section 3, replacing (B 1,1 , • 1,1 ) with (B, • ) and (B 0,2 , • 0,2 ) with (B w , • w ). We have included this material to make the relevant references to [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF] transparent. [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]Theorem 2.8] implies that the associated transfer operator L T is quasicompact on B, 1 is a simple eigenvalue and there are no other eigenvalues of modulus 1. This in particular implies (using the terminology as in [START_REF] Conze | Limit theorems for sequential expanding dynamical systems on [0, 1[END_REF]Definition 2.6]) that L T is exact in {h ∈ B : h(1) = 0}.

Let Γ B * and X ǫ be the sets of maps as defined in [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]Section 2.4]. By applying [5, Lemma 6.1], we find that there exists C > 0 such that

sup h ≤1 |(L T ′ -L T )h| w ≤ Cǫ β for T ′ ∈ X ǫ .
It then follows from [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]Lemma 3.5] and the discussion on [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]Section 2.4] that there exist ǫ, A > 0 and c ∈ (0, 1) such that for any T ′ ∈ Γ ǫ , we have that

• the unit ball in B is relatively compact in B w ; • |L n T ′ h| w ≤ A|h| w for each n ∈ N and h ∈ B; • L n T ′ h ≤ Ac n h + A|h| w for each n ∈ N and h ∈ B.

Consider now a family of operators

P = {L T ′ : T ′ ∈ X ǫ }.
It then follows from [4, Proposition 2.10] (applied to the case where

• = |•| w and |•| v = • )
that there exists 0 < ǫ ′ ≤ ǫ, D, λ > 0 such that for any T 1 , . . . , T n ∈ X ǫ ′ , we have that

L Tn • • • L T 2 L T 1 h ≤ De -λn h for h ∈ B satisfying h(1) = 0, (60)
where L T i denotes the transfer operator associated with T i . From now on, we replace ǫ ′ with ǫ so that (60) holds for T 1 , . . . , T n ∈ X ǫ . We now build our cocycle by prescribing that for each ω ∈ Ω, T ω ∈ X ǫ ′ and we consider L ω which is the transfer operator associated to T ω . Then, it follows readily from (60) that

L (n) ω h ≤ De -λn h for any ω ∈ Ω, n ∈ N and h ∈ B, h(1) = 0. (61
) In addition, by decreasing ǫ if necessary, we have (see the proof of [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF]Lemma 6.3]) that there exist a ∈ (0, 1) and B > 0 such that

|L (n) ω h| w ≤ B|h| w and L (n) ω h ≤ Ba n h + B|h| w , (62)
for every ω ∈ Ω, n ∈ N and h ∈ B. In particular, there exists K > 0 such that

L ω h ≤ K h for ω ∈ Ω and h ∈ B. (63)
9.2.1. P-continuity of ω → L ω . We assume Ω is a Borel subset of a complete separable metric space, F is the Borel sigma-algebra and σ is a homeomorphism. Unfortunately, in this (piecewise-hyperbolic) setting we are unable to establish strong measurability of the map ω → L ω under the assumption that ω → T ω is measurable. In order to be able to apply the weaker version of MET from [START_REF] Froyland | A semi-invertible Oseledets theorem with applications to transfer operator cocycles[END_REF], we ask instead that ω → T ω is measurable and that it has a countable range. 9.2.2. Quasi-compactness of the cocycle L and existence of Oseledets splitting. Similarly to the description at the end of Section 3.2, by Lemma 2.1 [START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF], the inequalities (62) and (63) imply that the cocycle R is quasi-compact. By quasi-compactness and P-continuity of L, the multiplicative ergodic theorem (Theorem 17, [START_REF] Froyland | A semi-invertible Oseledets theorem with applications to transfer operator cocycles[END_REF]) yields the existence of a unique P-continuous Oseledets splitting

B 1,1 = l j=1 Y j (ω) ⊕ V (ω),
where each component of the splitting is equivariant under L ω . The Y j (ω) are finitedimensional and have corresponding (finite or infinite) sequence of Lyapunov exponents 0 = λ 1 > λ 2 > . . .

Ω × X → R such that g(ω, •) ∈ C 1 for ω ∈ Ω and M := ess sup ω∈Ω g(ω, •) C 1 < ∞. (64)
We assume that g is ω-fibrewise centred: for P-a.e. ω ∈ Ω, h 0 ω (g(ω, •)) = 0. For g ∈ C 1 (X, C) and h ∈ B, we can introduce g • h ∈ B as in Section 2. Furthermore, for ω ∈ Ω, θ ∈ C, and h ∈ B set L θ ω (h) = L ω (e θg(ω,•) h). We will need the following lemma, which is analogous to Lemma 3.2 [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF]. Lemma 9.2. For h ∈ B and g ∈ C 1 (X, C), we have that gh ≤ C|g| C 1 h , for some C > 0, independent of g and h.

Proof. It is sufficient to establish the desired conclusion for h ∈ C 1 (X, C). Note that gh = gh s + b gh u .

(65)

We have

gh s = sup W ∈Σ sup ϕ∈C 1 (W,C) |ϕ| W,α,q ≤1 W hϕg dm ≤ |g| C 1 • h s , ( 66
) since |ϕg| W,α,q = |W | α |ϕg| C q (W,C) ≤ |W | α |ϕ| C q (W,C) |g| C 1 = |ϕ| W,α,q |g| C 1 ≤ |g| C 1 .
Furthermore,

gh u = sup ǫ≤ǫ 0 sup W 1 ,W 2 ∈Σ d Σ (W 1 ,W 2 )≤ǫ sup |ϕ i | C 1 (W i ,C) ≤1 dq(ϕ 1 ,ϕ 2 )≤ǫ 1 ǫ β W 1 hϕ 1 g dm - W 2 hϕ 2 g dm .
Using the notation as in [5, p.12] we have that

1 ǫ β W 1 hϕ 1 g dm - W 2 hϕ 2 g dm ≤ 1 ǫ β W 1 hϕ 1 g dm - W 2 h((ϕ 1 g) • Φ) dm + 1 ǫ β W 2 h((ϕ 1 g) • Φ) dm - W 2 hϕ 2 g dm =: (I) + (II),
where

Φ := G F 1 • G -1 F 2 ,
and F 1 and F 2 are respectively the parametrization of W 1 and W 2 in the local charts.

Let us first estimate term (I). Note that

|ϕ 1 g| C 1 (W 1 ,C) ≤ |ϕ 1 | C 1 (W 1 ,C) • |g| C 1 (W 1 ,C) ≤ |g| C 1 . (67)
We now bound the term |(ϕ 1 g) • Φ| C 1 (W 2 ,C) . In the estimates that follow C > 0 will denote an arbitrary positive number independent of g and h. Observe that

|(ϕ 1 g) • Φ| C 0 (W 2 ,C) ≤ |ϕ 1 g| C 0 (W 1 ,C) ≤ |ϕ 1 | C 0 (W 1 ,C) • |g| C 0 ≤ |ϕ 1 | C 1 (W 1 ,C) • |g| C 1 ≤ |g| C 1 . Furthermore, 6 Lip 1,W 2 ((ϕ 1 g) • Φ) ≤ Lip 1,W 1 (ϕ 1 g) • Lip 1,W 2 (Φ) ≤ CLip 1,W 1 (ϕ 1 g), since sup F max{|G F | C 1 , |G -1 F | C 1 } < ∞. Moreover, since Lip 1,W 1 (ϕ 1 g) ≤ |ϕ 1 | C 0 (W 1 ,C) Lip 1,W 1 (g) + |g| C 0 (W 1 ,C) Lip 1,W 1 (ϕ 1 ), it follows that Lip 1,W 1 (ϕ 1 g) ≤ C|g| C 1 . Consequently, |(ϕ 1 g) • Φ| C 1 (W 2 ,C) ≤ C|g| C 1 .
(68) Finally, we observe that

d q (ϕ 1 g, (ϕ 1 g) • Φ) = |(ϕ 1 g) • G F 1 -(ϕ 1 g) • Φ • G F 2 | C q (Ir 1 ,C) = 0, ( 69
) since Φ • G F 2 = G F 1 .
Hence, (67), ( 68) and (69) imply that

(I) ≤ C|g| C 1 h u . (70)
In order to estimate (II), we need to bound

|(ϕ 1 g) • Φ -ϕ 2 g| W 2 ,α,q = |W 2 | α • |(ϕ 1 g) • Φ -ϕ 2 g| C q (W 2 ,C) ≤ C|(ϕ 1 g) • Φ -ϕ 2 g| C q (W 2 ,C) . Note that |(ϕ 1 g) • Φ -ϕ 2 g| C q (W 2 ,C) = |(ϕ 1 g) • Φ • G F 2 • G -1 F 2 -(ϕ 2 g) • G F 2 • G -1 F 2 | C q (W 2 ,C) ≤ C|(ϕ 1 g) • Φ • G F 2 -(ϕ 2 g) • G F 2 | C q (Ir 1 ,C) = |(gϕ 1 ) • G F 1 -(gϕ 2 ) • G F 2 | C q (Ir 1 ,C) ≤ |(g • G F 1 )(ϕ 1 • G F 1 ) -(g • G F 1 )(ϕ 2 • G F 2 )| C q (Ir 1 ,C) + |(g • G F 1 )(ϕ 2 • G F 2 ) -(g • G F 2 )(ϕ 2 • G F 2 )| C q (Ir 1 ,C) ≤ |g • G F 1 | C q (Ir 1 ,C) • d q (ϕ 1 , ϕ 2 ) + |g • G F 1 -g • G F 2 | C q (Ir 1 ,C) • |ϕ 2 • G F 2 | C q (Ir 1 ,C) ≤ ǫ|g • G F 1 | C 1 (Ir 1 ,C) + |g • G F 1 -g • G F 2 | C q (Ir 1 ,C) • |ϕ 2 • G F 2 | C 1 (Ir 1 ,C) . 6 Lip q,W (f ) := sup x =y x,y∈W |f (x)-f (y)| |x-y| q . Since sup F |G F | C 1 (Ir,C) < ∞, we have that |g • G F 1 | C 1 (Ir 1 ,C) ≤ C|g| C 1 and |ϕ 2 • G F 2 | C 1 (Ir 1 ,C) ≤ C.
Finally, it remains to estimate

|g • G F 1 -g • G F 2 | C q (Ir 1 ,C) .
Since d Σ (W 1 , W 2 ) ≤ ǫ, it follows from the mean-value theorem that and setup from [5, p.12

• G F 1 (t) -g • G F 2 (t) -g • G F 1 (s) + g • G F 2 (s)| |t -s| q ≤ 2Cǫ|g| C 1
|t -s| q . On the other hand, by applying the mean-value theorem we have

|g • G F 1 (t) -g • G F 2 (t) -g • G F 1 (s) + g • G F 2 (s)| |t -s| q ≤ 2C|g| C 1 |t -s| |t -s| q .
One can now proceed as in [5, p.20] to show that

sup t =s |g • G F 1 (t) -g • G F 2 (t) -g • G F 1 (s) + g • G F 2 (s)| |t -s| q ≤ C|g| C 1 ǫ 1-q .
This together with (71) implies that

|g • G F 1 -g • G F 2 | C q (Ir 1 ,C) ≤ C|g| C 1 ǫ 1-q .
We conclude that

|(ϕ 1 g) • Φ -ϕ 2 g| C q (W 2 ,C) ≤ Cǫ 1-q |g| C 1 ≤ Cǫ β |g| C 1 .
Thus, (II) ≤ C|g| C 1 h s .

(72) The conclusion of the lemma follows directly from (65), (66), (70) and (72).

The following proposition is analogous to Proposition 4.3. σ N-j ω (L θ σ N-1-j ω -L σ N-1-j ω )L (N -1-j) ω .

It follows from ( 63) and (73) that L (N -1-j) ω ≤ K N -1-j and L θ,(j) σ N-j ω ≤ K(θ) j . Furthermore, using (63) and Lemma 9.2, we have that for any h ∈ B and P-a.e. ω ∈ Ω, (L θ ω -L ω )(h) = L ω (e θg(ω,•) hh) ≤ K (e θg(ω,•) -1)h ≤ CK|e θg(ω,•) -1| C 1 h . On the other hand, using (64) and applying the mean value theorem for the map z → e θz , it is easy to verify that there exists C ′ > 0 such that for θ ∈ B C (0, 1), |e θg(ω,•) -1| C 1 ≤ C ′ |θ| for P-a.e. ω ∈ Ω.

(74)

Hence, there exists C > 0 such that L θ ω -L ω ≤ C|θ|, for P-a.e. ω ∈ Ω. We conclude that

L θ,(N ) ω -L (N ) ω ≤ C|θ| N -1 j=0 K N -1-j K(θ) j ,
and therefore there exists γ ∈ (0, 1) such that for any θ sufficiently close to 0 and h ∈ B, L θ,(N )

ω h ≤ γ h + B|h| w . (75)
Similarly, one can show that there exists B > 0 such that for any θ sufficiently close to 0 and h ∈ B, |L θ ω h| w ≤ B|h| w .

(76)

The conclusion of the proposition follows from (75) and (76) by arguing as in [9, Theorem 3.12]. 9.4. Regularity of the top Oseledets space, convexity of Λ. The regularity of the top Oseledets space of the twisted cocycles follows identically as in Section 5, with Lemma 9.2 used in place of Lemma 3.2 [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF] in the proof of Lemma 5.1. Moreover the family of probability measures h 0 ω will allow us to define the fibred measure µ ω as we did in Section 6.3. difference: its derivative DT becomes infinite near singularities. This fact will induce a slight change in the definition of the norms. The latter are defined exactly as in section 9 but the norm on the test function ϕ (54) is now modified as |ϕ| W,α,q = |W | α • cos W • |φ| C q (W,C) , where cos W = 1 m W (W ) W cos θdm W , being m W the unnormalized Lebesgue measure on the stable curve W. With this precaution, the Banach space B is defined as the completion of C 1 (M, C) with respect to the norm given by the sum of the strong norm defined in (55) and the strong unstable norm given by (56). In [START_REF] Demers | Spectral analysis of the transfer operator for the lorentz gas[END_REF], Demers and Zhang established the Lasota-Yorke inequality and the associated spectral picture in the deterministic setting of a single billiard map T . The main technical difference with [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF] was the control of distortion which required additional cuts at the boundaries of homogeneity strips with the consequence of generating a countably infinite number of curves in both the finite and infinite horizon cases. In [START_REF] Demers | A functional analytic approach to perturbations of the lorentz gas[END_REF] the same authors introduced a distance between maps, see section 3.4 in [START_REF] Demers | A functional analytic approach to perturbations of the lorentz gas[END_REF]. Then they consider a family F of billiard maps such that by taking composition of maps close in F , that composition has the same hyperbolic and distortion properties of the iterates of a single map, see the discussion in section 5.3 in [START_REF] Demers | A functional analytic approach to perturbations of the lorentz gas[END_REF]. Moreover they defined a random walk on M by choosing the sequence of maps in F in an i.i.d. way with a prescribed density (see their g(ω, •) in section 2.3), where ω belongs to the probability space (Ω, ν). By averaging the Perron-Fröbenius operators associated to the maps in the sequence over ν they finally defined an averaged transfer operator and applied to it standard perturbation theory, thus getting the annealed limit theorems stated in their Theorem 2.6. Our approach is devoted to quenched results; for that and by eventually reducing the family F by the Conze-Raugi criterion which we already employed in the previous sections, we get the quenched Lasota-Yorke inequality and the exponential decay expressed by the bound (61), which are the bases of our theory. Theorems A,B and C then follows for the billiards maps associated with perturbations of the periodic Lorentz gas described above.

Lemma 5 . 5 .

 55 Let D = {θ ∈ C : |θ| < ǫ}×B S (0, R) be as in Lemma 5.3. Then, F : D → S is C ∞ and the equation F (θ, W) = 0 (36) has a unique solution O(θ) ∈ S, for every θ in a neighborhood of 0. Furthermore, O(θ) is a C ∞ function of θ.

6 . 2 .

 62 of part 3 is identical to the proof ofLemma 3.10 [9], using differentiability of H and O in Lemmas 5.2 and 5.5.The proof of part 4 is identical to proof of Lemma 3.11[START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF]. One-dimensionality of Y θ 1 (ω) and differentiability of Λ. Let Y θ 1 (ω) denote the top Oseledets subspace of the cocycle (L θ ω ) ω∈Ω . The proof of part 1 of the following result can be obtained by repeating the argument as in [9, Theorem 3.12], using Proposition 4.4. Part 2 follows by arguing as in [9, Corollary 3.14]. Proposition 6.2. For θ ∈ C near 0

) Remark 7 . 1 .

 71 The differentiability of θ → φ θ follows similarly to the presentation in[START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] Appendix C]. The proofs of Lemmas C.4 and C.6 make use of regularity estimates (90) and (99) in terms of variation; in the present work, these estimates may be replaced with C 1 estimates. In the proof of Lemma C.2, the expression v 0 ω 1 may be replaced with |h 0 ω (1)| and bounded by[START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF] in the present work.In addition, letB 1,1 = Y θ ω ⊕ H θ ω and (B 1,1 ) * = Y * θ ω ⊕ H * θ ωbe the Oseledets splitting of cocycles (L θ ω ) ω∈Ω and ((L θ ω ) * ) ω∈Ω respectively into a direct sum of the top space and the sum of all other Oseledets subspaces. 7.1. Large deviation principle. The following lemmas link the limits of characteristic functions of Birkhoff sums to the function Λ. Lemma 7.2. Let θ ∈ C be sufficiently close to 0 and h ∈ B 1,1 be such that h / ∈ H θ ω , i.e. φ θ ω (h) = 0. Then, lim n→∞ 1 n log h(e θSng(ω,•) ) = Λ(θ).

Proposition 9 . 3 .Proposition 9 . 4 .≤

 9394 There exists a continuous function K : C → (0, ∞) such thatL θ ω h ≤ K(θ) h , for h ∈ B, θ ∈ C and P-a.e. ω ∈ Ω. (73)Proof. Note that it follows from (63) and Lemma 9.2 thatL θ ω h = L ω (e θg(ω,•) h) ≤ K e θg(ω,•) h ≤ CK|e θg(ω,•) | C 1 h , for h ∈ B, θ ∈ C and P-a.e. ω ∈ Ω. Furthermore, observe that (64) implies that |e θg(ω,•) | C 0 ≤ e M |θ| for P-a.e. ω ∈ Ω.Similarly, it follows from the mean-value theorem (applied for a map z → e θz ) and (64) thatsup x =y |e θg(ω,x)e θg(ω,y) | |x -y| ≤ |θ|e 2M |θ| sup x =y |g(ω, x)g(ω, y)| |x -y| ≤ M|θ|e 2M |θ| .The desired conclusion follows directly from the above estimates.Analogously to Proposition 4.4 we have: For θ close to 0, the cocycle (L θ ω ) ω∈Ω is quasicompact. Proof. We follow closely[START_REF] Dragičević | A spectral approach for quenched limit theorems for random expanding dynamical systems[END_REF] Lemma 3.13]. Observe (62) and choose N ∈ N such that γ := Ba N < 1. Hence, L θ,(N ) ω h ≤ L(N ) ω h + L θ,(N ) γ h + B|h| w + L θ,(N )

  9.2.3. One-dimensionality of the top Oseledets space. The material in section 3.3 of the present work can be reused verbatim in the piecewise hyperbolic setting, replacing (B 1,1 , • 1,1 ) with (B, • ) and (B 0,2 , • 0,2 ) with (B w , • w ). In particular we can construct a unique family of probability measures (h 0 ω ) ω∈Ω ⊂ B such that for P-a.e. ω ∈ Ω, L ω h 0

	h 0 σω .	ω =

9.3. The twisted cocycle. Our observable will be a map g :

  ] that |g • G F 1 (t)g • G F 2 (t)| ≤ Cǫ|g| C 1 for t ∈ I r 1 .(71) Indeed, we have that|g • G F 1 (t)g • G F 2 (t)| ≤ |g| C 1 |G F 1 (t) -G F 2 (t)| = |g| C 1 |χ j (x 1 + (t, F 1 (t)))χ j (x 2 + (t, F 2 (t)))| ≤ C|g| C 1 (|x 1x 2 | + |F 1 (t) -F 2 (t)|) ≤ C|g| C 1 d Σ (W 1 , W 2 ),for t ∈ I r 1 which implies (71). Take now t, s ∈ I r 1 . Then, (71) implies that |g

See the discussion at the beginning of §7 in[START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF].

10. Acknowledgements DD was supported by Croatian Science Foundation under the project IP-2014-09-2285 and by the University of Rijeka under the project number 17.15.2.2.01. GF and CGT thank AMU, CPT and CIRM (Marseille) for hospitality during this research. GF is partially supported by an ARC Discovery project. CGT is supported by an ARC DECRA. SV was supported by the MATH AM-Sud Project Physeco, and by the project APEX Systèmes dynamiques: Probabilitès et Approximation Diophantienne PAD funded by the Région PACA (France). SV warmly thanks the LabEx Archiméde (AMU University, Marseille), and INdAM (Italy).

9.5. Large deviation principle and central limit theorem. The results of Sections 6 and 7 follow verbatim with the obvious modifications. We thus obtain our main results for piecewise hyperbolic dynamics.

Theorem D (Quenched large deviations theorem). In the setting of Section 9, there exists ǫ 0 > 0 and a non-random function c : (-ǫ 0 , ǫ 0 ) → R which is nonnegative, continuous, strictly convex, vanishing only at 0 and such that

, for 0 < ǫ < ǫ 0 and P-a.e. ω ∈ Ω.

Theorem E (Quenched central limit theorem). In the setting of Section 9, assume that the non-random variance Σ 2 , defined in (2) satisfies Σ 2 > 0. Then, for every bounded and continuous function φ : R → R and P-a.e. ω ∈ Ω, we have

(The discussion in §6.3 deals with the degenerate case Σ 2 = 0). 9.6. Local central limit theorem.

Theorem F (Quenched Local central limit theorem). In the setting of Section 9, suppose that condition (L) holds, where the functional norm in (L) is now B.

Then, for P-a.e. ω ∈ Ω and every bounded interval J ⊂ R, we have

The LCLT can also be obtained under the assumptions HK A1, A2, A3 and the hypothesis of Lemma 8.1 with the obvious change of the functional space which is now B. The Lasota-Yorke inequality for the twisted operator follows now by adapting the analogous proof in [START_REF] Demers | Stability of statistical properties in two-dimensional piecewise hyperbolic maps[END_REF] for the usual operator. So we have the analogous statement as in Theorem 8.4. 9.7. Billiards. The results of this section also apply to the billiard map associated with both a finite and infinite horizon Lorentz gas having smooth scatterers with strictly positive curvature: we refer in the following to the papers by Demers and Zhang [START_REF] Demers | Spectral analysis of the transfer operator for the lorentz gas[END_REF][START_REF] Demers | A functional analytic approach to perturbations of the lorentz gas[END_REF].

We first recall the setting. Let us consider on the bidimensional torus T 2 a finite number of pairwise disjoint and simply connected convex regions {Γ} d i=1 , which moreover have C 3 boundary curves ∂Γ i with strictly positive curvature. We denote by intA the interior of the set A; then the billiard table Q is defined as Q = T 2 \ ∪ i intΓ i . On the phase space M = Q × S 1 / ∼, with the conventional identifications at the boundaries, we define the billiard flow, which is induced by a particle traveling at unit speed and undergoing elastic collisions at the boundaries. We will be concerned instead with the billiard map T : M → M as the Poincaré map corresponding to collisions with scatterers and defined on M = ∪ i ∂Γ i × [-π/2, π/2]. We put on M the coordinates (r, θ), where r ∈ ∪ i ∂Γ i is parametrized by arc length and θ is the angle formed by the unit tangent vector at r with the normal pointing into the domain Q. The map T preserves a probability measure µ defined by dµ = c cos θdrdθ, where c is the normalizing constant. If we denote by τ (x) the first non-tangential collision time of the orbit flow starting at x, then T is defined when τ (x) < ∞ and in this case it is uniformly hyperbolic. In particular T has a finite horizon if τ is bounded from above, otherwise T has infinite horizon. The map T shares the same properties of the piecewise hyperbolic maps studied in this section, with a relevant