Periodic Lorentz gas with small scatterers

We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}ρ may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{n\log n}$$\end{document}nlogn scaling (i) for fixed infinite horizon configurations—letting first \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document}n→∞ and then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \rightarrow 0$$\end{document}ρ→0—studied e.g. by Szász and Varjú (J Stat Phys 129(1):59–80, 2007) and (ii) Boltzmann–Grad type situations—letting first \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \rightarrow 0$$\end{document}ρ→0 and then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document}n→∞—studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016) .


Introduction
In this paper we are interested in limit laws for infinite horizon planar periodic Lorentz gases with small scatterers.The Lorentz gas, a popular model of mathematical physics introduced by H. Lorentz in 1905 [23], is a dynamical system on the infinite billiard table obtained by removing strictly convex scatterers from R 2 .We study the periodic model when these scatterers are round disks of radius ρ ∈ (0, 1/2) positioned at the points of the Euclidean lattice Z 2 .This table can be split up into countably many compact cells, each congruent to the unit square, which can be also regarded as the 2-dimensional flat torus.As usual, a point particle on the table moves with a unit velocity vector along straight lines inside the table, and collides elastically -angle of incidence equals angle of reflection -at the scatterers.This billiard flow produces a billiard map for the Poincaré section of outgoing collisions.The phase space of the billiard map in a single cell is where O is a round disk at the origin with radius ρ.The phase space representing all cells together is M = M × Z 2 and the displacement function κ ρ : M → Z 2 indicates the difference in cell numbers going from one collision to the next.As O is strictly convex, the billiard is dispersing, and the dynamics has good hyperbolicity properties.
For any ρ ∈ (0, 1/2), the horizon is infinite which means that the time between two consecutive collisions -and accordingly, κ ρ : M → Z 2 -is unbounded.This corresponds to corridors, that is, infinite strips on R 2 parallel to some direction ξ ∈ Z 2 \ {0} which do not intersect any of the scatterers of the infinite billiard table.Both the number and the geometry of these corridors depend on ρ.Hence, the value of the parameter ρ strongly affects the asymptotic properties of the dynamics, and in particular, plays a central role in our exposition.
1.1 Recalling limit laws for fixed ρ ∈ (0, 1/2) as time n → ∞ A consequence of the infinite horizon is the superdiffusive behaviour of κ ρ with ρ ∈ (0, 1/2) fixed, captured in the first place in the Central Limit Theorem (CLT) with non-standard normalization.
To recall this result along with its refinements, we introduce some notation to be used throughout this paper.Let T ρ : M → M be the billiard map, recall that it preserves the canonical invariant probability measure µ.Set Choosing the initial point on M according to µ, we can regard κ n,ρ as a family of random variables.Throughout we let =⇒ stand for convergence in distribution.We recall the CLT with non-standard normalization: for every ρ ∈ (0, 1  2 ) there exists a positive definite matrix Σ ρ such that: For fixed ρ ∈ (0, 1/2), This result was conjectured by Bleher [5] and proved rigorously via two different methods: Szász & Varjú in [31], and Chernov & Dolgopyat in [11].In the setting above, the requirement of having two non-parallel corridors (present in [31,11]) is automatically satisfied because the scatterers are positioned at the lattice points.
It is important to note that there is an explicit formula for Σ ρ , which involves the scatterer geometry for fixed ρ, see for example [11,Formula (2.1)].A computation (similar to our proof of Lemma A.4) shows that lim where Σ is the diagonal matrix given in (1).To compare these results with our Theorem A below, we point out the following direct consequence of (2) and (3): The method of proof in [31] relies on the existence of a Young tower for T ρ as in [34,9] and an abstract result of Bálint & Gouëzel [4] along with several additional properties of (κ ρ , T ρ ) established in [31].One notable feature of this method is that it provides a refinement of the CLT (2), namely the Local Limit Theorem (LLT): For fixed ρ ∈ (0, 1/2), (n log n)µ(κ n,ρ = 0) → Φ Σρ (0) as n → ∞, (5) where Φ Σρ is the density of the Gaussian random variable on the r.h.s of (2).The method of proof in [11] exploits exponential mixing for the sequence {κ ρ • T n ρ } n≥1 .The authors of that work develop an argument based on standard pairs to establish a bound on the correlations for κ ρ : For fixed ρ ∈ (0, 1/2), there exist θρ ∈ (0, 1) and Ĉρ > 0 so that M κ ρ • κ ρ • T n ρ dµ ≤ Ĉρ • θn ρ for all n ≥ 1.
The dynamics Tρ preserves the measure μ = µ × Leb Z 2 , where Leb Z 2 denotes the counting measure.
1.2 Recalling results as first ρ → 0 and then n → ∞ (Boltzmann-Grad limit) In [24,25], Marklof & Strömbergsson studied the Boltzmann-Grad limit of the periodic Lorentz gas.This corresponds to letting the scatterer size ρ → 0 and investigating the displacement in the rescaled continuous time T = ρt (so that the mean free path remains bounded).In particular, [24] proves that, in this Boltzmann-Grad limit, the displacement of the particle converges, on any finite time interval, to an explicitly given Markov process.Marklof & Tóth [26] then studied the large time asymptotic of this Markov process, and obtained the CLT and the WIP with non-standard normalization √ T log T .These results on the Boltzmann-Grad limit scenario hold in any dimension, not just in d = 1, 2 as the results mentioned in the previous subsection.For more details, we refer to the original references.What is most relevant for us is that [26,Theorem 1.1] and [26,Theorem 1.3] are reduced to discrete time statements that can be formulated in terms of the behavior of κ n,ρ in the limits ρ → 0 first and then n → ∞.In particular, [26,Theorem 1.2] states for d = 2 that: κ n,ρ ( √ 4πρ) −1 √ n log n =⇒ N (0, Σ) as ρ → 0 followed by n → ∞, where κ n,ρ and Σ are as in (1) 1 , while [26,Theorem 1.4] is the corresponding WIP which, when d = 2, reads as (7) with the main difference of the limit paths: ρ → 0 followed by n → ∞, as opposed to fixed ρ.
In [26], the authors state that an open problem is to consider the joint limit ρ → 0 and n → ∞.In the Boltzmann-Grad limit scenario with diffusive behaviour, this type of question is answered by Lutsko & Tóth in [22] for random Lorentz gases, in dimension d = 3, where, on top of the initial condition, additional randomness comes from the random placement of the scatterers.However, their model is very different from the model considered in [26] and it is characterized by diffusion (Brownian motion with standard normalization).
1.3 Main results as ρ → 0 and n → ∞ in the joint limit Our main result takes a step in answering the open question in [26] for the planar periodic Lorentz gas.It reads as follows.
A precise expression of M (ρ) is given in Theorem 7.1 in Section 7. At this stage we mention that M (ρ) depends on the rate of correlation decay for Hölder observables as ρ → 0. How this decay rate depends on ρ is not known and we do not attempt to study this in the present paper.However, we comment on some relevant aspects of correlation decay below.
In the remainder of this section, we make some further comments on how our results compare to various other works, and on some key ingredients of our argument.
Comments on the rate of correlation decay.Statistical limit laws in dynamical settings in general, and our results in particular, are strongly related to effective bounds on time correlations.For several decades, it has been a major problem to prove exponential decay of correlations for Hölder observables in dispersing billiards, that is, bounds of the form: where ψ 1 : M → R and ψ 2 : M → R are centered, Hölder continuous observables, and θρ < 1 may depend on the Hölder exponent, while C ρ (ψ 1 , ψ 2 ) > 0 on the Hölder norm of these functions, and both constants depend also on ρ (i.e. on the billiard table).Several powerful methods have been designed to prove bounds of the form (10), in particular using quasi-compactness of the transfer operator on Young towers [34] or anisotropic Banach spaces [14], coupling of standard pairs [10,Chapter 7] or most recently, Birkhoff cones [13].However, each of these methods involve some non-constructive compactness argument which is the reason why there is no explicit information available on how the rate of decay (i.e.C ρ and θρ ) depends on ρ.For instance, in the framework of quasi-compact transfer operators, this corresponds to having effective bounds on the essential spectral radius, but not on the spectral gap.
In fact, depending on the method, ψ 1 and ψ 2 may belong to a larger space (that contains Hölder functions), however, these spaces do not contain the unbounded observable κ ρ .Hence, even for fixed ρ, it requires additional effort to obtain correlation bounds for unbounded observables, in particular, to derive (6).As mentioned above, in our context of the infinite horizon Lorentz gas, (6) was proved by Chernov and Dolgopyat in [11,Proposition 9.1], which is an important reference for our work.Let us also mention [3, Lemma 3.2] on a similar bound for the induced return time arising in dispersing billiards with cusps, and the more recent paper [32] where correlation bounds for unbounded observables are studied in an axiomatic framework that includes further billiard models.Nonetheless, all these works consider the large time asymptotics of a fixed billiard system.To treat the simultaneous scaling of ρ → 0 and n → ∞, in Appendix C of the present paper we extend [11,Proposition 9.1] in two directions.On the one hand, on top of the mere existence of some Ĉρ > 0 and θρ < 1 in (6), we explicitly relate these constants to C ρ and θρ of (10), as expressed in (66). 2n the other hand, to exploit correlation bounds of the type (6) when taking the joint limit, these have to be combined with the action of the perturbed transfer operator R ρ (t) (introduced below) as stated in our Proposition C.1.
Comparison with results on the random Lorentz gas.To compare our Theorem A with the results of Lutsko & Tóth on the random Lorentz gas, it is important to emphasize that although both [22] and our paper consider a joint limit of scatterer radius tending to 0 and time tending to infinity simultaneously, the settings of these two papers are quite different.In particular, the starting point of Lutsko & Tóth is the Boltzmann Grad limit of the random Lorentz gas, and accordingly, [22] can handle situations when time tends to infinity at a sufficiently slow pace in relation to the scatterer size tending to 0. In contrast, the starting point of our work is the superdiffusive limit in the infinite horizon periodic Lorentz gas with fixed scatterer size (see subsection 1.1 for a summary of previous results), and accordingly we can handle situations when time tends to infinity at a sufficiently fast pace in relation to the scatterer size tending to 0.
It is also important to note that under the condition which shows that our Theorem A is indeed a direct analogue of both ( 4) and (9).To simplify the exposition, we omit the case d = 1 (i.e., the Lorentz tube), but believe that similar results can be obtained by the same arguments.
Further comments on some corollaries of our result and some elements of our proofs.A main advantage of the current method of proof via spectral methods is that it allows us to obtain (with no additional effort) the LLT (5) and the mixing statement (8) with appropriate limit paths ρ → 0 simultaneously with n → ∞, as opposed to fixed ρ.For the LLT we refer to Theorem 7.3 and for the mixing result we refer to Corollary 7.5.
We mention up front that unlike in the fixed ρ scenario with main results recalled in Subsection 1.1, we cannot exploit the existence of a Young tower because it seems undoable to build such a tower in a fashion that it depends continuously on ρ.Instead, we prove Theorem A via the Nagaev method on Banach spaces of distributions introduced by Demers & Zhang [14,15,16] in the spirit of the spaces constructed in Demers & Liverani [12].See Aaronson & Denker [1,2] for a classical reference on the Nagaev method in (Gibbs Markov) dynamics beyond the CLT with standard normalization (that is √ n).However, as we shall explain below, the standard pairs argument in [11] plays a crucial role in our proof.We end this introduction summarizing the main steps and challenges of our proofs.A main difficulty comes from the fact that as ρ → 0, more and more corridors open up and controlling their number and geometry is a non-trivial task.Another challenge for the proofs of Theorem A and the LLT in Theorem 7.3 comes from the fact that the spaces in [14,15,16] cannot be used in a straightforward way even in the infinite horizon case with fixed ρ.
The Nagaev method requires: 1) the existence of a Banach space (B, B ) on which the transfer operator R ρ of T ρ has a spectral gap; 2) that the perturbed transfer operator (R ρ (t)ψ = R(e itκρ • ψ) for ψ ∈ B) has sufficiently good continuity estimates R ρ (t) − R ρ (0) B ≤ C|t| ν ; the larger ν > 0, the better.
Regarding 1), using a Lasota-Yorke inequality on a strong space B and a weak space B w , Demers & Zhang [14,15,16] established the spectral gap for every fixed ρ, see Section 4. This is the main reason why we resorted to the use of such Banach spaces.
Regarding 2), as in Keller & Liverani [20], one could work with the weak space.For infinite horizon billiards, continuity estimates in the strong or weak Banach spaces in [14,15,16] have not been obtained previously.In Section 5.2, we give continuity estimates in such Banach spaces (strong or weak); the estimates there rely heavily on a version of the growth lemma, namely Proposition 3.1.These continuity estimates are O(|t| ν ) for ν < 1/2 with explicit dependence on ρ, in both the strong and the weak spaces.This exponent ν is too small to obtain the asymptotics of the leading eigenvalue of R ρ (t) directly.Therefore we resort to a decomposition of the eigenvalue in several pieces (see the proof of Proposition 6.3) and exploit the standard pairs arguments in [11] to deal with some parts of the estimate, see Appendix C. Along the way, we give a new proof of the LLT (5) for fixed ρ which is new at an abstract level as well, namely by working on the Banach spaces [14,15,16] in the absence of good continuity estimates but in the presence of exponential decay of correlations.
The paper is organised as follows: In Section 2 we recall some basic properties of hyperbolic billiards and also estimate widths of corridors that open up as ρ → 0. Section 3 gives the Growth Lemmas, following [14,15,16] but with estimates made explicit in terms of ρ, and including sums over unbounded number of corridors (this is the reason why the estimates are worse than for the usual Growth Lemmas).Section 4 introduces the Banach spaces and recalls the proof of the spectral gap property for the unperturbed transfer operator R ρ , showing that the ρ-dependence can be controlled.Section 5 is devoted to the continuity estimates of the perturbed transfer operator R ρ (t) and Section 6 gives the asymptotics of the corresponding leading eigenvalue.The precise statements and proofs of the limit theorems are gathered in Section 7.
The appendices give further technical details on corridor sums (Appendix A), distortion (Appendix B) and decay of correlations by a combination of tower and standard pair arguments (Appendix C).
Acknowledgements: We thank the anonymous referees for useful comments.HB was supported by the FWF Project P31950-N35.PB was supported by National Research, Development and Innovation Office --NKFIH, Projects K123782, K142169 and KKP144059.PB and HB acknowledge the support of Stiftung A ÖU Project 103öu6.DT was partially supported by EPSRC grant EP/S019286/1.
We would like to thank M. Weber for his helpful correspondence on the results established in [33].
2 Preliminaries on Lorentz gas on Z 2 Our general reference on hyperbolic billiards is Chernov & Markarian [10], the conventions of which are followed in our exposition, except for some minor differences.In particular, we use coordinates ] on M, where • θ ∈ S 1 in clockwise orientation describes the collision point on the scatterer (so the corresponding point on ∂O is (ρ sin θ, ρ cos θ)); denotes the outgoing angle that the billiard trajectory makes after a collision at a point with coordinate θ with the outward normal vector N θ at this point (so φ = π 2 corresponds to an outgoing trajectory tangent to O in the positive θ-direction).
In these coordinates (θ, φ), the measure µ has the same form dµ = 1 4π cos φ dφ dθ for all values of the radius ρ > 0. Integrals involving the displacement function κ ρ , however, do depend on ρ.If the flight between (x, ℓ) and (T ρ (x), ℓ + κ ρ (x)) goes through a corridor for a long time before hitting a scatterer at the boundary of this corridor, then the angle at which the second scatterer is hit is close to ± π 2 .This sparks another long flight in the same corridor, i.e., κ ρ (T ρ x) is large, too.In the remainder of this section, we record some properties of T ρ and κ ρ .In Subsection 2.1 the geometry of corridors is described, with special emphasis on the asymptotics of small ρ.In Subsection 2.2 we focus on the singularities, which, in addition to strong hyperbolicity, are the other main feature of the map T ρ : M → M. In Subsection 2.3, the hyperbolic properties of T ρ : M → M are discussed.Some lemmas of technical character are moved to Appendices A and B.
Notation: For functions (or sequences) f and g, we use the Vinogradov notation f ≪ g and the Landau big O notation interchangeably: there is a constant C > 0 such that f ≤ Cg.Similarly f ≍ g means that there exists C > 1 such that C −1 g ≤ f ≤ Cg.

Corridors and their widths
Let O ℓ denote the circular scatterer of radius ρ placed at lattice point ℓ ∈ Z 2 .The computation of ) is based on the division of the phase space in corridors.These are infinite strips in rational directions given by ξ ∈ Z 2 \ {0} for ρ sufficiently small, that are disjoint from all scatterers (but maximal with respect to this property), and they are periodically repeated under integer translations.As soon as ρ < 1  2 , there are infinite corridors parallel to the coordinate axes.If ρ < 1 Given 0 = ξ ∈ Z 2 and ρ > 0 sufficiently small, there are two corridors simultaneous tangent to O 0 and O ξ , one corridor on either side of the arc connecting 0 and ξ.The widths of the corridors are denoted by d ρ (ξ) and dρ (ξ), see Figure 1.
In the sequel, we usually only need one of these two ξ-corridors, and we take the one with ξ ′ in its other boundary.
The vertical distance between ξ ′ and the arc L is |q ′ − p ′ q p | = 1 p |q ′ p − p ′ q| = 1 p .The vertical distance between L and ξ ′′ is The corridor's diameter is perpendicular to ξ, so d 0 (ξ) is computed from this vertical distance as the inner product of the vector (0, 1/p) T and the vector ξ = (p, q) T rotated over 90 • : 1 The computation for d0 (ξ) = |ξ| −1 is the same.

Singularities of the billiard map
In the coordinates (θ, φ, ℓ) ∈ S 1 ×[− π 2 , π 2 ]×Z 2 (or ×Z if it is a Lorentz tube), the size of the scatterers ρ doesn't appear, but it comes back in the formula of the billiard map T ρ and in its hyperbolicity.Also the curvature of the scatterers is K ≡ 1/ρ.We recall some notation from the Chernov & Markarian book [10] (going back to the work of Sinaȋ), bearing in mind that we have to redo several of their estimates to track the precise dependence on ρ.The phase space is M = M × Z 2 = ℓ∈Z 2 M ℓ , where each M ℓ is a copy of the cylinder S 1 × [− π 2 , π 2 ], see Figure 2.
Let S 0 = {φ = ± π 2 } be the discontinuity of the billiard map corresponding to grazing collisions.The forward and backward discontinuities are We line the curve S 0 with homogeneity strips The standard value is r 0 = 2, but as distortion results and some other estimates improve when r 0 is larger, we choose the optimal value of r 0 later.The set S −1 consists of multiple curves inside M 0 , one for each scatterer from which a particle can reach O 0 in the next collision.In Figure 2 we consider the corridor in the direction of ξ ∈ Z 2 , and drew the parts of S −1 coming from scatterers O ξ , O −ξ and O −κρ for some scatterer on the other side of this corridor.Lemma 2.3 For the ξ-corridor, let (θ −ξ , π 2 ) ∈ M 0 be the point of intersection of S 0 and the part of S −1 associated to the scatterer O −ξ , and (θ κρ , π 2 ) ∈ M 0 , κ ρ = ξ ′ − M ξ, be the point of intersection of S 0 and the part of S −1 associated to the scatterer O κρ = O ξ ′ −M ξ at the other side (i.e., the ξ ′ -boundary) of the ξ-corridor, see Figure 3. Let (θ ′ κρ , φ ′ κρ ) be the intersection of the parts of S −1 associated to the scatterers O −ξ and the scatterer O κρ , see Figure 2. Then Proof.The angle θ −ξ refers to the point where the common tangent line of O 0 and O −ξ touches O 0 .For the value θ κρ , κ ρ = ξ ′ − M ξ, we take the common tangent line to O 0 and O κρ which has slope The triangle △P O 0 Q has angles φ ′ κρ , α + π 2 and θ ′ κρ , which add up to π. Hence as claimed.

Hyperbolicity of the Lorentz gas with small scatterers
The derivative DT ρ : T M → T M preserves the unstable cone field This is [10, page 74] in the coordinates θ = r/2πρ, and we can sharpen this cone by replacing τ min by τ (x), the flight time at x before the next collision.The derivative of the inverse of the billiard map preserves the stable cone field Clearly, these cone-fields are transversal uniformly over M, and S n is a unstable (or stable) curve if n > 0 (or n < 0).
In the billiard literature it is common to use a pseudo-norm, the p-norm for unstable vectors, defined as dx p = cos φ dr.When restricted to the unstable cone, the p-norm is non-degenerate.With the notation R(x) = 2 ρ cos φ , the expansion/contraction factor Λ on unstable vectors in the p-norm satisfies This proves uniform hyperbolicity of the billiard map.
In our coordinates the p-norm can be also expressed as dx p = 2π ρ cos φ dθ, and it is related to the standard Euclidean norm as The expansion of DT ρ of unstable vectors is uniform in the p-norm, see [10, Formula (3.40)]: Expressed in Euclidean norm, this gives, for DT ρ (dx) = (dθ 1 , dφ 1 ), For later use, if T ρ (x) is in the homogeneity strip H k , then cos φ 1 ≈ k −r 0 .

Growth lemmas
As already mentioned in the introduction, the main line of our argument uses perturbed transfer operators acting on the Banach spaces constructed in [14] and [16].These works, as essentially all other methods studying statistical properties of hyperbolic billiards, rely on appropriately formulated growth lemmas, which quantify the competition of the two main dynamical effects, singularities and expansion, in these systems.The constructions of [14] and [16] involve several exponents, which thus are present in our setting, too.Additionally, we have to introduce some further exponents as we study perturbed transfer operators.Before stating the growth lemmas, here we include a table summarizing the role and the interrelation of these exponents.Essentially, we use the same notation as in [14] except for some subscripts 0 , and in fact some of the constants reduce to their value in [14] if is the exponent of the homogeneity strips: the exponent of κ ρ in the continuity estimate for the transfer operator, upper bound on ς in the Jensenized growth lemma, see (25), We use a class W s of admissible stable leaves defined as C 2 leaves W in the phase space such that all its tangent lines are in the stable cone bundle, their second derivative is uniformly bounded, W is contained in a single homogeneity strip, κ ρ (x) is constant on W and there is a ρ-dependent upper bound on |W |, namely sup where the small c > 0, to be fixed below, is independent of ρ.
Let W ∈ W s be an admissible stable leaf.The preimage T −1 ρ (W ) is cut by the discontinuity lines S 1 and boundaries of homogeneity strips into at most countably many pieces V i .Note that we may have to cut the pieces V i further into curves W i of length ≤ δ 0 .

The growth lemma in terms of V i
The particle can reach the scatterer O 0 at the origin from corridors in all directions, indexed by (ξ, ξ ′ ) ∈ Ψ, see Figure 3.If the previous scatterer is ±ξ itself, we call this a trajectory from the ξ-boundary; if the previous scatterer is at lattice point ξ ′ − M ξ, the trajectory comes in from the ξ ′ -boundary, see Remark 2.2.To each such scatterer and homogeneity strip H k belongs at most one V i , and the contraction |T ρ V i |/|V i | is governed by (14), where the distortion T ρ : Then there is a constant C > 0, uniform in ρ, ν and r 0 such that for ρ sufficiently small, and c chosen appropriately small.In addition, we assume that for distortion constant C d from Lemma B.2; (ii) As later we will need ν > 1 3 , we can take r 0 = 5 and ν = 3 8 .
Proof.The homogeneous admissible preimage curves T −1 ρ W = ∪ i V i are obtained by partitioning according to

• incoming corridors ξ;
• for a fixed corridor ξ, the scatterer on which V i is located.Accordingly, κ ρ (V i ) = M ξ − ξ ′ for some M ∈ N, and the summation is over M ; • for a fixed scatterer, the homogeneity strip containing V i , that is, If W is on the scatterer O 0 and V i is on the scatterer O ξ ′ −M ξ , then both of these scatterers are tangent to the same corridor.The trajectory makes and angle ∼ dρ(ξ) M |ξ| with the corridor and there is a lower bound on the collision angle given by (11).This puts restrictions on how M is related to k; as reflected by allowed intersections of homogeneity strips and M -cells on Figure 2. In particular which determines the range of k for M fixed.
We sum over the homogeneity strips for ξ and M fixed on the ξ ′ boundary.
where we used that the exponent 1 2 − 1 2r 0 of d ρ (ξ) is non-negative.By our assumption that ν < 1 2 − 1 2r 0 , this expression is summable over M , and therefore the sum over the ξ ′ -boundary of the entire ξcorridor is The sum over homogeneity strips for ξ fixed on the ξ-boundary is no different: Next we sum over all opened-up corridors, indexed by all the "visible" lattice points inside a sector of angle |W |/ √ 1 + 4π 2 , because only trajectories from scatterers within such a narrow sector can hit O 0 at coordinates in W .The "visible" corridors will be denoted by Ψ W .It can happen that a single corridor, or even a single scatterer in a corridor blocks the entire sector, and we reserve one term for |ξ| ≥ 1 (which is the worst case because the contraction of T ρ is the weakest).Apart from this corridor, and since we need an upper bound, we can replace we replace |W | by a stable curve of length δ 0 , and apply Lemma A.6 for a = 1 − ν and a = 3 2 − ν − 1 2r 0 .This gives Since δ 0 = cρ ν and ν < 1 2 , this completes the proof.

The growth lemma in terms of W i
The pieces of preimage leaf V i ⊂ T −1 ρ (W ) emerge by natural cutting at the discontinuity set S 1 and the homogeneity strips, but even so, their lengths can be larger than δ 0 , the bound of admissible stable leaves.We therefore need to cut them into shorter pieces, denoted as W i .In the worst case, each V i needs to be cut into δ −1 0 pieces, which gives the estimate Although this estimate suffices for some purposes, it is not always good enough for larger iterates T n ρ .The next lemma (which follows [14,Lemma 3.2] or [16,Lemma 3.3]) achieves an estimate, uniform in n, for ν = 0.
For the next lemma we recall some notation used in [16].For W ∈ W s , we construct the components G k (W ) of T −k ρ W inductively on k = 0, . . ., n.That is G 0 (W ) = {W }, and to obtain G k+1 (W ) first we apply Proposition 3.1 to each curve in G k (W ), and then we partition curves that are longer then δ 0 into pieces of length between δ 0 and δ 0 /2.We enumerate the leaves of the k-th generation G k (W ) as W k i .
Lemma 3.3 There is a constant C s > 0, independent of ρ, such that and for all ς ∈ [0, 1).
Proof.Define L k as the collection of indices such that as the collection indices of W n i such that their most recent long ancestor is . If for some W n i 1 no such long ancestor exists, then set k(i 1 ) = 0 and n need not be cut artificially (they are already short), and due to the distortion bound from Lemma B.2, Recall that by our assumption δ 1 is so small that θ 1 < 1.In the estimate below, we group W n i ∈ G n (W ) according to their most recent long ancestors.
where we have used that for fixed k and which proves the second statement.

Banach spaces and spectral gap
For the exponents p 0 and q 0 defined in (15) we define the Banach spaces (of distributions) C p 0 , B, B w , (C q 0 ) ′ in analogy to [16]. 3 We recall that (C q 0 ) ′ is the topological dual of C q 0 .Given W ∈ W s , let m W be the Lebesgue measure on W , and define , and H p 0 W (ψ) the Hölder constant of ψ along W . Also let d W (W 1 , W 2 ) stand for the distance between leaves as in [14, Section 3.1] or [16,Section 3.1]; in particular, if W 1 and W 2 belong to the same homogeneity strip, d W (W 1 , W 2 ) is the C 1 distance of their graphs in the (θ, φ) coordinates, and otherwise infinite.
Given W ∈ W s and h ∈ C 1 (W ), define the weak norm4 h Bw := sup With q 0 < p 0 fixed we define the distance between functions d(ψ 1 , ψ 2 ) in the same way as in [14, Section 3.1].We define the strong stable norm by Choosing ε 0 ∈ (0, δ 0 ) and β 0 ∈ (0, min{α 0 , p 0 − q 0 }), we define the strong unstable norm by The strong norm is defined by h B = h s + c u h u , where we will fix c u ≪ 1 (but independent of ρ) at the beginning of Subsection 5.2.Since C p 0 ⊂ B ⊂ B w ⊂ (C q 0 ) ′ (see Subsection 4.1), we have As in [16], we define B to be the completion of C 1 in the strong norm and B w to be the completion in the weak norm.

Transfer operator on B
Throughout we let R ρ : L 1 (m) → L 1 (m) be the transfer operator of the billiard map T ρ .We recall that [14, Lemmas 3.7-3.10]ensure that: i) R ρ (C 1 ) ⊂ B and as a consequence R is well defined on B; B w ; ii) the unit ball of B is compactly embedded in B w , and iii) It follows that R ρ is well defined on B and B w , and we also let R ρ denote the extension of this transfer operator to B w .

Lasota-Yorke inequalities
Using Proposition 3.1 with ν = 0 and Lemma 3.3 we obtain the analogue of the Lasota-Yorke inequality [16,Proposition 2.3].As our set-up fits [16], our only concern is the dependence on ρ.It is important to point out that our all estimates in Section 3 and Appendix B are independent of ρ, except that δ 1 < δ 0 ≪ ρ ν .Lemma 4.1 (Weak norm) There exists a uniform constant C > 0 so that for all h ∈ B and for all n ≥ 0, R n ρ h Bw ≤ C • C s h Bw , where C s is given by (20).
Using the present definition of the weak norm, From here on the argument goes almost word for word as the argument in [16, Section 4.1], except for the use of equation ( 20) (the analogue of [16, Lemma 3.3(a)] with ς = 0).
Lemma 4.2 (Strong stable norm) Take δ 1 as in (17) and θ 1 as in (22).There exists a uniform constant C > 0 so that for all h ∈ B and all n ≥ 0, where we have used the distortion bounds of Appendix B and Formula (21) (with ς = α 0 ).ii) To obtain the analogue of [16, Equation (4.6)], as in [16, Section 4.2], we split the sum ).
As in [16], dealing with the strong unstable norm is the most delicate part of the Lasota-Yorke inequality.The only difference from [16, Argument in Section 4.3] is that we apply (20) (instead of [16,Lemma 3.3 (b)]) multiple times.Note that our bound in (20) is independent of ρ, so no ρ-dependence arises here.Lemma 4.4 (Strong unstable norm) There exists a uniform constant C > 0 so that for all h ∈ B and for all n ≥ 0, ≤ ε, we may identify matched and unmatched pieces in The estimates of [16] on the length of the unmatched pieces apply, thus we may estimate their contribution by the strong stable norm using (20) (instead of [16,Lemma 3.3
To bound the contribution of the matched pieces we use, on the one hand, the strong unstable norm (as in [16, Formula (4.14)]) and, on the other hand, the strong stable norm (as in [16, Formula (4.17)]).Here again we rely on equation (20) which plays the role of [16,Lemma 3.3 (b)].β 0 < p 0 −q 0 ensures that after division by ε β 0 the proof of Lemma 4.4 can be completed.

Perturbed transfer operators
A standard way of obtaining limit theorems for dynamical systems is via the perturbed transfer operator method.In Section 7 we will use the spectral properties of the family of perturbed transfer operators Rρ (t), t ∈ R with Rρ (t)h = R(e itκρ h), h ∈ L 1 (m).
The argument goes parallel to Subsection 4.2, except that this time we need the estimates (i) for ν > 0 and (ii) only for n = 1, we rely on ( 19) and ( 25) instead of Lemma 3.3.Lemma 5.1 Assume (16).Then there exists a uniform constant C > 0 so that for all h ∈ B, Proof.The argument goes similarly to the argument in [16, Section 4.1] restricted to the case n = 1.More precisely, for Using the definition of the weak norm and the inequality From here on the proof goes the same as the argument in [16, Section 4.1] except for the use of equation ( 19) instead of [16,Lemma 3.3 (b)].
Lemma 5.2 There exists a uniform constant C > 0 so that for all h ∈ B and for all n ≥ 0, Proof.This time we are only concerned with n = 1, and do not need a contraction of the strong stable norm.Hence, an argument analogous to the proof of Lemma 5.1 suffices, with the weak norm replaced by the strong stable norm.Accordingly, we use (25) with ς = α 0 instead of [16,Lemma 3.3 (b)].
Lemma 5.3 There exists a uniform constant C > 0 so that for all h ∈ B, Proof.As with the proof of Lemma 4.4, the argument goes similar to [16, Argument in Section 4.3], restricted to the case n = 1.The matched and unmatched pieces can be again identified, this time for T −1 ρ W ℓ , ℓ = 1, 2. Then, as in the proof of Lemma 5.1, the factors |t| ν and |κ ρ | ν arise.Clearly κ ρ is constant on each of the (matched or unmatched) pieces, and takes the same value on any two pieces that are matched.Accordingly, the various contributions can be estimated in the same way as in proof of Lemma 4.4, with the only difference that, by the presence of the factor |κ ρ | ν , throughout the argument (19) is used instead of (20).
Equation (29) follows from the definition of the norm in B together with Lemmas 5.1, 5.2 and 5.3.

Peripheral spectrum and spectral gap
, Λ −q 0 }.By Lemmas 4.1, 4.2 and 4.4 and arguing as in [16, Equation (2.14)], we obtain the traditional Lasota-Yorke inequality for some N ≥ 1, provided c u in the definition of B (below ( 28)) is chosen small enough in terms of N .That is, Combined with the properties collected in Subsection 4.1 (that is, the relative compactness of the unit ball of B in B w ), equation (30) shows that the essential spectral radius of R ρ is bounded by σ and that the spectral radius is 1.Let Π ρ be the eigenprojection (that is, the projection on the eigenspace of R ρ ) corresponding to the eigenvalue 1.In particular, Π ρ µ = µ is the invariant measure for T ρ .Since for every ρ, T ρ is mixing, the peripheral spectrum of R ρ consists of just the simple eigenvalue at 1. Thus, for every ρ > 0, the eigenprojection Π ρ corresponding to the eigenvalue 1 of R ρ can be also characterized by for all h ∈ B. Let Q ρ be complementary spectral projection.From here onwards, we exploit that for every ρ > 0, there exist γ ρ ∈ (0, 1) and C ρ > 0 so that for every m ≥ 1.Altogether, R m ρ = Π ρ + Q m ρ , where Q ρ satisfies (32).

Asymptotics of the dominant eigenvalue
To establish limit theorems (such as Theorem 7.1 below) we study the asymptotics of E µ (e itκm,ρ 1) = E µ ( Rρ (t) m 1), as t → 0 and m → ∞ via the properties of Rρ (t)h = R ρ (e itκρ h), h ∈ B.
We already know that for every ρ ∈ ( 1 3 , 1 2 ), 1 is a simple eigenvalue of Rρ (0) = R ρ when viewed as an operator from B to B. Due to (29), Rρ (t) is C ν (in t) from B to B. It follows that for t in a neighbourhood of 0, Rρ (t) has a dominant eigenvalue λ ρ (t) (with λ ρ (0) = 1).
Let γ ρ be as in equation (32).The continuity properties together with (32) ensure that for any δ ∈ (0, γ ρ ) and t ∈ B δ (0), for some for all t small enough.A standard consequence of ( 29) and ( 32) is that for every δ ∈ (0, γ ρ ) and for all u so that |u − 1| = δ, The rest of this section is allocated to the study the asymptotics of λ ρ (t) as t → 0. The following property was used in [21,6,7] (see [7, assumption (H2)]) for the study of eigenvalues of perturbed transfer operators in the Banach spaces introduced in [12].Here we use it to obtain an adequate analogue for the present set-up.Lemma 6.1 Take s 0 = 1−α 0 (r 0 +1) 2r 0 as in (15).Let h ∈ B and v ∈ C p 0 .For every corridor with boundaries determined by O ξ and O ξ ′ , there exists a constant C > 0 independent of ρ and ξ so that Proof.Let {W ℓ } ℓ∈L be the foliation of the set {κ ρ = ξ ′ +ξN } into stable leaves.We can parametrise these leaves by their endpoints (ℓ, π 2 ) in S 0 , then L is an interval of length c ≪ ρ+τ min τ min < w ′ (φ) < − 1 2π because of the direction of the stable cones, see (13).
Let ν be a measure on L that produces the decomposition of Lebesgue measure m on {κ ρ = ξ ′ + ξN } along stable leaves.We have ν ≪ m L (and dν/dm L is bounded above).Since we need to partition stable leaves W ℓ by the homogeneity strips , as claimed.
Using (35), Lemma A.2 and Lemma 6.1 we obtain the asymptotics of the eigenvalue in Proposition 6.3 below.Proof.The coordinate axes p = 0 and q = 0, and the two diagonals p = q and p = −q divide the plane into eight sectors.Here we count counter-clockwise with the first sector Ψ 1 directly above the positive p-axis.Let γ = γ(t, ξ) be the angle between the vectors t and ξ.Let α = arctan q/p and θ be the polar angles of ξ and t ∈ R 2 respectively, so γ = θ − α.For the first sector Ψ 1 , taking into account that for every ξ there are two ξ ′ , we have The eighth sector Ψ 8 directly below the positive p-axis gives the same result with −q instead of q, and sectors Ψ 4 and Ψ 5 above and below the negative p-axis give the same results as sectors Ψ 8 and Ψ 1 .Therefore The same result holds the remaining sectors with cos θ replaced by sin θ and vice versa.Putting the results on all eight sectors together, we get by Lemma A.4 Hence Σt, t = lim ρ→0 ρ 2 Ā(t, ρ) = |t| 2 π , as required.
For the result on the asymptotics of the eigenvalue in Proposition 6.3, we will also assume some correlation decay type results.Namely, we assume that there exist ρ-dependent constants γρ ∈ (0, 1) and Ĉρ > 0 so that for every j ≥ 1, More generally, we assume that that there exist ρ-dependent constants γρ ∈ (0, 1) and Cρ > 0 so that for every j ≥ 1 and every m ≥ 0  36) and (37), and let Ā(t, ρ) be as defined in Lemma 6.2.Let ν ∈ ( 1 3 , 1 2 ) and δ ∈ (0, 1 2 min{γ ρ , γρ }), ensuring that (33) holds.Then for any δ 0 ≤ δ 4/(3ν−1) and t ∈ B δ 0 (0), where for Cρ and γρ as in (37) and some uniform constant C. Remark 6.4 It is possible to shrink δ 0 further to δ 0 < e − max{ Cρ γ−2 ρ ,ρ −2 } leading to E(t, ρ) = o(|t| 2 log |1/t|).This would mean that in the proof of main results in Section 7 we would work on this very small neighborhood and obtain the same range of n and ρ in the final statements.We find it more convenient to work on the neighborhood B δ 0 (0) as in the statement above.Remark 6.5 Let q ρ be the flight function taking values in R 2 as opposed to the displacement function κ ρ taking values in Z 2 .A similar statement holds for the the dominant eigenvalue of the perturbed operator R ρ (e itqρ ).The proof is similar to the one below using that |q ρ − κ ρ | ≤ 1.
Proof of Proposition 6.3.In the notation of Banach spaces of distributions (see, for instance, [21]) for h ∈ C q 0 we write h, 1 = 1, h = h1 dm and m, h = h dm, where 1 is both an element of B and of (C q 0 ) ′ .Let v ρ (t) = Πρ(t)1 Πρ(t)1,1 and recall that v ρ (0) = 1.Recall also that for every ρ, λ ρ (t)v ρ (t) = Rρ (t)v ρ (t) for t small enough, and that λ ρ (0 With the meaning of inner product clarified, for ease of notation from here on we will write V (t, ρ) = M (e itκρ − 1)(v ρ (t)− 1) dm.We recall the terminology in Remark 2.2.For ξ = (p, q) with gcd(p, q) = 1, we let ξ ′ = (p ′ , q ′ ) be the point uniquely determined by ξ in the sense that p ′ /q ′ is convergent preceding p/q in the continued fraction expansion of p/q; in particular |ξ ′ | ≤ |ξ|.Recall that Ψ is the set of all such pairs (ξ, ξ ′ ) with |ξ| ≤ 1/(2ρ).With this specified, we write Using the fact that κ ρ dµ = 0, we compute that where the involved constants in the last big O are independent of ρ.Further, using Lemma A.4, Hence, with Ā(t, ρ) as in Lemma 6.2, Hence, Estimating I 1 (t, ρ).Since M 0 κ ρ dµ = 0, we have Now, by (35) and Lemma 6.1, for some uniform C. Using also that |e ix − 1 − ix| ≤ x y , for any y ∈ (0, 2], where in the last inequality we have used Lemma A.5.Note that for |t| ∈ B δ 0 (0) with δ 0 ≤ γ (32) holds and that δ is chosen so that (34) holds.Using the definition of Π ρ (t) and noting that for every ρ, (u − Rρ (0) Thus, for where and We first treat K 2 (t, ρ).Note that for u in the chosen contour, (u This together with Lemma 6.1 gives that Estimating J 1 (t, ρ) in (39) and K 1 (t, ρ) in (40).These terms are in, some sense, independent of the Banach space B (see the explanation below) and can be analysed either via the correlation function (36) or its generalization (37).The rest of the proof is allocated to this type of analysis.
Our main contribution lies in characterizing the limit paths allowed as ρ → 0; this is done up to the unknown γ ρ , C ρ in (32) and Cρ , γρ as in (37).
Throughout this section, the notation is the same in Subsection 1.1.In particular, b n,ρ = √ , and the variance matrix Σ are defined as in (1), in agreement with Lemma 6.2.We recall that =⇒ stands for convergence in distribution with respect to the invariant measure µ.Theorem 7.1 Let γ ρ , C ρ be as in (32), let γρ , Cρ be as in (37) and let C be as in Proposition 6.3. Set Remark 7.2 A similar statement holds for the flight function q ρ .The only change in the proof is the use of Remark 6.5 instead of Proposition 6.3.
The next result gives a local limit theorem as ρ → 0, again up to the unknown γ ρ , C ρ and γρ , Cρ .This is possible due to the present proof based on spectral methods which produces the fine control of the eigenvalue in Proposition 6.3.The present proof of local limit theorem for the infinite horizon is new even for ρ fixed.We recall that the only proof of such a local limit is given in [31] via the abstract results in [4] for Young towers.Our proof relies on Proposition 6.3, which is new in the set-up of the Banach spaces considered here and it relies heavily on Appendix C and on Proposition 3.1 (which provides useful continuity estimates).
In the notation of Theorem 7.1 we let Φ Σ be the density of a Gaussian random variable distributed according to N (0, Σ) and recall from Section 4.1 that C p 0 ⊂ B. Theorem 7.3 Assume the assumptions and notation of Theorem 7.1; in particular M (ρ) is defined in the same way.Let v ∈ C p 0 (M) and w ∈ L a (M), for a > 1.
Let ρ → 0 and simultaneously n → ∞ in such a way that M (ρ) = o(log n).
Remark 7.4 A similar statement holds for the flight function q ρ .By a similar argument, using Remark 6.5 instead of Proposition 6.3, we obtain A uniform LLT for q ρ can be obtained by, for instance, a straightforward adaptation of the argument used in [27, Proof of Theorem 2.7].
It is known that for every ρ > 0, κ ρ is aperiodic, i.e., there exists no non-trivial solution to the equation e itκρ g • T ρ = g.The aperiodicity of κ ρ has been used in [31] (in fact, in [30]) to provide LLT for fixed ρ.Given Proposition 6.3 and the aperiodicity of κ ρ , the proof of Theorem 7.3 is classic, see [1] and for a variation of it that provides the uniformity in N , see, for instance, [28, First part of Proof of Theorem 2].The proof below recalls the main elements needed to obtain the range of n in the statement.
Proof of Theorem 7.3.Let δ 0 = δ 4/(3ν−1) be so that (34), (32) and Proposition 6.3 hold for all |t| ∈ B δ 0 (0).Since κ ρ is aperiodic, a known argument (see [Lemma 4.3 and Theorem 4 Recall that w ∈ L a , a > 1 and set b = a/(a − 1).Using the Hölder inequality, Recall v ∈ B. Using ( 35), ( 29) and Lemma 6.1 and proceeding as in equation ( 38), for some uniform C and some ε > 0. In the last inequality we have used that |t| < δ 0 .Thus, With a change of variables, Given the range of n in the statement, we use (41) to obtain lim To deal with the big O term in (43), we use that by (41) there exists a uniform constant C so that as ρ → 0. Putting these together and using (43), This together with (42) gives that as ρ → 0, where in the last equation we used that M (ρ) = o(log n).This concludes the proof.
It is known that the local limit theorem for κ ρ and the billiard map T ρ (with ρ fixed) implies mixing for the planar Lorentz map Tρ (again ρ fixed), see [28].In fact, sharp error rates in local limit theorems and mixing are also known, see [28] for the finite horizon case and [29] for the infinite horizon case.
We recall from Section 1 that the Lorentz map Tρ defined on M = M×Z 2 is given by Tρ (θ, Remark 7. 6 The class of functions in Corollary 7.5 is rather restrictive as the functions v, w are supported on the cell M. Given the work [28] (see also [29,Section 6]), it is very plausible that the present mixing result can be generalized to a suitable class of dynamically Hölder functions supported on the whole of M. Since the involved argument is rather delicate and not a main concern of the present work, we omit this.

A Estimates on corridors
] of initial conditions x such that the next collision occurs at a scatterer on the opposite side of the corridor.For this situation, [31] use the coordinates (α, z), where α is the angle the trajectory of some x ∈ ∂O 0 makes with the tangent line at x 0 , and the intersection point is y = x 0 + zξ, see Figure 5. the angle of the trajectory to the normal vector at the collision point.When this section is the line y = x 0 + xξ, we have α = π 2 − ϕ, so cos ϕ = sin α.But to get the correct normalizing constant, we give a more extensive argument.From Figure 5 we have After making α and z subject of these equations, we see that the change of coordinates involved is The Jacobian determinant is + cos θ .
Thus, using (44) and some trigonometric formulas, The following is [31,Proposition 6] in more detail: Lemma A.2 Suppose that the scatterers have radius ρ > 0 and the width of the corridor given by ξ is d ρ (ξ).Then where ξ ′ as in Remark 2.2 is the integer vector on the boundary of the corridor opposite to the ξ-boundary.
tangent line of O 0 and O κρ−ξ .For z = z 1 there is also is only one line, namely the common tangent line of O ξ and O κρ , see Figure 6.These two lines are obtained from each other by translation over one unit ξ, so z 1 ξ − z 0 ξ = |ξ|.However, if ρ is small compared to N , these two tangent lines are the common tangent lines at the upper sides of O 0 and O κρ and at the lower sides of O 0 and O κρ .In this case This also shows that the transition between the two cases is when 2ρ = dρ(ξ) N .For each z ∈ [z 0 , z 1 ], the range of possible values of α is again bounded by the α's obtained at the tangent lines to O κρ−ξ and O κρ .Therefore, see Figure 7, 2) and z ≤ |ξ| as well, each α in this interval satisfies α = dρ(ξ) Integrating the density given in Lemma A.1 for the case 2ρ ≥ N , see Figure 7 with small version of O κρ , we have Figure 7: The parameter interval [α 0 (z), α 1 (z)] given by angles between two tangent lines.

A.2 Corridors sums
Let ϕ be Euler's totient function, i.e., the number of integers 1 ≤ q ≤ p coprime with p.The following lemma is classical number theory, but we couldn't locate a proof of the full statement.
Lemma A.3 For every a > −2, we have where Proof.Let µ be the Möbius function.A standard equality is ϕ where we used the Dirichlet series identity ∞ d=1 µ(d) As an aside, there are asymptotic formulas for s > see [19,Theorem 288].
In the course of this paper we denote, for a fixed value of ρ, the set of corridors that are "visible" from the origin by Ψ.As described in Lemma 2.1, these can be characterized by pairs (ξ, ξ ′ ) ∈ Z 2 ×Z 2 where ξ = (p, q), gcd(p, q) = 1 and |ξ| ≤ (2ρ) −1 , while ξ ′ may denote either the first or the second convergent preceding ξ in the continued fraction expansion of p/q, see Remark 2.2.Sums of the type in the following lemma are used throughout the paper.
Lemma A. 4 We have Proof.Using the two coordinate axes and their bisectrices, we divide the plane into eight sectors and for each sector, we sum the scatterers in S. Circular sections of radius R have asymptotically π 4 as many points as triangular sectors with base R. Also, every corridor direction in this sector comes with two corridors (ξ, ξ ′ ) and (ξ, ξ ′′ ).By Lemma A.3, their sum is, for a > −2, If a = −2, then a similar computation gives ≍ | log ρ|, and for a < −2, the series is summable: Lemma A.5 For p ∈ [1, 2), the p-norm of the displacement function satisfies Proof.Take p ∈ [1, 2).We estimate over all ξ-corridors similarly as in Lemma A.4: Taking the p-th root gives the result.
Lemma A.6 Let W ∈ W s be a stable leaf, and let Ψ W stand for all lattice points ξ = (p, q) ∈ Ψ that can be reached from O 0 with coordinates in W . Then for every a ∈ ( 1 2 , 1), Proof.There is an arc W ∈ S 1 of length | W | ≪ |W | such that every lattice point that can be reached from O 0 with coordinates in W has its polar angle in W . Due to the symmetries in the Z 2 , it suffices to study W ⊂ [0, π/2], so the lattice point ξ = (p, q) in this sector satisfy 0 ≤ q ≤ p and tan( W ) ⊂ [0, 1].In fact, we will start by assuming that tan( W ) ∈ [ 1 10 , 9  10 ].Because p 2 + q 2 ≥ 2pq for all (p, q) = ξ, we have (pq) a/2 1 W ( q p ).We will apply an estimate from [33, Theorem 2.2], which, in our terminology, reduces to where C a is a constant depending only on a, and c ψ (ℓ) is the ℓ-th Fourier coefficient of x → ψ(x)x −a .If ψ = 1 W , then these Fourier coefficients are not summable, so we first smoothen 1 W to a function ψ with supp(ψ) concentric to W and | supp(ψ)| = | W | = 3|W |.On W itself, ψ ≡ 1 and on the two interval components ψ is a translated copy of the function Then ψ dx = 2|W | and integrating by parts twice gives an estimate of the Fourier coefficients of because supp(ψ) is bounded away from {0, 1} (so x −a doesn't blow up) and (ψ(x)x −a ) ′′ = 0 outside supp(ψ).
The Dirichlet series of the Möbius function can be estimated as We use this and the fact that ℓ ∈ N has O(ℓ 1/2 ) divisors to estimate the last big O-term in (48).

B Distortion properties
Throughout, a uniform constant is a constant that is independent of ρ.
Let us recall some terminology and notations from [10,Chapter 4].Unstable curves generate dispersing wavefronts, which are evolved by the free flight, and then leave traces of unstable curves on the scatterer at the next collision.For wavefronts it is convenient to use the Jacobi coordinates (dξ, dω), and an important quantity5 Ω = dω dξ , the curvature of the wavefront.Let Ω − and Ω + denote its value immediately before and after a particular collision, respectively.
On the scatterer, the traditional coordinates are (r, φ) yet, we prefer to use the ρ-independent (θ, φ) and take advantage of d dθ = (2πρ) d dr .
First we relate Ω − to the slope of the unstable curve: 1 2π dφ dθ = ρΩ − cos φ + 1. Differentiating with respect to θ gives 1 2π Lemma B.1 There exists a uniform constant C > 0 such that for any C 2 smooth unstable curve W there exists n W such that for n ≥ n W on all components of T n ρ W we have Thus we may restrict to the class of regular unstable curves for which (50) holds.Also, this shows that as ρ → 0, the unstable curves limit in a C 2 sense to straight lines of slope 2π.
Proof.The properties of the free flight are not effected by shrinking the scatterers or using the θ-coordinate.Thus 0 ≤ Ω − ≤ (τ min ) −1 and, by (49), it is enough to show dΩ − dθ ≤ C to prove the lemma.Now dΩ − dθ = (2πρ) dΩ − dr , and the evolution of dΩ − dr is discussed in [10, section 4.6].Following the notation there, introduce and use superscripts − and + to denote pre-and post-collision values of these quantities, respectively.
Hence it follows that is the value of F 1 between the n-th and the (n + 1)st collision.This implies that there exists C 2 > 0 and n W (depending on the curve W ) such that for any n ≥ n W we have It follows that regular unstable curves can be parametrised by the coordinate θ, and for any smooth function f : W → R, df dθ ≍ df dx , where x is (Euclidean) arc-length along the curve -dx 2 = dθ 2 + dφ 2 (not to be confused with the arc-length r along the scatterer).
Let us also recall that an unstable curve is homogeneous if it is regular and contained in one of the homogeneity strips For such curves, analogous to [10, Formula (5.13)], we have for some uniform constant C > 0, where φ corresponds to any point of W . (This follows as the slope of the curve is uniformly bounded away from 0 and ∞.) Distortion bounds are stated as follows.Let W be a homogeneous unstable curve, and assume that for some N ≥ 1, W n = T −n ρ W is a homogeneous unstable curve for n = 0, 1, . . ., N .For x ∈ W , let x n = T −n ρ x ∈ W n .Let J W T −n ρ (x) and J Wn T −1 ρ (x n ) denote the respective Jacobians.
Lemma B.2 Consider W and N as above and y, z ∈ W arbitrary.There exists a uniform constant Proof.The lemma relies on the inequality for some uniform C > 0, cf.[10,Formula (5.8)].
Using this formula the argument in the proof of [10,Lemma 5.27] can be repeated literally: where we have used the chain rule, (52), and (51) and the uniform hyperbolicity.It remains to prove (52).Here we essentially follow [10, pp. 106-107].We have . We consider the derivatives of these terms separately.As noted above, differentiation with respect to θ n and x n can be interchanged.By Lemma B.1, the derivative of the second term w.r.t.θ n is uniformly bounded.The same applies to the derivative of the third term with respect to θ n+1 , while is uniformly bounded from above.The first term gives the main contribution: as cos φ n is not bounded away from 0, the derivative of its logarithm is The fourth term is the logarithm of the quantity 2ρ −1 τ n+1 + cos φ n+1 (1 + τ n+1 Ω − n+1 ) which is bounded from below, but not from above.It is thus (more than) enough to show that, when taking the derivative, all contributions to the numerator are uniformly bounded.This holds immediately by the previous discussion for all the terms except 2 ρ dτ n+1 dxn which requires further investigation.Note that τ n+1 = dist(P (x n ), P (x n+1 )) where P (x n ) and P (x n+1 ) are points on the billiard table (and thus on R 2 ) associated to the points x n ∈ W n and x n+1 ∈ W n+1 on the two scatterers, respectively.In an appropriate reference frame P (x n ) = (ρ cos θ n , ρ sin θ n ) hence the θ n -derivatives of both coordinates are ≪ ρ, and the same holds for the θ n+1 -derivatives of the coordinates of P (x n+1 ).Thus which is sufficient for our purposes.
C Decay of correlation for κ ρ .
The main result of this section is the justification of (37), that is Proposition C.1 There exist Ĉρ > 0 and θρ < 1 such that • furthermore, there exist Cρ > 0 and θρ < 1 such that for any j, ℓ ≥ 1 we have The ρ-dependence of this exponential rate gives the main source of unknown dependence on ρ in the main results of our paper.During the proof we will point out the exact sources of unknown dependence of Cρ > 0 and θρ < 1 on ρ.
Let us make some comments on the relations of the two estimates of Proposition C.1.We will first prove (54) with some Ĉρ > 0 and θρ < 1 that we can explicitly relate to the correlation decay rates of the map T ρ on Hölder functions, as expressed in (66) below.Then we extend our argument to obtain (55) for some Cρ > Ĉρ and θρ ∈ ( θρ , 1).Obtaining relations similar to (66) for the constants Cρ and θρ seems quite difficult and we do not push this point.
The proof of Proposition C.1 consists in: a) reconsider [11, Proposition 9.1]; b) only for (55), work with a version of R ρ with spectral gap in a Banach space embedded in some L p space with p > 1. Item a) is needed in order to obtain the bound |t| 2 and the decay of correlation in j.Item b) is needed to obtain the joint decay in j and ℓ.Item b) is possible because for every ρ > 0, there exists a Young tower ∆ ρ and a tower map T ∆ρ associated with the billiard map T ρ ; this is ensured by the construction in [9,34].We emphasize that we will not exploit any fine dependence on ρ of T ∆ρ (the mere existence is enough), which is why this part of our arguments can be worked on the Young tower ∆ ρ .
• Since c > c 1 (q) = q+1 2−q , the power of H in the second term of (ii) is negative.
• Choosing Ĥ = H c , the power of H in (i) is also negative, again for c > c 1 (q) = q+1 2−q .
Standard pairs and families.Let us recall some terminology related to standard pairs, see also [11, page 29].A standard pair ℓ = (W, h W ) is a regular unstable curve W that supports a dynamically log-Hölder continuous probability density h W .As such, it can be regarded as a probability measure on the phase space M, which will be denoted by ℓ, too.
A standard family is a collection of standard pairs G = {ℓ a }, a ∈ A equipped with a probability factor measure λ G on A. This induces a probability measure P G on M.
For a standard pair ℓ = (W, h W ) any x ∈ W splits W into two subcurves, let r W (x) denote the length of the shorter, and let Z ℓ = sup ε>0 ε −1 ℓ(r W ≤ ε).By Hölder continuity of log h W , ℓ is equivalent to the normalized Lebesgue measure on W and thus Z ℓ ≍ |W | −1 .This generalizes for the Z-function of a standard family Z G ≍ λ G (a) |Wa| dm W .The T ρ -image of a standard pair is a countable collection of standard pairs.Hence, the image of a standard family is a standard family.Given a standard family G, for n ≥ 1, G n denotes the T n ρ -image of G.It follows from the growth lemma (Proposition 3.1) that there exists ϑ < 1 and where δ 0 ≍ ρ ν (see (16) and Remark 3.2, part (i)).As consequence, for any standard pair and n ≥ 1 Cells.For ξ ∈ Z 2 such that the corridor is opened up, and for m ∈ Z let D ξ,m ⊂ M denote the set of points for which κ ρ = mξ + ξ ′ .The geometric properties of D ξ,m and its image T ρ D ξ,m will play an important role in the argument.T ρ D ξ,m is depicted in Figure 2. A similar description applies to D ξ,m ; it is delimited by a long singularity curve, decreasing in the (θ, ϕ) coordinates, which is connected to the boundary of M by two shorter decreasing singularity curves, of length ≍ (|ξ|ρm) −1/2 , running at a distance ≍ (|ξ|m) −2 from each other.Further properties: Our argument below follows the proof of [11,Proposition 9.1] taking into account that the corridor structure depends on ρ.
Proof of Lemma C.2.For item (i), using µ(T −j ρ D ξ, m) ≪ ρ −1 m−3 | ξ| −3 as well as Lemma A.4 several times, we get We will take Ĥ = H c for c > 0 to be determined.To get a negative power of H, we need q < 2 and c > q+1 2−q .
For the proof of (ii), we need to estimate For different ranges of the indices, we will use two different estimates to bound µ(D ξ,m ∩T −j ρ D ξ, m).On the one hand, as before, we have as the expansion factor of T ρ on W k is ≍ ρ −1 |ξ|mk r 0 .Equipped with the conditional measure induced by µ, W is a standard pair ℓ = (W, h W ), and its image is a standard family T ρ ℓ associated to the This analysis applies to all the curves in the foliation.Accordingly, µ conditioned on D ξ,m can be regarded as a standard family G, and the Z-function of its T ρ -image satisfies For further iterates, it follows form (56) that Z Gn ≤ Cρ −ν (m|ξ|) 2r 0 ρ c(q−1)−q−2−ν , where we have used that because q + c(q − 1) < 3 2 − 1 2r 0 , the contribution of m is summable (this condition is equivalent to c < c 2 (q) = 1− 1 r 0 2q−2 , cf.Remark C.3).Note that if q = 1 then this contribution is independent of c; however, there is an additional factor of | log ρ| • log H.
First we may fix q such that 3 2 − 1 2r 0 > q + q + 1 2 − q (q − 1) = 2q − 1 2 − q ⇔ q < 2 − 6 7 − 1/r 0 and then we can fix c slightly larger than q+1 2−q , such that the conditions are still met.The range of allowed q depends on r 0 , it can never exceed 8  7 ; for the traditional r 0 = 2 the upper bound is 14  13 , while for r 0 = 5 the upper bound is 19  17 .
Before proceeding to the proof of Proposition C.1, we recall one property of the norm the Young space B ∆ρ ⊂ L p (µ ∆ρ ), p > 1 that we shall need in the proof below.(The precise definition of B ∆ρ is not important in the proof below, and we omit it.)For any function that is constant on the partition elements of the Young tower, the involved seminorm is zero.This is the case for κ′ ρ and thus, κ′ ρ B ∆ρ ≤ H (and a similar version holds for κ′′′ ρ ).
Proof of Proposition C.1.We first prove the statement for the case when ℓ = 0 and point out the required modifications when ℓ ≥ 1.
Case ℓ = 0. Given (61), in this case we need to show that for some ρ-dependent constants θρ < 1 and Ĉρ > 0. Throughout this proof, we let κ ′ ρ , κ ′′ ρ , κ ′′′ ρ , κ ′′′′ ρ also denote their corresponding versions on the tower ∆ ρ and the context in which they appear will make it clear which version we are referring to.For I 3 (t, ρ) we use the exponential decay of correlation (see Remark C.4 below for the case that the roof function r of the tower has gcd(r) > 1).This gives the only source of unknown dependence on ρ in the case m = 0.More precisely, for every ρ > 0, there exists θρ < 1 and C ρ > 0 so that  (66) Case ℓ ≥ 1.The main differences in this case come down to dealing with integrals containing unbounded terms κ ′′ ρ and κ ′′′′ ρ in such a way that can gain exponential decay in ℓ and then proceed as in the case ℓ = 0 treated above.To do this, we exploit that B ∆ρ ⊂ L p (µ ∆ρ ).
Using (61), we need to estimate for some ρ-dependent C ρ and θρ < 1.This is the first source of unknown dependence on ρ.Since B ∆ρ ⊂ L p (µ ∆ρ ), for some ρ-dependent C 0 ρ .This is the second source of unknown dependence on ρ.
With these specified, we can write

4 √ 2 ,
then corridors at angles of ±45 • open up, and the smaller ρ becomes, the more corridors open up at rational angles.

Figure 3 :
Figure 3: A corridor collision map from O −ξ and O −κρ to O 0 .

Figure 4 : 3
Figure 4: Illustration of the proof of Lemma 2.3

2 M
where Leb Z 2 is the counting measure on Z 2 .An immediate consequence of Theorem 7.3 is Corollary 7.5 Assume the assumptions and notation of Theorem 7.3.Let ρ → 0 and simultaneously n → ∞ in such a way that M (ρ) = o(log n).Then lim ρ→0 (b n,ρ )
|κ ρ | ≍ |ξ|m on D ξ,m , the truncation κ ′ ρ restricts κ ρ to the cells D ξ,m with m ≤ H|ξ| −1 .The result we will use in the proof of Proposition C.1 below is Lemma C.2 For any c 0 > 2 we have m −3 (due to the factor cos φ in the measure);• an unstable curve may intersect D ξ,m in a subcurve of length ≤ C(|ξ|m) −2 ;• T ρ D ξ,m intersects homogeneity strips of index k ≥ C(ρ|ξ|m) If ℓ = (W, h W )is a standard pair, then it can intersect D ξ,m in a subcurve of length ≤ C(|ξ|m) −2 , thus the intersection has probability bounded above by C(|ξ|m) −2 |W | −1 ≍ Z ℓ (|ξ|m) −2 .It follows that for a standard family G we have

)
For the other estimate, foliate D ξ,m with unstable curves |W | of length ≍ (|ξ|m) −2 .The image of any such curve stretches along T ρ D ξ,m , crossing homogeneity strips with indices k ≥ C(ρ|ξ|m) 1 2r 0 .The piece of T ρ W in the k-th homogeneity strip will be denoted by T ρ W k , it has length ≍ k −r 0 −1 , and its preimage has length |W curves T ρ W k .To obtain the Z function, we use that the weight of|T ρ W k | within this family is |W k | |W |