Local large deviations for periodic infinite horizon Lorentz gases

We prove local large deviations for the periodic infinite horizon Lorentz gas viewed as a ${\mathbb Z}^d$-cover ($d=1,2$) of a dispersing billiard. In addition to this specific example, we prove a general result for a class of nonuniformly hyperbolic dynamical systems and observables associated with central limit theorems with nonstandard normalisation.


Introduction
Local large deviations (LLD) for one dimensional i.i.d.random variables that do not satisfy the classical central limit theorem (with the standard normalisation) but are in the domain of a stable law were recently obtained by Caravenna & Doney [9, Theorem 1.1] and refined by Berger [6,Theorem 2.3].Such results have been extended to multivariate i.i.d.random variables in the domain of the stable laws by Berger in [7].Roughly speaking, an LLD measures the probability that the sum of the random variables assumes precise, but asymptotically large values.In the absence of second and even first moments, the proofs are considerably harder.
For dynamical systems, the first LLD results in the absence of the classical central limit theorem were obtained in [17]; they are as optimal as [6,Theorem 2.3].The main shift in that paper is an analytic proof which overcomes the restriction of having independence.Although promising, the results in [17] are limited to the Gibbs Markov maps.The aim of this paper is to prove an LLD estimate for infinite horizon periodic Lorentz maps, which were shown to satisfy a central limit theorem with nonstandard normalisation by Szász & Varjú [21].A crucial new ingredient of the proofs of the present LLD results consists of a new operator renewal technique on the Young tower for the billiard map.
Periodic dispersing billiards and Lorentz gases were introduced into ergodic theory and studied by [20].For a general reference, see [11].We recall that the classical central limit theorem was proved in the finite horizon case by [8] and local, moderate and large deviations were recently obtained in Dolgopyat & Nándori [13].In the same work [13] the authors designed a strategy to prove the local limit theorem and mixing properties for group extensions (such as Z d ) of probability preserving flows by free flight functions with finite second moments.For a similar strategy but weaker results we refer to [3].The strategy in [13] consists of the systematic use of local large and moderate deviations for the underlying probability preserving Poincaré map.Their result applies to the finite horizon Lorentz flow.In that case, both the free flight and the roof function are bounded.The LLD obtained in this paper (see Theorem 1.1 below) is close to optimal (see Remark 1.2 below) and we believe that it can be used to prove the local limit theorem and mixing properties for the infinite horizon Lorentz flow.
An infinite horizon periodic Lorentz map ( T , M , μ) is a Z d -cover of an infinite horizon periodic dispersing billiard (T, M, µ) where μ and µ are the Liouville measures on M and M. The notation for the infinite horizon dispersing billiard is recalled in Section 2. We consider the cases d = 1 (tubular billiard) and d = 2 (planar billiard).
Let κ : M → Z d denote the free flight function between collisions, and define κ n = n−1 j=0 κ • T j .For the Lorentz gas, geometrically κ n ∈ Z d denotes the cell in the infinite measure phase space M where the n'th collision takes place for initial conditions starting in the 0'th cell.Set a n = n log n.
The central limit theorem with nonstandard normalisation proved in [21] says that a −1 n κ n converges in distribution to a d-dimensional normal distribution.In fact, [21] proves a stronger result, namely the corresponding local limit theorem.Our main result is: 1   Theorem 1.1 (LLD for the dispersing billiard) There exists C > 0 such that for all n ≥ 1, N ∈ Z d , Again, there is the geometric interpretation that µ(κ n = N) represents the probability that an initial condition in the 0'th cell of M lies in the N'th cell after n collisions.(c) For the dispersing billiard, the improved bound in (a) follows from a uniform version [18] of the local limit theorem [21] in the range N ≪ √ n log n.Hence, the principal novelty of Theorem 1.1 lies in the range N ≫ √ n log n.We note that, as in [17], the approach in this paper does not rely on the local limit theorem and extends to situations where the local limit theorem fails, see Theorem 7.1.
The approach in this paper, following [17], is Fourier analytic and relies on smoothness properties of the leading eigenvalues and their spectral projections for the appropriate transfer operator.We show how to obtain C r control for all r < 2, going considerably beyond previous estimates of [4,18].The methods developed in Section 5 to obtain this control in the context of exponential Young towers are the main technical advance of this paper and should have other applications, not only to LLD.
In Section 2, we recall the setting for dispersing billiards.In Section 3, we prove Theorem 1.1 in the range n ≪ log |N|.Sections 4 to 6 treat the complementary range log |N| ≤ ǫ 1 n where ǫ 1 is chosen sufficiently small.Key technical estimates are stated in Section 4 and proved in Section 5.In Section 6, we complete the proof of Theorem 1.1.In Section 7, we state and prove an abstract version, Theorem 7.1, of our main result, giving an LLD for a general class of nonuniformly hyperbolic systems modelled by Young towers with exponential tails.

Notation We use "big O" and ≪ notation interchangeably, writing b
We write B r (x) to denote the open ball in R d and C of radius r centred at x.

Set up
Recall that the Lorentz map ( T , M , μ) is a Z d -cover (here d = 1, 2) of the dispersing billiard (T, M, µ) by the free flight function κ : M → Z d .The measures μ and µ are T -invariant (respectively T -invariant) and are the Liouville measures on M and M, normalised so that µ is a probability measure.The dispersing billiard (T, M, µ) is the collision map on the billiard domain Q = T2 \Ω where T 2 = R 2 /Z 2 and Ω is a finite union of convex obstacles with C 3 boundaries and nonvanishing curvature.The two-dimensional phase space (position in ∂Ω and unit velocity) is given by M = ∂Ω × (−π/2, π/2).
Let Q denote the lifted domain inside An important part of the proof of Theorem 1.1 is that (T, M, µ) is modelled by a two-sided Young tower (f, ∆, µ ∆ ) with exponential tails [10,22].We briefly recall the notion of Young tower. 2  Let (Y, µ Y ) be a probability space with an at most countable measurable partition α, and let F : Y → Y be an ergodic measure-preserving transformation.For θ ∈ (0, 1), define the separation time s(y, y ′ ) to be the least integer n ≥ 0 such that F n y and F n y ′ lie in distinct partition elements in α.It is assumed that the partition α separates trajectories, so s(y, y ′ ) = ∞ if and only if y = y ′ ; then d θ (y, y ′ ) = θ s(y,y ′ ) is a metric.We say that F is a (full-branch) Gibbs-Markov map if Let F : Y → Y be a Gibbs-Markov map and let σ : Y → Z + be constant on partition elements such that µ Y (σ > n) = O(e −an ) for some a > 0, We define the one-sided Young tower with exponential tails ∆ = Y σ and tower map f : ∆ → ∆ as follows: Let σ = Y σ dµ Y .Then μ∆ = (µ Y ×counting)/σ is an ergodic f -invariant probability measure on ∆.We say that (T, M, µ) is modelled by a Young tower (f, ∆, µ ∆ ) with exponential tails if there exist a one-sided Young tower ( f , ∆, μ∆ ) and measure-preserving semiconjugacies π : ∆ → M, π : ∆ → ∆.
We end this subsection by recalling some results about transfer operators and perturbed transfer operators on the one-sided tower.Let P : L 1 ( ∆) → L 1 ( ∆) be the transfer operator for ( f , ∆, μ∆ ), so [4,Section 3.3], there is a Banach space B ′ ⊂ L 1 (called H in [4]) such that P : B ′ → B ′ is quasicompact.(The definition of B ′ is not used in this paper.)In particular, the intersection of the spectrum of P : B ′ → B ′ with the unit circle consists of finitely many eigenvalues λ 0 , . . ., λ q−1 of finite multiplicity and these are the q'th roots of unity λ k = e 2πik/q .By ergodicity, these eigenvalues are simple.
We consider the perturbed family of transfer operators where • denotes the standard scalar product on R d .Applying results of [16], it is shown in [4, Section 3.3.2]that there exists δ > 0 so that t → P t : B ′ → L 3 is continuous for t ∈ B δ (0).Moreover, there are continuous families of simple isolated eigenvalues t → λ k,t for P t : denote the corresponding spectral projections on B ′ .Then where Corollary 2], there exist C > 0 and γ ∈ (0, 1) such that sup where Σ ∈ R d×d is a positive-definite matrix.
3 The range n ≪ log |N |.
In this section, we prove Theorem 1.1 in the range n ≪ log |N|.This estimate holds at the level of T : M → M and κ : M → Z d (without requiring consideration of Young towers).Recall that d ∈ {1, 2}.

Key estimates on the one-sided tower
To prove Theorem 1.1, it remains by Lemma 3.1 to consider the range log |N| ≤ ǫ 1 n where ǫ 1 is chosen sufficiently small.Since µ(κ n = N) = μ∆ (κ n = N), it suffices to work on the one-sided tower ∆.To simplify the notation, we write (f, ∆, µ ∆ ) for the one-sided tower map, and κ : ∆ → Z d for the free flight function on the one-sided tower.
As clarified in [18, Lemma 5.1], the derivative of P t at t = 0 is not a bounded operator from B ′ → L 1 .In Section 5, we define a more suitable Banach space B ⊂ B ′ ∩ L ∞ such that we have better control on Π t : B → L 1 .To apply the method from [17], we require the following lemmas.Let Furthermore, there exist C > 0, δ > 0, b > 0 such that for all t, h ∈ B δ (0), Remark 4.3 Using the asymptotic in (2.3), we obtain the stronger conclusion λ k − λ k,t ∼ λ k Σt • t L(t) as t → 0. However, this stronger conclusion is not used in this paper and we consider the weaker conclusion since it generalises to the abstract setting of Theorem 7.1.
Proof By Lemma 4.2, after shrinking δ, for all t ∈ B 2δ (0), and so , which agrees with the definition of a n used here up to an inconsequential constant factor.)

Proof of Lemmas 4.1 and 4.2
We continue to work on the one-sided tower ∆.Fix θ ∈ (0, 1) and recall the definition of the metric d θ on Y from Section 2. We define the Banach space B = B(∆) of dynamically Hölder observables v : ∆ → R with v B < ∞, where In this section, we often write B(∆) and L 1 (∆) for the function spaces on the Young tower ∆, to distinguish them from related function spaces defined on the base Y .

Renewal operators
For y ∈ Y and a ∈ α, let y a denote the unique preimage y a ∈ a such that F y a = y.Recall that (Rv)(y) = a ξ(y a )v(y a ) and that there is a constant C > 0 such that for all y, y ′ ∈ Y , a ∈ α. (Standard references for properties of the transfer operator R for a Gibbs-Markov map include [1,2].)Define the Banach space Proof Since u is constant on partition elements, we write u(a) = u| a .By (5.1), By (5.1), , and the result follows.
For z ∈ C with |z| ≤ 1 and t ∈ R d , define Proposition 5.2 There exists δ > 0 such that, regarded as operators on B 1 (Y ), Proof It suffices to show that there exist a > 0, C > 0 such that Since κ σ ∈ L r (Y ) for all r < 2 and σ has exponential tails, there exists a > 0 such that 1 {σ=n} κ σ 1 ≪ e −an .
Note that σ and κ σ are constant on partition elements.By Proposition 5.1, Proposition 5.3 There exists δ > 0 such that regarded as operators from As in the proof of Proposition 5.2, it suffices to obtain exponential estimates for A t,n and ∂ j A t,n .
There exists a > 0 such that Proposition 5.4 There exists δ > 0 such that regarded as operators from . Again, it suffices to obtain exponential estimates for B t,n and ∂ j B t,n .
We can write B t,n v = R(u t,n v n ) where Note that u t,n is constant on partition elements and v n B 1 (Y ) ≤ v B .Also, there exists a > 0 such that 1 {σ>n} ψ L 1 (Y ) = O(e −an ).By Proposition 5.1, Similarly, where (E t,n v)(y, ℓ) = 1 {ℓ>n} (P n t v)(y, ℓ).
Proof We have z n 1 {ℓ=n} e it•κn(y,0) v(y) = z ℓ e it•κ ℓ (y,0) v(y). Hence We now proceed as in the proof of Proposition 5.6, except that there is one less factor of n (hence one less factor of L(h)).

Proof
We have This is the same as in Proposition 5.8 except that σ = n is replaced by σ > n (which makes no difference given the exponential tails).
and only if z q = 1 in which case 1 is a simple eigenvalue with eigenfunction 1.
Proof By quasicompactness, 1 ∈ spec R(z, 0) if and only if 1 is an eigenvalue.Recall also that 1 is a simple eigenvalue for R(1, 0) with eigenfunction 1 and hence this also holds for R(z, 0) when z q = 1.
Finally, it is standard that 1 is an eigenvalue for R(z, 0) if and only if z is an eigenvalue for P 0 which, as noted in Section 2, is the case if and only if z q = 1.By Proposition 5.2(b), for k = 0, . . ., q − 1, the eigenvalue 1 for R( λk , 0) extends to a C 1 family of simple isolated eigenvalues (z, t) → τ k (z, t) on B δ ( λk ) × B δ (0), for some δ > 0, with τ k ( λk , 0) = 1.Let σ = Y σ dµ Y .Recall that L(t) = log(1/|t|). where Hence it suffices to consider the first term I k .
Step 1 Write It follows from Proposition 5.2(a) that π k is analytic and hence that H is analytic. where It follows from Proposition 5.2(b) that ∂ z π k is jointly C 1 .Also, g is C 1 .Hence, by Corollary 5.12(b), Next, we note that G 2 , with π k changed to τ k , was estimated in (5.6), and the identical argument shows that Step 2 Write Again, it follows from Proposition 5.2(a) that τ k is analytic and hence that b is analytic.Also, it follows from Proposition 5.
By Proposition 5.11(a), |β(λ k , 0)| = σ > 0 and we can shrink δ if necessary so that β is bounded away from zero on B δ ( λk ) × B δ (0).Let β(z, t) = β(z, t) −1 .Then, we can write where q is analytic and jointly C 0 and Step 3 Combining Steps 1 and 2, we obtain (5.7) with The desired regularity properties of πk and H k follow immediately from the regularity properties established in Steps 1 and 2.

Completion of the proof of Lemmas 4.1 and 4.2
By [19], we have the renewal equation T = (I − R) −1 .Also, by [14], P = A T B + E.
Proposition 6.2There exist C > 0, δ > 0 such that Proof Set s = t − h and relabel so that k,t , it follows from Corollary 4.4 that The other integrals in (6.1) are also estimated using Corollary 4.4 and we obtain (Here, we used that log | so by the mean value theorem, for some u between t and s.By Lemma 4.1, Hence by Corollary 4.4, This completes the proof.
Proof In this proof we abbreviate B δ (0) to B δ and suppress dt.Let I = B δ e −it•N r(t)λ n k,t Π k,t .Integrating by parts, where Recall that r is C 2 and that t → λ k,t , t → Π k,t are C 1 by Lemma 4.1.Hence by Corollary 4.4, To estimate I 3 , we use a modulus of continuity argument (see for instance [15, Chapter 1]).
Set s = t − h where h = πN −1 j e j and notice that I 3 = − 1 iN j B 2δ e −it•N r(s)∂ j (λ n Π) k,s .Hence , we obtain

Now,
To complete the proof, we show that I 4 ≪ n Proof By Corollary 6.1, Hence we can reduce to the case n ≤ By (2.2), there exists γ ∈ (0, 1) such that B δ (0) e −it•N r(t)Q n t dt ≪ γ n .Together with Lemma 6.3, this implies that Proof of Theorem 1.1 By Lemma 3.1, it remains to consider the range log |N| ≤ ǫ 1 n.By [17, Lemma 3.9], there exists an even C 2 function r : . By Corollary 6.4, we obtain the desired estimate for log |N| ≤ ǫ 1 n.

LLD for nonuniformly hyperbolic systems modelled by Young towers
In this section, we state and prove an abstract version of Theorem 1.1 for systems modelled by a Young tower with exponential towers for a general class of observables κ.The observables take values in Z d where there is no restriction on the value of d ≥ 1.Let (T, M, µ) be a general nonuniformly hyperbolic map modelled by a two-sided Young tower ∆ with exponential tails (as in Section 2).Let κ : M → Z d be an integrable observable with M κ dµ = 0. Define the lifted observable κ = κ • π : ∆ → Z d .We require that κ is constant on π−1 (a × {ℓ}) for each a ∈ α, ℓ ∈ {0, . . ., σ(a).Then κ projects to an observable κ : ∆ → Z d constant on the partition elements a × {ℓ} of the one-sided tower ∆.
Define P t , λ k,t and so on as in Section 2. Properties (2.1) and (2.2) remain valid.Our further assumptions in the abstract setting are that there exist continuous slowly varying3 functions ℓ 1 , ℓ 2 : [0, ∞) → (0, ∞) and a constant δ > 0 such that µ(|κ| > x) ≤ x −2 ℓ 1 (x) for all x > 1 (7.1) u −1 ℓ 1 (u/ log u) du. 4 We require that there is a constant C > 0 such that The slowly varying function ℓ 3 can be determined by modifying the proof of Theorem 1.1.Some of the steps are indicated below.
In the case of billiards, assumptions (7.1) and (7.2) hold with ℓ 1 ≡ 1 and ℓ 2 (x) = log x.We note that even with these ℓ 1 , ℓ 2 and d ≤ 2, obtaining ℓ 3 (x) = log x and ℓ 3 (x) = log x log log x in Theorem 1.1 requires extra structure for billiards beyond the abstract setting of Theorem 7.1.This extra structure was used in Proposition 2.1 and Lemma 3.1.Similarly, assumption (7.3) is not required in the billiard setting due to the extra structure.
Remark 7.3 (a) In the simpler situation of Gibbs-Markov maps studied in [17], the underlying assumption is that µ(|κ| > x) ∼ x −2 ℓ 1 (x) and a consequence is that 1 − λ 0,t ∼ c|t| 2 ℓ 2 (1/|t|) where c > 0 and ℓ (b) As in [17], the proof of Theorem 7.1 does not rely on aperiodicity assumptions and hence the result applies in situations where the local limit theorem fails.(In fact, it may even be the case that a −1 n κ n fails to converge in distribution in the generality of Theorem 7.1.)(c) More generally, one could consider LLD in the abstract setting with for α ∈ (0, 2), where the underlying limit laws are stable laws (rather than normal distributions with nonstandard normalisation).We already mentioned that the study of such stable LLD started with [9] and [6] in the i.i.d.case for d = 1, extended to d ≥ 2 [7].The Gibbs-Markov case was studied in [17] for α ∈ (0, 1)∪(1, 2] and general d ≥ 1.We expect that Theorem 7.1, in the abstract setting where M is modelled by a Young tower with exponential tails, extends to the cases α ∈ (0, 1) ∪ (1, 2) with minor (and obvious) modifications.However, for purposes of readability we do not pursue this extension here.
In the remainder of this section, we sketch the proof of Theorem Since a n is regularly varying of index 1 2 , and r > 2 is arbitrary, it follows that n r−1 a d n ≪ n In particular, ψ ∈ L r (Y ) for all r < 2.
Proof of Theorem 7.1 The arguments are identical to those in Section 6 up to slowly varying factors.Various simplifications no longer hold as the slowly varying functions ℓ 1 , ℓ 2 , l1 and log are less well related, so the exact formulas are rather complicated and hence are omitted.

Remark 1 . 2 1+|N | 2
(a) A slightly stronger version of Theorem 1.1 for unrestricted d with bound for all n and N was proved in the much easier set up of Gibbs-Markov maps (and also for i.i.d.random variables) in [17, Corollary 3.3].(b) It follows from the arguments in this paper that the extra factor of log log |N| for d = 2 in Theorem 1.1 can be removed in the range |N| ≤ e ǫ 1 n , for ǫ 1 sufficiently small as chosen in Section 5.The same holds true in the range n ≪ log N/ log log N (see the end of the proof of Lemma 3.1).

2 .a d n 1 |N | 2 .a d n log |N| 1 +
The first integral on the right-hand side was estimated in Proposition 6.2 while the second and third are dominated by n The same calculation that was used for the integral K in Proposition 6.2 shows thatB 2δ |λ k,s | n ∂ j Π k,t − ∂ j Π k,s ≪ 1a d n log |N | |N | .The desired estimate for I 4 follows.Corollary 6.4There exist ǫ 1 > 0, C > 0 such that A n,N ≤ C n |N| 2 for all n ≥ 1, N ∈ Z d with |N| ≤ e ǫ 1 n .
1 2 (1 + |N| 2 )/ log |N|.Then we can suppose without loss that |N| > π/δ.In this way, we reduce to proving A n,N ≪ n Since a 2 n / log a n ∼ n/2, the last constraint can be weakened to a 2 n / log a n ≤ |N| 2 / log |N|, equivalently a n ≤ |N|.