1 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 Lorentz in 1905 [23], is a dynamical system on the infinite billiard table obtained by removing strictly convex scatterers from \({{\mathbb {R}}}^2\). We study the periodic model when these scatterers are round disks of radius \(\rho \in (0,1/2)\) positioned at the points of the Euclidean lattice \({{\mathbb {Z}}}^2\). 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 \({{\mathcal {M}}}= \partial O \times [-\frac{\pi }{2}, \frac{\pi }{2}]\), where O is a round disk at the origin with radius \(\rho \). The phase space representing all cells together is \({\widehat{{{\mathcal {M}}}}} = {{\mathcal {M}}}\times {{\mathbb {Z}}}^2\) and the displacement function \(\kappa _\rho :{{\mathcal {M}}}\rightarrow {{\mathbb {Z}}}^2\) 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 \(\rho \in (0,1/2)\), the horizon is infinite which means that the time between two consecutive collisions—and accordingly, \(\kappa _\rho :{{\mathcal {M}}}\rightarrow {{\mathbb {Z}}}^2\)—is unbounded. This corresponds to corridors, that is, infinite strips on \({{\mathbb {R}}}^2\) parallel to some direction \(\xi \in {{\mathbb {Z}}}^2\setminus \{0\}\) which do not intersect any of the scatterers of the infinite billiard table. Both the number and the geometry of these corridors depend on \(\rho \). Hence, the value of the parameter \(\rho \) 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 \(\rho \in (0,1/2)\) as time \(n \rightarrow \infty \)

A consequence of the infinite horizon is the superdiffusive behaviour of \(\kappa _\rho \) with \(\rho \in (0,1/2)\) 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 \(T_{\rho }:{{\mathcal {M}}}\rightarrow {{\mathcal {M}}}\) be the billiard map, recall that it preserves the canonical invariant probability measure \(\mu \). Set

$$\begin{aligned} \kappa _{n,\rho }=\sum _{j=0}^{n-1}\kappa _\rho \circ T_\rho ^j, \qquad \Sigma =\frac{1}{\pi } \begin{pmatrix} 1&{}0\\ 0&{}1 \end{pmatrix}. \end{aligned}$$

Choosing the initial point on \({{\mathcal {M}}}\) according to \(\mu \), we can regard \(\kappa _{n,\rho }\) as a family of random variables. Throughout we let \(\implies \) stand for convergence in distribution. We recall the CLT with non-standard normalization: for every \(\rho \in (0,\frac{1}{2})\) there exists a positive definite matrix \(\Sigma _{\rho }\) such that:

$$\begin{aligned} \text {For fixed }\rho \in (0,1/2),\quad \frac{\kappa _{n,\rho }}{\sqrt{n\log n}}\implies \mathcal {N}(0,\Sigma _{\rho }) \text { as } n\rightarrow \infty . \end{aligned}$$

This result was conjectured by Bleher [5] and proved rigorously via two different methods: Szász and Varjú [31], and Chernov and Dolgopyat [11]. In the setting above, the requirement of having two non-parallel corridors (present in [11, 31]) is automatically satisfied because the scatterers are positioned at the lattice points.

It is important to note that there is an explicit formula for \(\Sigma _{\rho }\), which involves the scatterer geometry for fixed \(\rho \), see for example [11, Formula (2.1)]. A computation (similar to our proof of Lemma A.4) shows that

$$\begin{aligned} \lim _{\rho \rightarrow 0} (4\pi \rho ^2) \Sigma _{\rho }=\Sigma , \end{aligned}$$

where \(\Sigma \) 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):

$$\begin{aligned} \frac{\kappa _{n,\rho }}{(\sqrt{4\pi }\rho )^{-1}\sqrt{n\log n}} \implies \mathcal {N}(0,\Sigma ) \text { as first } n\rightarrow \infty \text { and then } \rho \rightarrow 0. \end{aligned}$$

The method of proof in [31] relies on the existence of a Young tower for \(T_\rho \) as in [9, 34] and an abstract result of Bálint and Gouëzel [4] along with several additional properties of \((\kappa _\rho , T_\rho )\) 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):

$$\begin{aligned} \text {For fixed }\rho \in (0,1/2),\quad (n\log n)\mu (\kappa _{n,\rho }=0)\rightarrow \Phi _{\Sigma _{\rho }}(0)\text { as } n\rightarrow \infty , \end{aligned}$$

where \(\Phi _{\Sigma _{\rho }}\) 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 \(\{\kappa _\rho \circ T_\rho ^n\}_{n\ge 1}\). The authors of that work develop an argument based on standard pairs to establish a bound on the correlations for \(\kappa _\rho \):

$$\begin{aligned} \begin{array}{l} \text {For fixed }\rho \in (0,1/2),\text { there exist }\hat{\vartheta }_{\rho }\in (0,1) \text { and }\hat{C}_\rho > 0\\ \text {so that }\left| \int _{{{\mathcal {M}}}} \kappa _\rho \cdot \kappa _\rho \circ T_\rho ^n\,d\mu \right| \le \hat{C}_\rho \cdot \hat{\vartheta }_{\rho }^n \text { for all } n\ge 1.\\ \end{array} \end{aligned}$$

Using (6), the CLT (2) is proved in [11, Proof of Theorem 8 a)] via blocking type arguments; we refer to Denker [17] for a classical reference. Furthermore, as shown in [11, Proof of Theorem 8], the limit law (2) together with a tightness argument for a truncated version of \(\kappa _\rho \) provides another refinement of the CLT, namely, the Weak Invariance Principle (WIP):

$$\begin{aligned} \begin{array}{l} \text {For fixed }\rho \in (0,1/2) \text { and } s\in (0,1), \ \frac{\kappa _{\lfloor ns \rfloor ,\rho }+ \{ns\}(\kappa _{\lfloor ns \rfloor +1,\rho }- \kappa _{\lfloor ns \rfloor ,\rho }) }{\sqrt{n\log n}} \text { converges as }\\ n\rightarrow \infty \text { to a Brownian motion with mean } 0\text { and variance }\Sigma _{\rho }. \end{array} \end{aligned}$$

Similar versions of the CLT (2) and the WIP (7) hold for the flight time function taking values in \({{\mathbb {R}}}^2\), see [11].

In a different direction, a further important consequence of the LLT (5) established in [31] is that it allows one to study mixing of the infinite measure preserving billiard dynamics on the entire lattice \({\widehat{{{\mathcal {M}}}}} = {{\mathcal {M}}}\times {{\mathbb {Z}}}^2\). This can be modelled by a \({{\mathbb {Z}}}^2\) extension

$$\begin{aligned} {\hat{T}}_{\rho }^n(\theta ,\phi ,\ell ) = ( T_{\rho }^n(\theta ,\phi ), \ell +\kappa _{n,\rho }(\theta ,\phi )), \qquad (\theta ,\phi ) \in {{\mathcal {M}}},\ \ell \in {{\mathbb {Z}}}^2. \end{aligned}$$

The dynamics \({\hat{T}}_\rho \) preserves the measure \({\hat{\mu }}=\mu \times \text{ Leb}_{{{\mathbb {Z}}}^2}\), where \(\text{ Leb}_{{{\mathbb {Z}}}^2}\) denotes the counting measure. Let us introduce the notation \({{\mathcal {M}}}_{\xi }:={{\mathcal {M}}}\times \{\xi \} \subset {\widehat{{{\mathcal {M}}}}}\) for \(\xi \in {{\mathbb {Z}}}^2\), and furthermore, for brevity, let \({{\mathcal {M}}}_0:={{\mathcal {M}}}\times \{(0,0)\}\), where the label 0 refers here to the origin in \({{\mathbb {Z}}}^2\). An immediate consequence of (5) is:

$$\begin{aligned}&\text {For fixed }\rho \in (0,1/2),\ (n\log n) \mu (\kappa _{n,\rho }=0) =(n\log n) {\hat{\mu }}({{\mathcal {M}}}_0\cap {\hat{T}}_\rho ^{-n}{{\mathcal {M}}}_0) \nonumber \\&\quad \rightarrow \Phi _{\Sigma _{\rho }}(0) \text { as } n\rightarrow \infty . \end{aligned}$$

A first refinement of the LLT (5) and of the mixing statement (8) was obtained by Pène [28] who proved the analogue of these statements for the class of dynamically Hölder observables. Later on, Pène and Terhesiu [29], building on the results in [4], obtained sharp error rates in the LLT and the mixing for dynamically Hölder observables, including observables supported on compact sets. Furthermore, [29] establish optimal error rates for mean zero observables.

1.2 Recalling results as first \(\rho \rightarrow 0\) and then \(n \rightarrow \infty \) (Boltzmann–Grad limit)

In [24, 25], Marklof and Strömbergsson studied the Boltzmann–Grad limit of the periodic Lorentz gas. This corresponds to letting the scatterer size \(\rho \rightarrow 0\) and investigating the displacement in the rescaled continuous time \(T=\rho t\) (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 and Tóth [26] then studied the large time asymptotic of this Markov process, and obtained the CLT and the WIP with non-standard normalization \(\sqrt{T\log T}\).

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 \(\kappa _{n,\rho }\) in the limits \(\rho \rightarrow 0\) first and then \(n\rightarrow \infty \). In particular, [26, Theorem 1.2] states for \(d=2\) that:

$$\begin{aligned} \frac{\kappa _{n,\rho }}{(\sqrt{4\pi }\rho )^{-1}\sqrt{n\log n}} \implies \mathcal {N}(0,\Sigma ) \qquad \text { as } \rho \rightarrow 0\text { followed by } n\rightarrow \infty , \end{aligned}$$

where \(\kappa _{n,\rho }\) and \(\Sigma \) are as in (1),Footnote 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: \(\rho \rightarrow 0\) followed by \( n\rightarrow \infty \), as opposed to fixed \(\rho \).

In [26], the authors state that an open problem is to consider the joint limit \(\rho \rightarrow 0\) and \(n\rightarrow \infty \). In the Boltzmann–Grad limit scenario with diffusive behaviour, this type of question is answered by Lutsko and 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 \(\rho \rightarrow 0\) and \(n \rightarrow \infty \) 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.

Theorem A

Let \(\kappa _{n,\rho }\) and \(\Sigma \) be as in (1), and let

$$\begin{aligned} b_{n,\rho }=\frac{\sqrt{n\log (n/\rho ^2)}}{\sqrt{4\pi }\ \rho }. \end{aligned}$$

There exists a function \(M(\rho )\) with \(M(\rho )\rightarrow \infty \) as \(\rho \rightarrow 0\) such that,

$$\begin{aligned} \frac{\kappa _{n,\rho }}{b_{n,\rho }}\implies \mathcal {N}(0,\Sigma ), \text { as } n\rightarrow \infty \text{ and } \rho \rightarrow 0 \text { such that } M(\rho )=o(\log n). \end{aligned}$$

A precise expression of \(M(\rho )\) is given in Theorem 7.1 in Sect. 7. At this stage we mention that \(M(\rho )\) depends on the rate of correlation decay for Hölder observables as \(\rho \rightarrow 0\). How this decay rate depends on \(\rho \) 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:

$$\begin{aligned} \left| \int _{{{\mathcal {M}}}} \psi _1\cdot \psi _2\circ T_\rho ^n \,d\mu \right| \le C_\rho (\psi _1,\psi _2)\cdot \hat{\theta }_{\rho }^n \text { for all } n\ge 0, \end{aligned}$$

where \(\psi _1:{{\mathcal {M}}}\rightarrow {{\mathbb {R}}}\) and \(\psi _2:{{\mathcal {M}}}\rightarrow {{\mathbb {R}}}\) are centered, Hölder continuous observables, and \(\hat{\theta }_{\rho }<1\) may depend on the Hölder exponent, while \(C_\rho (\psi _1,\psi _2)>0\) on the Hölder norm of these functions, and both constants depend also on \(\rho \) (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. \(C_\rho \) and \(\hat{\theta }_{\rho }\)) depends on \(\rho \). 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, \(\psi _1\) and \(\psi _2\) may belong to a larger space (that contains Hölder functions), however, these spaces do not contain the unbounded observable \(\kappa _{\rho }\). Hence, even for fixed \(\rho \), 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 \(\rho \rightarrow 0\) and \(n\rightarrow \infty \), 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 \(\hat{C}_\rho > 0\) and \(\hat{\vartheta }_{\rho }<1\) in (6), we explicitly relate these constants to \(C_\rho \) and \(\hat{\theta }_{\rho }\) of (10), as expressed in (66).Footnote 2 On 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 \(R_{\rho }(t)\) (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 and 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 and 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 Sect. 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 \(M(\rho )=o(\log n)\) we have

$$\begin{aligned} \frac{b_{n,\rho }}{(\sqrt{4\pi }\rho )^{-1}\sqrt{n\log n}}\rightarrow 1, \end{aligned}$$

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 \(\rho \rightarrow 0\) simultaneously with \(n\rightarrow \infty \), as opposed to fixed \(\rho \). 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 \(\rho \) scenario with main results recalled in Sect. 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 \(\rho \). Instead, we prove Theorem A via the Nagaev method on Banach spaces of distributions introduced by Demers and Zhang [14,15,16] in the spirit of the spaces constructed in Demers and Liverani [12]. See Aaronson and Denker [1, 2] for a classical reference on the Nagaev method in (Gibbs Markov) dynamics beyond the CLT with standard normalization (that is \(\sqrt{n}\)). 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 \(\rho \rightarrow 0\), 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 \(\rho \).

The Nagaev method requires: (1) the existence of a Banach space \(({{\mathcal {B}}}, \Vert \,\Vert _{{{\mathcal {B}}}})\) on which the transfer operator \(R_\rho \) of \(T_\rho \) has a spectral gap; (2) that the perturbed transfer operator (\(R_\rho (t) \psi =R(e^{it\kappa _\rho }\cdot \psi )\) for \(\psi \in {{\mathcal {B}}}\)) has sufficiently good continuity estimates \(\Vert R_\rho (t)-R_\rho (0)\Vert _{{{\mathcal {B}}}}\le C |t|^\nu \); the larger \(\nu > 0\), the better.

Regarding 1), using a Lasota–Yorke inequality on a strong space \({{\mathcal {B}}}\) and a weak space \({{\mathcal {B}}}_w\), Demers and Zhang [14,15,16] established the spectral gap for every fixed \(\rho \), see Sect. 4. This is the main reason why we resorted to the use of such Banach spaces.

Regarding 2), as in Keller and 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 Sect. 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 \(O(|t|^\nu )\) for \(\nu <1/2\) with explicit dependence on \(\rho \), in both the strong and the weak spaces. This exponent \(\nu \) is too small to obtain the asymptotics of the leading eigenvalue of \(R_\rho (t)\) 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 \(\rho \) 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 Sect. 2 we recall some basic properties of hyperbolic billiards and also estimate widths of corridors that open up as \(\rho \rightarrow 0\). Section 3 gives the Growth Lemmas, following [14,15,16] but with estimates made explicit in terms of \(\rho \), 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 \(R_\rho \), showing that the \(\rho \)-dependence can be controlled. Section 5 is devoted to the continuity estimates of the perturbed transfer operator \(R_\rho (t)\) and Sect. 6 gives the asymptotics of the corresponding leading eigenvalue. The precise statements and proofs of the limit theorems are gathered in Sect. 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).

2 Preliminaries on Lorentz gas on \({{\mathbb {Z}}}^2\)

Our general reference on hyperbolic billiards is Chernov and Markarian [10], the conventions of which are followed in our exposition, except for some minor differences. In particular, we use coordinates \((\theta ,\phi ) \in {{\mathbb {S}}}^1 \times [-\frac{\pi }{2}, \frac{\pi }{2}]\) on \({{\mathcal {M}}}\), where

  • \(\theta \in {{\mathbb {S}}}^1\) in clockwise orientation describes the collision point on the scatterer (so the corresponding point on \(\partial O\) is \((\rho \sin \theta , \rho \cos \theta )\));

  • \(\phi \in [-\frac{\pi }{2}, \frac{\pi }{2}]\) denotes the outgoing angle that the billiard trajectory makes after a collision at a point with coordinate \(\theta \) with the outward normal vector \(\textbf{N}_\theta \) at this point (so \(\phi = \frac{\pi }{2}\) corresponds to an outgoing trajectory tangent to O in the positive \(\theta \)-direction).

In these coordinates \((\theta ,\phi )\), the measure \(\mu \) has the same form \(d\mu = \frac{1}{4\pi } \cos \phi \, d\phi \, d\theta \) for all values of the radius \(\rho > 0\). Integrals involving the displacement function \(\kappa _\rho \), however, do depend on \(\rho \). If the flight between \((x,\ell )\) and \(( T_\rho (x), \ell + \kappa _\rho (x))\) 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 \(\pm \frac{\pi }{2}\). This sparks another long flight in the same corridor, i.e., \(\Vert \kappa _\rho (T_\rho x)\Vert \) is large, too.

In the remainder of this section, we record some properties of \(T_\rho \) and \(\kappa _\rho \). In Sect. 2.1 the geometry of corridors is described, with special emphasis on the asymptotics of small \(\rho \). In Sect. 2.2 we focus on the singularities, which, in addition to strong hyperbolicity, are the other main feature of the map \(T_{\rho }:{{\mathcal {M}}}\rightarrow {{\mathcal {M}}}\). In Sect. 2.3, the hyperbolic properties of \(T_{\rho }:{{\mathcal {M}}}\rightarrow {{\mathcal {M}}}\) 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 \(f \ll g\) and the Landau big O notation interchangeably: there is a constant \(C > 0\) such that \(f \le C g\). Similarly \(f \asymp g\) means that there exists \(C>1\) such that \(C^{-1} g \le f \le Cg\).

2.1 Corridors and their widths

Let \(O_\ell \) denote the circular scatterer of radius \(\rho \) placed at lattice point \(\ell \in {{\mathbb {Z}}}^2\). The computation of \(\mu (x \in \partial O_0 \times [-\frac{\pi }{2}, \frac{\pi }{2}]: \kappa _\rho (x) = (p,q))\) is based on the division of the phase space in corridors. These are infinite strips in rational directions given by \(\xi \in {{\mathbb {Z}}}^2 {\setminus } \{ 0 \}\) for \(\rho \) 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 \(\rho < \frac{1}{2}\), there are infinite corridors parallel to the coordinate axes. If \(\rho < \frac{1}{4} \sqrt{2}\), then corridors at angles of \(\pm 45^\circ \) open up, and the smaller \(\rho \) becomes, the more corridors open up at rational angles.

Given \(0 \ne \xi \in {{\mathbb {Z}}}^2\) and \(\rho > 0\) sufficiently small, there are two corridors simultaneous tangent to \(O_0\) and \(O_\xi \), one corridor on either side of the arc connecting 0 and \(\xi \). The widths of the corridors are denoted by \(d_{\rho }(\xi )\) and \(\tilde{d}_{\rho }(\xi )\), see Fig. 1.

Fig. 1
figure 1

Corridors tangent to the scatterers at 0 and \(\xi = (3,2)\)

Lemma 2.1

If \(\rho = 0\) and \(\xi = (p,q) \in {{\mathbb {Z}}}^2\) is expressed in lowest terms, then

$$\begin{aligned} d_0(\xi ) = {\tilde{d}}_0(\xi ) = \frac{1}{|\xi |}. \end{aligned}$$

For \(\rho > 0\), the actual width of the corridor is then \(d_{\rho }(\xi ) = {\tilde{d}}_{\rho }(\xi ) = \max \{0, |\xi |^{-1} - 2\rho \}\).

Remark 2.2

Let us call these two corridors in the direction \(\xi \) the \(\xi \)-corridors. They open up only when \(\rho < d_0(\xi )/2 = {\tilde{d}}_0(\xi )/2\). For \(\rho = 0\), the common boundary (called \(\xi \)-boundary) of the two \(\xi \)-corridors is the line through 0 and \(\xi \). The other boundaries are lines parallel to the \(\xi \)-boundary, going through lattice points that are called \(\xi '\) and \(\xi ''\) in the below proof. For \(\xi = (p,q)\) (with \(\gcd (p,q) = 1\)), these points \(\xi ' = (p',q')\), \(\xi '' = (p'',q'')\) are uniquely determined by \(\xi \) in the sense that \(p'/q'\) and \(p''/q''\) are convergents preceding p/q in the continued fraction expansion of p/q. In particular \(|\xi '|, |\xi ''| \le |\xi |\). In the sequel, we usually only need one of these two \(\xi \)-corridors, and we take the one with \(\xi '\) in its other boundary.


If \((p,q) = (0, \pm 1)\) or \((\pm 1, 0)\), then clearly \(d_0(\xi ) = {\tilde{d}}_0(\xi ) = 1\), so we can assume without loss of generality that \(p \ge q > 0\). Let L be the arc connecting (0, 0) to (pq). The corridors associated to \(\xi \) intersect \([0,p] \times [0,q]\) in diagonal strips on either side of L.

Let \(\frac{q}{p} = [0;a_1, \dots , a_n=a]\) be the standard continued fraction expansion with \(a \ge 1\), and the previous two convergents are denoted by \(q'/p'\) and \(q''/p''\), say \(q''/p''< q/p <q'/p'\) (the other inequality goes analogously). Therefore \(q'p-qp' = 1\) and \(q''p'-q'p'' = -1\). Also

$$\begin{aligned} \frac{(a-1)q' + q''}{(a-1)p'+p''}< \frac{q}{p} < \frac{q'}{p'} \end{aligned}$$

are the best rational approximations of q/p, belonging to lattice points \(\xi '\) above L and \(\xi ''\) below L. The vertical distance between \(\xi '\) and the arc L is \(|q'-p'\frac{q}{p}| = \frac{1}{p}|q'p-p'q| = \frac{1}{p}\). The vertical distance between L and \(\xi ''\) is

$$\begin{aligned} ((a-1)p'+p'')\frac{q}{p} - ((a-1)q'+q'')= & {} \frac{1}{p} ( (a-1)(qp'-q'p) + qp''-q''p )\\= & {} \frac{1}{p} ( 1-a + ( aq'+q'')p'' - (ap'+p'')q'' ) \\= & {} \frac{1}{p} ( 1-a + a(q'p'' - q''p')) = \frac{1}{p}. \end{aligned}$$

The corridor’s diameter is perpendicular to \(\xi \), so \(d_0(\xi )\) is computed from this vertical distance as the inner product of the vector \((0,1/p)^T\) and the vector \(\xi = (p,q)^T\) rotated over \(90^\circ \):

$$\begin{aligned} \frac{1}{\sqrt{p^2+q^2}} \left\langle \begin{pmatrix} 0 \\ 1/p \end{pmatrix} \, \ \begin{pmatrix} -q \\ p \end{pmatrix} \right\rangle = \frac{1}{\sqrt{p^2+q^2}} = \frac{1}{|\xi |}. \end{aligned}$$

The computation for \({\tilde{d}}_0(\xi ) = |\xi |^{-1}\) is the same. \(\square \)

2.2 Singularities of the billiard map

In the coordinates \((\theta ,\phi ,\ell ) \in {{\mathbb {S}}}^1 \times [-\frac{\pi }{2}, \frac{\pi }{2}] \times {{\mathbb {Z}}}^2\) (or \(\times {{\mathbb {Z}}}\) if it is a Lorentz tube), the size of the scatterers \(\rho \) doesn’t appear, but it comes back in the formula of the billiard map \(T_{\rho }\) and in its hyperbolicity. Also the curvature of the scatterers is \({{\mathcal {K}}}\equiv 1/\rho \). We recall some notation from the Chernov and Markarian book [10] (going back to the work of Sinaĭ), bearing in mind that we have to redo several of their estimates to track the precise dependence on \(\rho \). The phase space is \({\widehat{{{\mathcal {M}}}}} = {{\mathcal {M}}}\times {{\mathbb {Z}}}^2 = \bigcup _{\ell \in {{\mathbb {Z}}}^2} {{\mathcal {M}}}_\ell \), where each \({{\mathcal {M}}}_\ell \) is a copy of the cylinder \({{\mathbb {S}}}^1 \times [-\frac{\pi }{2}, \frac{\pi }{2}]\), see Fig. 2.

Fig. 2
figure 2

The parameter subset \({{\mathcal {M}}}_0\) with singularity lines and \(\kappa _\rho = \xi '-M\xi \)

Let \({{\mathcal {S}}}_0 = \{ \phi = \pm \frac{\pi }{2} \}\) be the discontinuity of the billiard map corresponding to grazing collisions. The forward and backward discontinuities are

$$\begin{aligned} {{\mathcal {S}}}_n = \cup _{i=0}^n T_{\rho }^{-i}({{\mathcal {S}}}_0) \quad \text { and } \quad {{\mathcal {S}}}_{-n} = \cup _{i=0}^n T_{\rho }^{i}({{\mathcal {S}}}_0), \end{aligned}$$

so that \(T_{\rho }^n:{{\mathcal {M}}}\setminus {{\mathcal {S}}}_n \rightarrow {{\mathcal {M}}}\setminus {{\mathcal {S}}}_{-n}\) is a diffeomorphism. We line the curve \({{\mathcal {S}}}_0\) with homogeneity strips \({{\mathbb {H}}}_k\) bounded by curves \(|\pm \frac{\pi }{2}-\phi | = k^{-r_0}\) and \(|\pm \frac{\pi }{2}-\phi | = (k+1)^{-r_0}\), \(k \ge k_0\), for a fixed number \(r_0 > 1\). 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.

Fig. 3
figure 3

A corridor collision map from \(O_{-\xi }\) and \(O_{-\kappa _\rho }\) to \(O_0\)

The set \({{\mathcal {S}}}_{-1}\) consists of multiple curves inside \({{\mathcal {M}}}_0\), one for each scatterer from which a particle can reach \(O_0\) in the next collision. In Fig. 2 we consider the corridor in the direction of \(\xi \in {{\mathbb {Z}}}^2\), and drew the parts of \({{\mathcal {S}}}_{-1}\) coming from scatterers \(O_\xi \), \(O_{-\xi }\) and \(O_{-\kappa _\rho }\) for some scatterer on the other side of this corridor.

Lemma 2.3

For the \(\xi \)-corridor, let \((\theta _{-\xi }, \frac{\pi }{2}) \in {{\mathcal {M}}}_0\) be the point of intersection of \({{\mathcal {S}}}_0\) and the part of \({{\mathcal {S}}}_{-1}\) associated to the scatterer \(O_{-\xi }\), and \((\theta _{\kappa _\rho }, \frac{\pi }{2}) \in {{\mathcal {M}}}_0\), \(\kappa _\rho = \xi '-M\xi \), be the point of intersection of \({{\mathcal {S}}}_0\) and the part of \({{\mathcal {S}}}_{-1}\) associated to the scatterer \(O_{\kappa _\rho } = O_{\xi '-M\xi }\) at the other side (i.e., the \(\xi '\)-boundary) of the \(\xi \)-corridor, see Fig. 3. Let \((\theta '_{\kappa _\rho }, \phi '_{\kappa _\rho })\) be the intersection of the parts of \({{\mathcal {S}}}_{-1}\) associated to the scatterers \(O_{-\xi }\) and the scatterer \(O_{\kappa _\rho }\), see Fig. 2. Then

$$\begin{aligned} |\theta _{-\xi } - \theta _{\kappa _\rho }| = \frac{d_\rho (\xi )}{|\xi | M} \left( 1+{{\mathcal {O}}}\left( \frac{\rho }{|\xi | M} \right) \right) \end{aligned}$$


$$\begin{aligned} \frac{\pi }{2}-\phi '_{\kappa _\rho } = \sqrt{ \frac{2 d_{\rho }(\xi )}{\rho M} } \left( 1-{{\mathcal {O}}}\left( \frac{\rho }{|\xi |} - \frac{1}{M} + \frac{\sqrt{d_{\rho }(\xi ) \rho }}{|\xi | \sqrt{M} }\right) \right) . \end{aligned}$$


The angle \(\theta _{-\xi }\) refers to the point where the common tangent line of \(O_0\) and \(O_{-\xi }\) touches \(O_0\). For the value \(\theta _{\kappa _\rho }\), \(\kappa _\rho = \xi '-M\xi \), we take the common tangent line to \(O_0\) and \(O_{\kappa _\rho }\) which has slope \(\frac{d_{\rho }(\xi )}{M |\xi |} \left( 1+{{\mathcal {O}}}(\frac{\rho }{|\xi | M} ) \right) \). This is then also \(|\theta _{-\xi }-\theta _{\kappa _\rho }|\).

Fig. 4
figure 4

Illustration of the proof of Lemma 2.3

Now for the other endpoint of this piece of \({{\mathcal {S}}}_{-1}\), consider the common tangent line to \(O_{-\xi }\) and \(O_{\kappa _\rho }\) which has slope \(\tan \alpha := \frac{d_{\rho }(\xi )}{(M-1) |\xi |} (1+{{\mathcal {O}}}(\frac{\rho }{|\xi |(M-1)}))\), hitting the scatterer \(O_0\) in point P and when extended inside \(O_0\) hits the vertical line through the center \(O_0\) in point Q. Let also R be the tangent point of \(O_0\) to the corridor, and \(Q'\) is the point on \(O_0R\) at the same horizontal height as P, see Fig. 4. Then \(|RQ| = |\xi |\sin \alpha \) whereas \(|O_0Q'| = \rho - (|\xi | - \rho \sin \theta '_{\kappa _\rho }) \sin \alpha = \rho \cos \theta '_{\kappa _\rho }\). The latter gives

$$\begin{aligned} \theta '_{\kappa _\rho } = \sqrt{ \frac{2|\xi |}{\rho } \sin \alpha \left( 1-{{\mathcal {O}}}(\frac{\rho }{|\xi |} \sin \theta )\right) } = \sqrt{ \frac{2d_\rho (\xi )}{\rho M} \left( 1-{{\mathcal {O}}}(\frac{\rho }{|\xi |} - \frac{1}{M} )\right) }. \end{aligned}$$

The triangle \(\triangle PO_0Q\) has angles \(\phi '_{\kappa _\rho }\), \(\alpha +\frac{\pi }{2}\) and \(\theta '_{\kappa _\rho }\), which add up to \(\pi \). Hence

$$\begin{aligned} \frac{\pi }{2} - \phi '_{\kappa _\rho } = \alpha +\theta '_{\kappa _\rho } = \sqrt{ \frac{2d_\rho (\xi )}{\rho M} } \left( 1-{{\mathcal {O}}}\left( \frac{\rho }{|\xi |} - \frac{1}{M} + \frac{\sqrt{d_{\rho }(\xi ) \rho }}{|\xi | \sqrt{M} }\right) \right) \end{aligned}$$

as claimed. \(\square \)

2.3 Hyperbolicity of the Lorentz gas with small scatterers

The derivative \(DT_{\rho }: {{\mathcal {T}}}{{\mathcal {M}}}\rightarrow {{\mathcal {T}}}{{\mathcal {M}}}\) preserves the unstable cone field

$$\begin{aligned} {{\mathcal {C}}}^u_x = \left\{ (d\theta , d\phi ) \in {{\mathcal {T}}}_x{{\mathcal {M}}}: 1 \le \frac{1}{2\pi } \frac{d\phi }{d\theta } \le 1 + \frac{\rho }{\tau _{\min }} \right\} . \end{aligned}$$

This is [10, p. 74] in the coordinates \(\theta = r/2\pi \rho \), and we can sharpen this cone by replacing \(\tau _{\min }\) by \(\tau (x)\), the flight time at x before the next collision. The derivative of the inverse of the billiard map preserves the stable cone field

$$\begin{aligned} {{\mathcal {C}}}^s_x = \{ (d\theta , d\phi ) \in {{\mathcal {T}}}_x{{\mathcal {M}}}: -1-\frac{\rho }{\tau _{\min }} \le \frac{1}{2\pi } \frac{d\phi }{d\theta } \le -1 \}. \end{aligned}$$

Clearly, these cone-fields are transversal uniformly over \({{\mathcal {M}}}\), and \({{\mathcal {S}}}_n\) 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 \(\Vert dx \Vert _p = \cos \phi \, dr\). When restricted to the unstable cone, the p-norm is non-degenerate. With the notation \({{\mathcal {R}}}(x) = \frac{2}{\rho \cos \phi }\), the expansion/contraction factor \(\Lambda \) on unstable vectors in the p-norm satisfies

$$\begin{aligned} \Lambda \ge 1 + \tau (x) {{\mathcal {R}}}(x) \ge 1 + \tau _{\min } {{\mathcal {R}}}_{\min } = 1 + \frac{2\tau _{\min }}{\rho }. \end{aligned}$$

This proves uniform hyperbolicity of the billiard map.

In our coordinates the p-norm can be also expressed as \(\Vert dx \Vert _p = 2\pi \, \rho \, \cos \phi \, d\theta \), and it is related to the standard Euclidean norm as

$$\begin{aligned} \Vert dx \Vert = \frac{ \sqrt{1+(\frac{d\phi }{dr})^2} }{\cos \phi } \Vert dx\Vert _p = \frac{ \sqrt{4\pi ^2 \rho ^2 +(\frac{d\phi }{d\theta })^2} }{2\pi \rho \cos \phi } \Vert dx\Vert _p. \end{aligned}$$

The expansion of \(DT_{\rho }\) of unstable vectors is uniform in the p-norm, see [10, Formula (3.40)]:

$$\begin{aligned} \frac{ \Vert DT_{\rho }(dx)\Vert _p}{\Vert dx\Vert _p} = 1+\frac{\tau (x)}{\cos \phi } ({{\mathcal {K}}}+ \frac{d\phi }{dr}) = \frac{\tau (x)}{\rho \cos \phi } \left( 1+\frac{1}{2\pi } \frac{d\phi }{d\theta } + \frac{\rho \cos \phi }{\tau (x)}\right) . \end{aligned}$$

Expressed in Euclidean norm, this gives, for \(DT_{\rho }(dx) = (d\theta _1, d\phi _1)\),

$$\begin{aligned} \frac{\Vert DT_{\rho }(dx)\Vert }{\Vert dx \Vert } = \sqrt{\frac{4\pi ^2\rho ^2 + (\frac{d\phi _1}{d\theta _1})^2}{4\pi ^2\rho ^2 + (\frac{d\phi }{d\theta })^2} } \ \frac{\tau (x)}{\rho \cos \phi _1}\ \left( 1+\frac{1}{2\pi } \frac{d\phi }{d\theta } + \frac{\rho \cos \phi }{\tau (x)} \right) . \end{aligned}$$

For later use, if \(T_{\rho }(x)\) is in the homogeneity strip \({{\mathbb {H}}}_k\), then \(\cos \phi _1 \approx k^{-r_0}\).

3 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 \(r_0=2\).

$$\begin{aligned} {\left\{ \begin{array}{ll} r_0 \ge 2&{} \text {is the exponent of the homogeneity strips:}\\ &{} \qquad {{\mathbb {H}}}_{\pm k} = \{ |\pm \frac{\pi }{2} - \varphi | \in [(k+1)^{-r_{0}}, k^{-r_{0}})\},\\ 0<\nu<\frac{1}{2}-\frac{1}{2r_{0}} &{} \text {the exponent of } \kappa _\rho \text { in the continuity estimate for the}\\ &{} \text {transfer operator,}\\ \varsigma _{0}=1-\frac{2r_{0}\nu }{r_{0-1}} &{} \text {upper bound on }\varsigma \text { in the Jensenized growth lemma, see } 25, \\ \alpha _{0}< \min \left( \frac{1}{2(r_{0+1})},\varsigma _{0}\right) &{} \text {needed for [14, Lemma 3.7] for general } r_{0},\\ s_0 = \frac{1-\alpha _{0}{(r_{0+1})}}{2r_{0}} > 0 &{} \text {used in Lemma}~6.1,\\ 0< q_{0}< p_{0}< \frac{1}{r_{0+1}} &{} \text {cf. Lemma}~{\textrm{B}}.2,\\ 0< \beta _{0} < \min \{\frac{\alpha _{0}}{2}, p_{0}-q_{0}\}. &{} \end{array}\right. }\nonumber \\ \end{aligned}$$

We use a class \({{\mathcal {W}}}^s\) 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, \(\kappa _\rho (x)\) is constant on W and there is a \(\rho \)-dependent upper bound on |W|, namely

$$\begin{aligned} \sup _{W \in {{\mathcal {W}}}^s} |W| =\delta _0:= c\rho ^{\nu }, \end{aligned}$$

where the small \(c > 0\), to be fixed below, is independent of \(\rho \).

Let \(W \in {{\mathcal {W}}}^s\) be an admissible stable leaf. The preimage \(T_{\rho }^{-1}(W)\) is cut by the discontinuity lines \({{\mathcal {S}}}_1\) 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 \(\le \delta _0\).

3.1 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 \((\xi , \xi ') \in {\Psi }\), see Fig. 3. If the previous scatterer is \(\pm \xi \) itself, we call this a trajectory from the \(\xi \)-boundary; if the previous scatterer is at lattice point \(\xi '-M\xi \), the trajectory comes in from the \(\xi '\)-boundary, see Remark 2.2. To each such scatterer and homogeneity strip \({{\mathbb {H}}}_k\) belongs at most one \(V_i\), and the contraction \(|T_{\rho }V_i|/|V_i|\) is governed by (14), where the distortion \(T_{\rho }:V_i \rightarrow T_{\rho }V_i\) is uniformly bounded, see Appendix B.

Proposition 3.1

Assume \(0 \le \nu < \frac{1}{2}-\frac{1}{2r_0}\). Then there is a constant \(C > 0\), uniform in \(\rho ,\nu \) and \(r_0\) such that

$$\begin{aligned} \sum _i |\kappa _\rho (V_i)|^{\nu } \ \frac{|T_{\rho }V_i|}{|V_i|} \le C \left( \rho + \rho ^{-\nu } \, \delta _0 \right) \end{aligned}$$

for every stable leaf \(W \in {{\mathcal {W}}}^s\).

Remark 3.2

(i) Since \(|W| \le \delta _0 \le c \rho ^{\nu }\), there is \(\theta _* < 1\) such that

$$\begin{aligned} \sum _{V_i} |\kappa _\rho (V_i)|^{\nu } \ \frac{|T_{\rho }V_i|}{|V_i|} \le 3C (\rho +c) \le \theta _*, \end{aligned}$$

for \(\rho \) sufficiently small, and c chosen appropriately small. In addition, we assume that

$$\begin{aligned} \delta _1 \in (0, \delta _0/2) \ \text { is such that } \ \theta _* e^{C_d \delta _1^{1/(r_0+1)} } =: \theta _1 < 1 \end{aligned}$$

for distortion constant \(C_d\) from Lemma B.2;

(ii) As later we will need \(\nu > \frac{1}{3}\), we can take \(r_0=5\) and \(\nu = \frac{3}{8}\).


The homogeneous admissible preimage curves \(T_{\rho }^{-1}W=\cup _i V_i\) are obtained by partitioning according to

  • incoming corridors \(\xi \);

  • for a fixed corridor \(\xi \), the scatterer on which \(V_i\) is located. Accordingly, \(\kappa _\rho (V_i)=M\xi -\xi '\) for some \(M \in {{\mathbb {N}}}\), and the summation is over M;

  • for a fixed scatterer, the homogeneity strip containing \(V_i\), that is, \(V_i\subset {{\mathbb {H}}}_k\) for some k.

If W is on the scatterer \(O_{0}\) and \(V_i\) is on the scatterer \(O_{\xi '-M\xi }\), then both of these scatterers are tangent to the same corridor. The trajectory makes and angle \(\sim \frac{d_{\rho }(\xi )}{M |\xi |}\) 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 Fig. 2. In particular

$$\begin{aligned} k \ge C(\rho d_{\rho }(\xi )^{-1} M)^{\frac{1}{2r_0}} \end{aligned}$$

which determines the range of k for M fixed.

We sum over the homogeneity strips for \(\xi \) and M fixed on the \(\xi '\) boundary.

$$\begin{aligned} \sum _{V_i \in {{\mathcal {M}}}_{\xi '-M\xi }} |\kappa _\rho (V_i)|^{\nu } \ \frac{|T_{\rho }V_i|}{|V_i|}&\ll \frac{\rho |\xi |^{\nu } M^\nu }{|\xi | M} \sum _{k \ge (\max \{ C(\frac{\rho M}{d_{\rho }(\xi )} , 1\} )^{\frac{1}{2r_0} }} \frac{1}{k^{r_0}} \\&\ll \rho ^{\frac{1}{2r_0}+\frac{1}{2}} |\xi |^{\nu -1} d_{\rho }(\xi )^{\frac{1}{2}-\frac{1}{2r_0}} M^{\nu -\frac{3}{2}+\frac{1}{2r_0} } \\&\ll \rho ^{\frac{1}{2r_0}+\frac{1}{2}} |\xi |^{\nu -\frac{3}{2}+\frac{1}{2r_0}} M^{\nu -\frac{3}{2}+\frac{1}{2r_0} }, \end{aligned}$$

where we used that the exponent \(\frac{1}{2}-\frac{1}{2r_0}\) of \(d_{\rho }(\xi )\) is non-negative. By our assumption that \(\nu <\frac{1}{2}-\frac{1}{2r_0}\), this expression is summable over M, and therefore the sum over the \(\xi '\)-boundary of the entire \(\xi \)-corridor is

$$\begin{aligned} \sum _{\text {corridor }\xi } |\kappa _\rho (V_i)|^{\nu } \ \frac{|T_{\rho }V_i|}{|V_i|}\ll & {} \rho ^{\frac{1}{2}+\frac{1}{2r_0} } |\xi |^{\nu -\frac{3}{2}+\frac{1}{2r_0} }. \end{aligned}$$

The sum over homogeneity strips for \(\xi \) fixed on the \(\xi \)-boundary is no different:

$$\begin{aligned} \sum _{V_i \in {{\mathcal {M}}}_{-\xi }} |\kappa _\rho (V_i)|^{\nu } \ \frac{|T_{\rho }V_i|}{|V_i|}&\ll \frac{\rho |\xi |^{\nu } }{ |\xi | } \sum _{k \ge 1} \frac{1}{k^{r_0}} \ll \rho |\xi |^{\nu -1}. \end{aligned}$$

Next we sum over all opened-up corridors, indexed by all the “visible” lattice points inside a sector of angle \(|W|/\sqrt{1+4\pi ^2}\), 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 \({\Psi }_W\). 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 \(|\xi | \ge 1\) (which is the worst case because the contraction of \(T_{\rho }\) 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 \(\delta _0\), and apply Lemma A.6 for \(a = 1-\nu \) and \(a = \frac{3}{2} - \nu - \frac{1}{2r_0}\). This gives

$$\begin{aligned} \sum _{V_i} |\kappa _\rho (V_i)|^{\nu } \ \frac{|T_{\rho }V_i|}{|V_i|}&\ll \rho + \sum _{(\xi ,\xi ')\in {\Psi }_W} \rho |\xi |^{\nu -1} + \rho ^{\frac{1}{2}+\frac{1}{2r_0}} |\xi |^{\nu -\frac{3}{2}+\frac{1}{2r_0}} \\&\ll \rho + \rho ^{-\nu } \delta _0 + \rho ^{1-\nu } \log (1/\rho ) + \rho ^{1-\nu } \delta _0^{-1} \\&\qquad + \rho ^{-\nu } \delta _0 + \rho ^{1-\nu } \log (1/\rho ) + \rho ^{2-\nu } \delta _0^{-1} \\&\ll \rho + \rho ^{-\nu } \delta _0 + \rho ^{1-\nu } \log (1/\rho ) + \rho ^{1-\nu } \delta _0^{-1}. \end{aligned}$$

Since \(\delta _0 = c\rho ^{\nu }\) and \(\nu < \frac{1}{2}\), this completes the proof. \(\square \)

3.2 The growth lemma in terms of \(W_i\)

The pieces of preimage leaf \(V_i \subset T_{\rho }^{-1}(W)\) emerge by natural cutting at the discontinuity set \({{\mathcal {S}}}_1\) and the homogeneity strips, but even so, their lengths can be larger than \(\delta _0\), 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 \(\delta _0^{-1}\) pieces, which gives the estimate

$$\begin{aligned} \sum _i |\kappa _\rho (W_i)|^{\nu } \ \frac{|T_{\rho }W_i|}{|W_i|} \le C \left( \rho \delta _0^{-1} + \rho ^{-\nu } \right) \ll \rho ^{-\nu }. \end{aligned}$$

Although this estimate suffices for some purposes, it is not always good enough for larger iterates \(T_{\rho }^n\). The next lemma (which follows [14, Lemma 3.2] or [16, Lemma 3.3]) achieves an estimate, uniform in n, for \(\nu =0\).

For the next lemma we recall some notation used in [16]. For \(W \in {{\mathcal {W}}}^s\), we construct the components \({{\mathcal {G}}}_k(W)\) of \(T_{\rho }^{-k}W\) inductively on \(k=0,\dots ,n\). That is \({{\mathcal {G}}}_0(W) = \{ W \}\), and to obtain \({{\mathcal {G}}}_{k+1}(W)\) first we apply Proposition 3.1 to each curve in \({{\mathcal {G}}}_k(W)\), and then we partition curves that are longer then \(\delta _0\) into pieces of length between \(\delta _0\) and \(\delta _0/2\). We enumerate the leaves of the k-th generation \({{\mathcal {G}}}_k(W)\) as \(W_i^k\).

Lemma 3.3

There is a constant \(C_s>0\), independent of \(\rho \), such that

$$\begin{aligned} \sum _{W_i^n\in {{\mathcal {G}}}_n(W)} \frac{|T_{\rho }^nW_i^n|}{|W_i^n|}\le C_s, \end{aligned}$$


$$\begin{aligned} \sum _{W_i^n\in {{\mathcal {G}}}_n(W)} \frac{|W_i^n|^\varsigma }{|W|^\varsigma } \, \frac{|T_{\rho }^nW_i^n|}{|W_i^n|}\le C_s^{1-\varsigma }, \end{aligned}$$

for all \(\varsigma \in [0,1)\).


Define \({{\mathcal {L}}}_k\) as the collection of indices such that \(W_i^k \in {{\mathcal {G}}}_k(W)\) is long, i.e., \(|W_i^k| \ge \delta _1\) for \(i \in {{\mathcal {L}}}_k\), and \({{\mathcal {I}}}_n(W_j^k)\) as the collection indices of \(W_i^n\) such that their most recent long ancestor is \(W_j^k \in {{\mathcal {G}}}_k(W)\). If for some \(W^n_{i_1}\) no such long ancestor exists, then set \(k(i_1)=0\) and \(W_{i_1}^n\) belongs to \({{\mathcal {I}}}_n(W)\); if \(W^n_{i_2}\) is itself long, then set \(k(i_2)=n\). Fix some \(j \in {{\mathcal {L}}}_k\). As for \(W^i_n\in {{\mathcal {I}}}_n(W_j^k)\) the preimages under \(T_{\rho }^{n-k}\) of \(T_{\rho }^{n-k}W^i_n\) need not be cut artificially (they are already short), and due to the distortion bound from Lemma B.2,

$$\begin{aligned} \sum _{i \in {{\mathcal {I}}}_n(W^k_j)} \frac{|T_{\rho }^{n-k}W_i^n|}{|W_i^n|}\le \theta _1^{n-k}, \quad \text { for } \theta _1 = \theta _*e^{C_d |\delta _1|^{\frac{1}{r_0+1}}}. \end{aligned}$$

Recall that by our assumption \(\delta _1\) is so small that \(\theta _1 < 1\). In the estimate below, we group \(W_i^n\in {{\mathcal {G}}}_n(W)\) according to their most recent long ancestors.

$$\begin{aligned} \sum _i \frac{|T_{\rho }^nW_i^n|}{|W^n_i|}= & {} \sum _{k=1}^n \sum _{W_j^k \in {{\mathcal {L}}}_k(W)} \sum _{i \in {{\mathcal {I}}}_n(W_j^k)} \frac{|T_{\rho }^nW_i^n|}{|W^n_i|} + \sum _{i \in {{\mathcal {I}}}_n(W)} \frac{|T_{\rho }^nW_i^n|}{|W^n_i|} \nonumber \\\le & {} \sum _{k=1}^n \sum _{W_j^k \in {{\mathcal {L}}}_k(W)} \left( \sum _{i \in {{\mathcal {I}}}_n(W_j^k)} \frac{|T_{\rho }^{n-k}W_i^n|}{|W^n_i|} \right) e^{\delta _1^{1/r_0+1} C_d} \frac{|T_{\rho }^kW_j^k|}{|W^k_j|} + \theta _1^n \nonumber \\\le & {} \sum _{k=1}^n \sum _{W_j^k \in {{\mathcal {L}}}_k(W)} \theta _1^{n-k} \delta _1^{-1} |T_{\rho }^kW_j^k| +\theta _1^n \nonumber \\\le & {} C\delta _1^{-1}|W| \sum _{k=1}^n \theta _1^{n-k} +\theta _1^n\le C_s, \end{aligned}$$

where we have used that for fixed k and \({W_j^k \in {{\mathcal {L}}}_k(W)}\), (i) \(|W_j^k|\ge \delta _1\), (ii) the \(T_{\rho }^kW_j^k\) are pairwise disjoint subcurves of W, and (iii) \(|W|\le \delta _1\). By Jensen’s inequality and (23),

$$\begin{aligned} \sum _i \frac{|W_i^n|^\varsigma }{|W|^{\varsigma }} \frac{|T_{\rho }^nW_i^n|}{|W^n_i|}= & {} \sum _i \left( \frac{|W|}{|W_i^n|} \right) ^{1-vs} \frac{|T_{\rho }^nW_i^n|}{|W|} \le \left( \sum _i \frac{|T_{\rho }^nW_i^n|}{|W^n_i|} \right) ^{1-\varsigma } \ll C_s^{1-\varsigma }, \end{aligned}$$

which proves the second statement. \(\square \)

It is worth including the following bound, which follows from (22) by Jensen inequality:

$$\begin{aligned} \sum _{i \in {{\mathcal {I}}}_n(W)} \frac{|W_i^n|^\varsigma }{|W|^{\varsigma }} \frac{|T_{\rho }^n W_i^n|}{|W_i^n|}\le \theta _1^{(1-\varsigma )n}, \qquad \text { for all } \varsigma \in [0,1). \end{aligned}$$

Remark 3.4

For further reference, we state a version of (21) for \(\nu >0\), \(n=1\). Let \(\varsigma _0=1-\frac{2r_0\nu }{r_0-1}\).

$$\begin{aligned} \sum _i |\kappa _\rho (W_i)|^{\nu } \ \frac{|T_{\rho }W_i|}{|W_i|} \frac{|W_i|^\varsigma }{|W|^{\varsigma }} \ll \rho ^{-\nu }, \qquad \text { for all } \varsigma \in [0,\varsigma _0). \end{aligned}$$

This follows by Jensen’s inequality from (19), applied with \(\frac{\nu }{1-\varsigma }\) in place of \(\nu \). The condition \(\varsigma <\varsigma _0\) ensures that \(\frac{\nu }{1-\varsigma }<\frac{1}{2}-\frac{1}{2r_0}\). For the choices \(r_0=5\), \(\nu =\frac{3}{8}\) we have \(\varsigma _0=\frac{1}{16}\).

4 Banach spaces and spectral gap

For the exponents \(p_0\) and \(q_0\) defined in (15) we define the Banach spaces (of distributions) \(C^{p_0}, {{\mathcal {B}}}, {{\mathcal {B}}}_w, (C^{q_0})'\) in analogy to [16],Footnote 3 We recall that \((C^{q_0})'\) is the topological dual of \(C^{q_0}\).

Given \(W\in {{\mathcal {W}}}^s\), let \(m_W\) be the Lebesgue measure on W, and define

$$\begin{aligned} |\psi |_{W,\alpha ,p_0}:= |W|^{\alpha } \cos W\, |\psi |_{C^{p_0}}, \qquad |\psi |_{C^{p_0}}:= |\psi |_{C^0} + H^{p_0}_W(\psi ),s \end{aligned}$$

for \(\alpha \ge 0\), \(\cos W = |W|^{-1} \int _W \cos \phi \, dm_W\) (note that \(\cos W \ll k^{-r_0}\) if \(W \subset {{\mathbb {H}}}_{\pm k}\)), and \(H^{p_0}_W(\psi )\) the Hölder constant of \(\psi \) along W. Also let \(d_W(W_1, W_2)\) stand for the distance between leaves as in [14, Sect. 3.1] or [16, Sect. 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 \((\theta ,\phi )\) coordinates, and otherwise infinite.

Given \(W\in {{\mathcal {W}}}^s\) and \(h\in C^1(W)\), define the weak norm.Footnote 4

$$\begin{aligned} \Vert h\Vert _{{{\mathcal {B}}}_w}:=\sup _{W\in {{\mathcal {W}}}^s}\;\sup _{{\mathop { |\psi |_{W,0,p_0}\le 1}\limits ^{|\psi | \in C^{p_0}(W) }} } \int _W h\psi \, dm_W. \end{aligned}$$

With \(q_0 < p_0\) fixed we define the distance between functions \(d(\psi _1,\psi _2)\) in the same way as in [14, Sect. 3.1]. We define the strong stable norm by

$$\begin{aligned} \Vert h\Vert _s:=\sup _{W\in {{\mathcal {W}}}^s}\;\sup _{{\mathop {|\psi |_{W,\alpha _0,q_0}\le 1}\limits ^{\psi \in C^{q_0}(W)}}} \int _W h\psi \, dm_W. \end{aligned}$$

Choosing \(\varepsilon _0 \in (0,\delta _0)\) and \(\beta _0 \in (0, \min \{ \alpha _0, p_0-q_0\})\), we define the strong unstable norm by

$$\begin{aligned} \Vert h\Vert _u:=\sup _{\varepsilon \le \varepsilon _0} \sup _{{\mathop { d(W_1, W_2)\le \varepsilon }\limits ^{W_1, W_2\in {{\mathcal {W}}}^s}}} \sup _{\begin{array}{c} {1}{\psi _i \in C^{p_0}(W),\ }\\ { |\psi _i|_{C^1(W)} \le 1 } \\ { d_{q_0}(\psi _1,\psi _2) \le \varepsilon } \end{array}} \frac{1}{\varepsilon ^{\beta _0}}\left| \int _{W_1} h\psi _1\, dm_{W} - \int _{W_2} h\psi _2\, dm_{W}\right| . \end{aligned}$$

The strong norm is defined by \(\Vert h\Vert _{{{\mathcal {B}}}}=\Vert h\Vert _s+c_u\Vert h\Vert _u\), where we will fix \(c_u\ll 1\) (but independent of \(\rho \)) at the beginning of Sect. 5.2.

Since \(C^{p_0}\subset {{\mathcal {B}}}\subset {{\mathcal {B}}}_w\subset (C^{q_0})'\) (see Sect. 4.1), we have \(\Vert h\Vert _{{{\mathcal {B}}}_w}+\Vert h\Vert _{{{\mathcal {B}}}}\le C\Vert h\Vert _{C^1}\). As in [16], we define \({{\mathcal {B}}}\) to be the completion of \(C^1\) in the strong norm and \({{\mathcal {B}}}_w\) to be the completion in the weak norm.

4.1 Transfer operator on \({{\mathcal {B}}}\)

Throughout we let \(R_\rho : L^1(m)\rightarrow L^1(m)\) be the transfer operator of the billiard map \(T_\rho \). We recall that [14, Lemmas 3.7\(-\)3.10] ensure that: i) \(R_\rho (C^1)\subset {{\mathcal {B}}}\) and as a consequence R is well defined on \({{\mathcal {B}}}\); \({{\mathcal {B}}}_w\); ii) the unit ball of \({{\mathcal {B}}}\) is compactly embedded in \({{\mathcal {B}}}_w\), and iii) \(C^{p_0}\subset {{\mathcal {B}}}\subset {{\mathcal {B}}}_w\subset (C^{q_0})'\).

It follows that \(R_\rho \) is well defined on \({{\mathcal {B}}}\) and \({{\mathcal {B}}}_w\), and we also let \(R_\rho \) denote the extension of this transfer operator to \({{\mathcal {B}}}_w\).

4.2 Lasota–Yorke inequalities

Using Proposition 3.1 with \(\nu =0\) 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 \(\rho \). It is important to point out that our all estimates in Sect. 3 and Appendix B are independent of \(\rho \), except that \(\delta _1<\delta _0\ll \rho ^{\nu }\).

Lemma 4.1

(Weak norm) There exists a uniform constant \(C>0\) so that for all \(h\in {{\mathcal {B}}}\) and for all \(n\ge 0\),

$$\begin{aligned} \Vert R_\rho ^n h\Vert _{{{\mathcal {B}}}_w}\le C\cdot C_s\,\Vert h\Vert _{{{\mathcal {B}}}_w}, \end{aligned}$$

where \(C_s\) is given by (20).


For \(W\in {{\mathcal {W}}}^s\), \(h\in C^1({{\mathcal {M}}}_0)\), \(\psi \in C^{p_0}(W)\) with \(|\psi |_{W,\alpha _0,p_0}\le 1\),

$$\begin{aligned} \int _W R_\rho ^n h \psi \, dm_W=\sum _{W_i^n\in {{\mathcal {G}}}_n(W)}\int _{W_i^n} h \frac{J_{W_i^n} T_\rho ^n}{|DT_{\rho }^n|}\psi \circ T_\rho ^n\, dm_W. \end{aligned}$$

Using the present definition of the weak norm,

$$\begin{aligned} \int _W R_\rho ^n h \psi \, dm_W\le \sum _{W_i^n\in {{\mathcal {G}}}_n(W)}\int _{W_i^n} \Vert h\Vert _{{{\mathcal {B}}}_w} \frac{|J_{W_i} T_\rho |_{C^{p_0}(W_i)}}{|DT_\rho |}|\psi \circ T_\rho |_{C^{p_0}(W_i)}\cos (W_i^n)\, dm_W. \end{aligned}$$

From here on the argument goes almost word for word as the argument in [16, Sect. 4.1], except for the use of equation (20) (the analogue of [16, Lemma 3.3(a)] with \(\varsigma =0\)). \(\square \)

Lemma 4.2

(Strong stable norm) Take \(\delta _1\) as in (17) and \(\theta _1\) as in (22). There exists a uniform constant \(C>0\) so that for all \(h\in {{\mathcal {B}}}\) and all \(n\ge 0\),

$$\begin{aligned} \Vert R_\rho ^n h\Vert _{s}\le C \left( \theta _1^{(1-\alpha _0)n}+C_s^{1-\alpha _0}\Lambda ^{-q_0n}\right) \Vert h\Vert _{s}+ C\delta _1^{-\alpha _0}\Vert h\Vert _{{{\mathcal {B}}}_w}. \end{aligned}$$

Remark 4.3

The compact term \(C\delta _1^{-\alpha _0}\Vert h\Vert _{{{\mathcal {B}}}_w}\) in Lemma 4.2 is the only point in the Lasota–Yorke inequalities where a \(\rho \)-dependence arises, via \(\delta _1 = c \rho ^{\nu }\).


The argument goes almost word for word as the [16, Argument in Sect. 4.2], except for the differences:

i) We use of equation (21) with \(\varsigma =\alpha _0\) instead of [16, Lemma 3.3 (b)] (also with \(\varsigma =\alpha _0\)) in [16, Equation (4.5)]. In particular, using the present definition of the stable norm, with the same notation as in [16, Sect. 4.2], we have the following analogue of [16, Equation (4.5)]:

$$\begin{aligned} \sum _{W_i^n\in {{\mathcal {G}}}_n(W)}&\int _{W_i^n} h \frac{J_{W_i^n} T_\rho ^n}{|DT_\rho ^n|}\left( \psi \circ T_\rho ^n-{\bar{\psi }}_i\right) \, dm_W\\&\ll \Lambda ^{-q_0n} \Vert h\Vert _{s}\sum _{W_i^n\in {{\mathcal {G}}}_n(W)} \frac{|W_i^n|^{\alpha _0}}{|W|^{\alpha _0}} \, \frac{|T_\rho ^nW_i^n|}{|W_i^n|} \, \ll \Lambda ^{-q_0n} \Vert h\Vert _{s}, \end{aligned}$$

where we have used the distortion bounds of Appendix B and Formula (21) (with \(\varsigma =\alpha _0\)).

ii) To obtain the analogue of [16, Equation (4.6)], as in [16, Sect. 4.2], we split the sum

$$\begin{aligned} \sum _{k=0}^n\sum _ {j\in L_k}\sum _{i\in {{\mathcal {I}}}_n(W_j^k)} |W|^{-\alpha _0} (\cos W)^{-1} \int _{W_i^n} h \frac{J_{W_i^n}T_{\rho }^n}{|DT_{\rho }^n|}\, dm_W \end{aligned}$$

into a term for \(k=0\) and further terms for \(k=1,\dots , n\). For \(k=0\), we use the strong stable norm and (24) (the analogue of [16, Lemma 3.3(a)]) with \(\varsigma =\alpha _0\), giving a contribution \(\ll \Vert h\Vert _{s} \theta _1^{n(1-\alpha _0)}\). For the terms \(k=1,\dots n\), we use the weak norm, (21) (the analogue of [16, Lemma 3.3(b)]) with \(\varsigma =\alpha _0\), and the fact that \(|W^k_j|\ge \delta _1\) for \(j\in {{\mathcal {L}}}_k(W)\), resulting in a contribution of \(O(\Vert h\Vert _{{{\mathcal {B}}}_w} \delta _1^{-\alpha _0})\). \(\square \)

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 Sect. 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 \(\rho \), so no \(\rho \)-dependence arises here.

Lemma 4.4

(Strong unstable norm) There exists a uniform constant \(C>0\) so that for all \(h\in {{\mathcal {B}}}\) and for all \(n\ge 0\),

$$\begin{aligned} \Vert R_\rho ^n h\Vert _{u}\le C\cdot C_s \cdot \Lambda ^{-\beta _0 n}\Vert h\Vert _{u}+ C\cdot C_s \cdot n\Vert h\Vert _{s}. \end{aligned}$$


Given \(W_1,W_2 \in {{\mathcal {W}}}^s\) with \(d(W_1,W_2)\le \varepsilon \), we may identify matched and unmatched pieces in \(T_\rho ^{-n}W_{\ell }\), \(\ell =1,2\). 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 (b)]). As the length estimates give \(\varepsilon ^{\alpha _0/2}\), \(\beta _0<\alpha _0/2\) is essential here (cf. [16, Formulas (4.10) and (4.11)], noting that \(\gamma =0\) in our case).

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)]. \(\beta _0<p_0-q_0\) ensures that after division by \(\varepsilon ^{\beta _0}\) the proof of Lemma 4.4 can be completed. \(\square \)

5 Perturbed transfer operators

A standard way of obtaining limit theorems for dynamical systems is via the perturbed transfer operator method. In Sect. 7 we will use the spectral properties of the family of perturbed transfer operators \({\hat{R}}_\rho (t), t\in {{\mathbb {R}}}\) with \({\hat{R}}_\rho (t) h =R(e^{it\kappa _\rho } h)\), \(h\in L^1(m)\).

5.1 Continuity properties

By definition, \({\hat{R}}_\rho (0)=R_\rho \). Take \(0 \le \nu < \frac{1}{2} - \frac{1}{2r_0}\) as in Proposition 3.1. In this subsection we show the following continuity estimate:

$$\begin{aligned} \Vert ({\hat{R}}_\rho (t)-{\hat{R}}_\rho (0))h\Vert _{{{\mathcal {B}}}} \le C \rho ^{-\nu } |t|^\nu \Vert h\Vert _{{{\mathcal {B}}}} \end{aligned}$$

for some uniform constant C.

The argument goes parallel to Sect. 4.2, except that this time we need the estimates (i) for \(\nu >0\) 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 \(h\in {{\mathcal {B}}}\),

$$\begin{aligned} \Vert R_\rho (e^{it\kappa _\rho }-1) h)\Vert _{{{\mathcal {B}}}_w} \le C \rho ^{-\nu } |t|^{\nu } \Vert h\Vert _{{{\mathcal {B}}}_w}. \end{aligned}$$


The argument goes similarly to the argument in [16, Sect. 4.1] restricted to the case \(n=1\). More precisely, for \(W\in {{\mathcal {W}}}^s\), \(h\in C^1({{\mathcal {M}}}_0)\), \(\psi \in C^{p_0}(W)\) with \(|\psi |_{W,\alpha _0,p_0}\le 1\),

$$\begin{aligned} \int _W R_\rho (e^{it\kappa _\rho }-1) h \psi \, dm_W=\sum _{i\in {{\mathcal {G}}}_1(W)}\int _{W_i} (e^{it\kappa _\rho }-1) h \frac{J_{W_i} T_\rho }{|DT_{\rho }|}\psi \circ T_\rho \, dm_W. \end{aligned}$$

Using the definition of the weak norm and the inequality \(|e^{ix}-1|\le x^\nu \),

$$\begin{aligned} \int _W R_\rho (e^{it\kappa _\rho }-1) h \psi \, dm_W&\le \ |t|^{\nu }\sum _{i\in {{\mathcal {G}}}_1(W)}\int _{W_i}\Vert h\Vert _{{{\mathcal {B}}}_w} |\kappa _\rho (W_i)|^\nu \\&\quad \times \frac{|J_{W_i} T_\rho |_{C^{p_0}(W_i)}}{|DT_\rho |}|\psi \circ T_\rho |_{C^{p_0}(W_i)}\cos (W_i)\, dm_W. \end{aligned}$$

From here on the proof goes the same as the argument in [16, Sect. 4.1] except for the use of equation (19) instead of [16, Lemma 3.3 (b)]. \(\square \)

Lemma 5.2

There exists a uniform constant \(C>0\) so that for all \(h\in {{\mathcal {B}}}\) and for all \(n\ge 0\),

$$\begin{aligned} \Vert R_\rho (e^{it\kappa _\rho }-1) h)\Vert _{s}\le C|t|^{\nu } \rho ^{-\nu } \Vert h\Vert _{s}. \end{aligned}$$


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 \(\varsigma =\alpha _0\) instead of [16, Lemma 3.3 (b)]. \(\square \)

Lemma 5.3

There exists a uniform constant \(C>0\) so that for all \(h\in {{\mathcal {B}}}\),

$$\begin{aligned} \Vert R_\rho (e^{it\kappa _\rho }-1) h)\Vert _{u}\le C|t|^{\nu }\left( \rho ^{-\nu } \cdot \Vert h\Vert _{u}+\rho ^{-\nu } \cdot \Vert h\Vert _{s}\right) . \end{aligned}$$


As with the proof of Lemma 4.4, the argument goes similar to [16, Argument in Sect. 4.3], restricted to the case \(n=1\). The matched and unmatched pieces can be again identified, this time for \(T_{\rho }^{-1}W_{\ell }\), \(\ell =1,2\). Then, as in the proof of Lemma 5.1, the factors \(|t|^{\nu }\) and \(|\kappa _\rho |^{\nu }\) arise. Clearly \(\kappa _\rho \) 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 \(|\kappa _\rho |^{\nu }\), throughout the argument (19) is used instead of (20). \(\square \)

Equation (29) follows from the definition of the norm in \({{\mathcal {B}}}\) together with Lemmas 5.15.2 and 5.3.

5.2 Peripheral spectrum and spectral gap

Choose \(1>\sigma >\max \{\Lambda ^{-\beta _0}, \theta _1^{(1-\alpha _0)}, \Lambda ^{-q_0}\}\). By Lemmas 4.14.2 and 4.4 and arguing as in [16, Eq. (2.14)], we obtain the traditional Lasota–Yorke inequality for some \(N\ge 1\), provided \(c_u\) in the definition of \(\Vert \ \Vert _{{{\mathcal {B}}}}\) (below (28)) is chosen small enough in terms of N. That is,

$$\begin{aligned} \Vert R_\rho ^N h\Vert _{{{\mathcal {B}}}}\le \sigma ^N\Vert h\Vert _{{{\mathcal {B}}}}+C \delta _1^{-\alpha _0} \Vert h\Vert _{{{\mathcal {B}}}_w}. \end{aligned}$$

Combined with the properties collected in Sect. 4.1 (that is, the relative compactness of the unit ball of \({{\mathcal {B}}}\) in \({{\mathcal {B}}}_w\)), equation (30) shows that the essential spectral radius of \(R_\rho \) is bounded by \(\sigma \) and that the spectral radius is 1.

Let \(\Pi _\rho \) be the eigenprojection (that is, the projection on the eigenspace of \(R_\rho \)) corresponding to the eigenvalue 1. In particular, \(\Pi _{\rho } \mu =\mu \) is the invariant measure for \(T_{\rho }\). Since for every \(\rho \), \(T_\rho \) is mixing, the peripheral spectrum of \(R_\rho \) consists of just the simple eigenvalue at 1. Thus, for every \(\rho >0\), the eigenprojection \(\Pi _{\rho }\) corresponding to the eigenvalue 1 of \(R_{\rho }\) can be also characterized by

$$\begin{aligned} \Pi _{\rho }h=\lim _{m\rightarrow \infty }R_{\rho }^m h, \end{aligned}$$

for all \(h\in {{\mathcal {B}}}\).

Let \(Q_{\rho }\) be complementary spectral projection. From here onwards, we exploit that for every \(\rho >0\), there exist \(\gamma _{\rho }\in (0,1)\) and \(C_\rho >0\) so that

$$\begin{aligned} \Vert Q_{\rho }^m\Vert _{{{\mathcal {B}}}}\le C_\rho (1-\gamma _{\rho })^m \end{aligned}$$

for every \(m\ge 1\). Altogether, \(R_{\rho }^m=\Pi _{\rho } +Q_{\rho }^m\), where \(Q_{\rho }\) satisfies (32).

6 Asymptotics of the dominant eigenvalue

To establish limit theorems (such as Theorem 7.1 below) we study the asymptotics of \( {{\mathbb {E}}}_\mu (e^{it\kappa _{m,\rho } } 1)={{\mathbb {E}}}_\mu ({\hat{R}}_{\rho }(t)^m 1), \) as \(t\rightarrow 0\) and \(m\rightarrow \infty \) via the properties of \({\hat{R}}_{\rho }(t) h= R_{\rho }(e^{it\kappa _\rho } h)\), \(h\in {{\mathcal {B}}}\).

We already know that for every \(\rho \in (\frac{1}{3},\frac{1}{2})\), 1 is a simple eigenvalue of \({\hat{R}}_{\rho }(0)=R_{\rho }\) when viewed as an operator from \({{\mathcal {B}}}\) to \({{\mathcal {B}}}\). Due to (29), \({\hat{R}}_{\rho }(t)\) is \(C^\nu \) (in t) from \({{\mathcal {B}}}\) to \({{\mathcal {B}}}\). It follows that for t in a neighbourhood of 0, \({\hat{R}}_{\rho }(t)\) has a dominant eigenvalue \(\lambda _{\rho }(t)\) (with \(\lambda _{\rho }(0)=1\)).

Let \(\gamma _{\rho }\) be as in equation (32). The continuity properties together with (32) ensure that for any \(\delta \in (0,\gamma _{\rho })\) and \(t\in B_\delta (0)\),

$$\begin{aligned} {\hat{R}}_{\rho }(t)^m=\lambda _{\rho }(t)^m\Pi _{\rho }(t)+Q_{\rho }(t)^m,\quad \Vert Q_{\rho }(t)^m\Vert _{{{\mathcal {B}}}}\le C_\rho (1-\gamma _{\rho })^m, \end{aligned}$$

for some \(C_\rho >0\) and \(\Pi _{\rho }(t)^2=\Pi _{\rho }(t)\), \(\Pi _{\rho }(t)Q_{\rho }(t)=Q_{\rho }(t)\Pi _{\rho }(t)=0\). Further, for all \(t\in B_\delta (0)\),

$$\begin{aligned} \Pi _{\rho }(t)=\int _{|u-1|=\delta }(u-{\hat{R}}_{\rho }(t))^{-1}\, du, \end{aligned}$$

for all t small enough. A standard consequence of (29) and (32) is that for every \(\delta \in (0,\gamma _{\rho })\) and for all u so that \(|u-1|=\delta \),

$$\begin{aligned} \Vert (u-{\hat{R}}_{\rho }(t))^{-1}-(u-{\hat{R}}_{\rho }(0))^{-1}\Vert _{{{\mathcal {B}}}}&\le C \rho ^{-\nu } |t|^{\nu }\Vert (u-{\hat{R}}_{\rho }(t))^{-1}\Vert _{{{\mathcal {B}}}}\Vert (u-{\hat{R}}_{\rho }(0))^{-1}\Vert _{{{\mathcal {B}}}} \nonumber \\&\le C \rho ^{-\nu }\gamma _{\rho }^{-2}|t|^{\nu }. \end{aligned}$$

Hence, \(\Vert \Pi _{\rho }(t)-\Pi _{\rho }(0)\Vert _{{{\mathcal {B}}}}\le C\rho ^{-\nu } |t|^{\nu }\rho ^{-\nu }\gamma _{\rho }^{-2}|t|^{\nu }\).

The rest of this section is allocated to the study the asymptotics of \(\lambda _{\rho }(t)\) as \(t\rightarrow 0\).

The following property was used in [6, 7, 21] (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 \(s_0 = \frac{1-\alpha _0(r_0+1)}{2r_0}\) as in (15). Let \(h\in {{\mathcal {B}}}\) and \(v\in C^{p_0}\). For every corridor with boundaries determined by \(O_\xi \) and \(O_{\xi '}\), there exists a constant \(C > 0\) independent of \(\rho \) and \(\xi \) so that

$$\begin{aligned} \left| \int h v 1_{\{ \kappa _\rho = \xi '+ N\xi \}} \, dm \right| \le C \Vert h \Vert _s |v|_{C^{q_0}} d_{\rho }(\xi )^{\frac{3}{2}-s_0} |\xi |^{-1} \rho ^{-\frac{1}{2}+s_0} N^{-\frac{5}{2}+s_0}. \end{aligned}$$


Let \(\{ W_\ell \}_{\ell \in L}\) be the foliation of the set \(\{\kappa _\rho = \xi '+\xi N\}\) into stable leaves. We can parametrise these leaves by their endpoints \((\ell , \frac{\pi }{2})\) in \({{\mathcal {S}}}_0\), then L is an interval of length \(c \ll \frac{d_{\rho }(\xi )}{N^2 |\xi |}\) according to Lemma 2.3. The lengths of these stable leaves \(|W_\ell | \le c'\) for another constant \(c' \ll \sqrt{\frac{2 d_{\rho }(\xi )}{\rho N}}\), again by Lemma 2.3. The measure \(dm_{W_\ell }\) is Lebesgue on the \(C^1\) stable leaf \(W_\ell \), and it can be parametrised as \((w_\ell (\phi ), \phi )\) where w is \(C^1\) with \(-\frac{1}{2\pi } \frac{\rho + \tau _{\min }}{\tau _{\min }}< w'(\phi ) <- \frac{1}{2\pi }\) because of the direction of the stable cones, see (13).

Let \(\nu \) be a measure on L that produces the decomposition of Lebesgue measure m on \(\{ \kappa _\rho = \xi '+\xi N\}\) along stable leaves. We have \(\nu \ll m_L\) (and \(d\nu /dm_L\) is bounded above). Since we need to partition stable leaves \(W_\ell \) by the homogeneity strips \({{\mathbb {H}}}_k\) near \({{\mathcal {S}}}_0\) into pieces \(W_{\ell ,k}:= W_\ell \cap {{\mathbb {H}}}_k\), we get an extra sum over \(k \ge k(c'):= \lfloor (c')^{-1/r_0} \rfloor \). Then

$$\begin{aligned} \left| \int h v 1_{\{ \kappa _\rho = N\xi +\xi ' \}} \, dm\right|= & {} \left| \int _L \sum _{k \ge k(c')} \int _{W_{\ell ,k}} h\, v \, dm_{W_\ell } \ d\nu (\ell ) \right| \\\ll & {} \left| \int _L |v|_{C^{q_0}} \sum _{k \ge k(c')} \int _{W_{\ell ,k}} h\, \frac{v}{|v|_{C^{q_0}}} \, dm_{W_\ell } \ d\ell \right| \\\le & {} |v|_{C^{q_0}} \Vert h\Vert _s \int _L \left( \sum _{k \ge k(c')} |W_{\ell ,k}|^{\alpha _0-1} \int _{W_{\ell ,k}} \!\!\!\!\!\!\! \cos \phi \, \sqrt{1+|w'(\phi )|^2} \, d\phi \right) \ d\ell \\\ll & {} |v|_{C^{q_0}} \Vert h\Vert _s \int _L \sum _{k \ge k(c')} |W_{\ell ,k}|^{\alpha _0} k^{-r_0-(r_0+1)\alpha _0} \, d\ell \\\le & {} |v|_{C^{q_0}}\, \Vert h\Vert _s\, c\, k(c')^{1-\alpha _0(r_0+1)-r_0} \\\ll & {} |v|_{C^{q_0}} \Vert h\Vert _s |\xi |^{-1} d_{\rho }(\xi )^{\frac{3}{2}-s_0} \rho ^{-\frac{1}{2}+s_0} N^{-\frac{5}{2}+s_0}, \end{aligned}$$

for \(s_0 = \frac{1-\alpha _0(r_0+1)}{2r_0}\), as claimed. \(\square \)

Using (35), Lemmas A.2 and  6.1 we obtain the asymptotics of the eigenvalue in Proposition 6.3 below.

Lemma 6.2

For \(t\in {{\mathbb {R}}}^2\), let \({\bar{A}}(t,\rho )=\sum _{|\xi | \le 1/(2\rho )} \frac{d_{\rho }(\xi )^2 \langle t, \xi \rangle ^2}{|\xi |}\). Then

$$\begin{aligned} \lim _{\rho \rightarrow 0} \frac{\rho }{2} {\bar{A}}(t,\rho ) = \frac{|t|^2}{\pi } = \langle \Sigma t,t \rangle \quad \text { for } \quad \Sigma = \begin{pmatrix} \frac{1}{\pi } &{} 0 \\ 0 &{} \frac{1}{\pi } \end{pmatrix} \quad \text { as defined in 1}. \end{aligned}$$


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 \({\Psi }_1\) directly above the positive p-axis. Let \(\gamma = \gamma (t,\xi )\) be the angle between the vectors t and \(\xi \). Let \(\alpha = \arctan q/p\) and \(\theta \) be the polar angles of \(\xi \) and \(t \in {{\mathbb {R}}}^2\) respectively, so \(\gamma = \theta -\alpha \). For the first sector \({\Psi }_1\), taking into account that for every \(\xi \) there are two \(\xi '\), we have

$$\begin{aligned} \sum _{(\xi ,\xi ') \in {\Psi }_1} \frac{d_{\rho }(\xi )^2 \langle t, \xi \rangle ^2 }{|\xi |}&= 2|t|^2 \sum _{(\xi ,\xi ') \in {\Psi }_1} \frac{d_{\rho }(\xi )^2 (|\xi | \cos \gamma )^2 }{|\xi |} \\&= 2|t|^2 \sum _{(\xi ,\xi ') \in {\Psi }_1} \frac{d_{\rho }(\xi )^2 (\cos \theta \cos \alpha |\xi | + \sin \theta \sin \alpha |\xi |)^2 }{|\xi |} \\&= 2|t|^2 \sum _{(\xi ,\xi ') \in {\Psi }_1} \frac{d_{\rho }(\xi )^2 (p \cos \theta + q\sin \theta )^2 }{|\xi |}. \end{aligned}$$

The eighth sector \({\Psi }_8\) directly below the positive p-axis gives the same result with \(-q\) instead of q, and sectors \({\Psi }_4\) and \({\Psi }_5\) above and below the negative p-axis give the same results as sectors \({\Psi }_8\) and \({\Psi }_1\). Therefore

$$\begin{aligned} \sum _{(\xi ,\xi ') \in {\Psi }_1 \cup {\Psi }_4 \cup {\Psi }_5 \cup {\Psi }_8} \frac{d_{\rho }(\xi )^2 }{|\xi |} = 4 |t|^2 \sum _{(\xi ,\xi ') \in {\Psi }_1} \frac{d_{\rho }(\xi )^2 }{|\xi |} (p^2 \cos ^2 \theta +q^2 \sin ^2 \theta ). \end{aligned}$$

The same result holds the remaining sectors with \(\cos \theta \) replaced by \(\sin \theta \) and vice versa. Putting the results on all eight sectors together, we get by Lemma A.4

$$\begin{aligned} \sum _{(\xi ,\xi ') \in {\Psi }} \frac{d_{\rho }(\xi )^2 \langle t, \xi \rangle ^2 }{|\xi |}= & {} |t|^2 \sum _{(\xi ,\xi ') \in {\Psi }} \frac{ (|\xi |^{-1}-2\rho )^2 }{|\xi |} (p^2+q^2) \\= & {} |t|^2 \sum _{(\xi ,\xi ') \in {\Psi }} |\xi |^{-1}-4\rho + 4\rho ^2|\xi | \\= & {} |t|^2 \frac{2\pi }{\zeta (2)} \frac{1}{2\rho } (1-\frac{2}{2}+\frac{1}{3}) (1+o(1)) = \frac{2|t|^2}{\rho \pi } (1+o(1)). \end{aligned}$$

Hence \(\langle \Sigma t, t\rangle = \lim _{\rho \rightarrow 0} \frac{\rho }{2} {\bar{A}}(t,\rho ) = \frac{|t|^2}{\pi }\), as required. \(\square \)

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 \(\rho \)-dependent constants \({\hat{\gamma }}_{\rho }\in (0,1)\) and \({\hat{C}}_\rho > 0\) so that for every \(j\ge 1\),

$$\begin{aligned} \left| \int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\, (e^{it\kappa _\rho }-1)\circ T_{\rho }^{j}\, d\mu -\Big (\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,d\mu \Big )^2 \right| \le {\hat{C}}_\rho |t|^2 (1-{\hat{\gamma }}_{\rho })^j.\nonumber \\ \end{aligned}$$

More generally, we assume that that there exist \(\rho \)-dependent constants \(\bar{\gamma }_{\rho }\in (0,1)\) and \(\bar{C}_{\rho }> 0\) so that for every \(j\ge 1\) and every \(m\ge 0\)

$$\begin{aligned} \Big |&\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\cdot R_\rho (0)^{m}(e^{it\kappa _\rho }-1)\, (e^{it\kappa _\rho }-1)\circ T_{\rho }^{j}\, d\mu \nonumber \\&-\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1) R_\rho (0)^{m}(e^{it\kappa _\rho }-1)\,d\mu \int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,d\mu \nonumber \\&-C\Big (\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,d\mu \Big ) \int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\, (e^{it\kappa _\rho }-1)\circ T_{\rho }^{j}\, d\mu \nonumber \\&+C\Big (\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,d\mu \Big )^3\Big |\le \bar{C}_{\rho }|t|^2 (1-\bar{\gamma }_{\rho })^{m+j}, \end{aligned}$$

where \(C=0\) if \(m=0\) and \(C=1\) if \(m\ge 1\). As justified in Proposition C.1 in Appendix C via the argument used in [11, Proof of Proposition 9.1], assumptions (36) and (37) are natural.

Proposition 6.3

Assume (32), (36) and (37), and let \({\bar{A}}(t,\rho ) \) be as defined in Lemma 6.2. Let \(\nu \in (\frac{1}{3},\frac{1}{2})\) and \(\delta \in (0,\frac{1}{2}\min \{\gamma _{\rho },\hat{\gamma }_{\rho }\})\), ensuring that (33) holds. Then for any \(\delta _0\le \delta ^{4/(3\nu -1)}\) and \(t\in B_{\delta _0}(0)\),

$$\begin{aligned} 1-\lambda _{\rho }(t)={\bar{A}}(t,\rho ) \frac{\log (1/|t|)}{8\pi \rho }+E(t, \rho ), \end{aligned}$$

where \(|E(t,\rho )|\le \bar{C}_{\rho }\,\bar{\gamma }_{\rho }^{-2}\,|t|^2 + C|t|^2 \rho ^{-2}\) for \(\bar{C}_{\rho }\) and \(\bar{\gamma }_{\rho }\) as in (37) and some uniform constant C.

Remark 6.4

It is possible to shrink \(\delta _0\) further to \(\delta _0<e^{-\max \{\bar{C}_{\rho }\bar{\gamma }_{\rho }^{-2},\rho ^{-2}\}}\) leading to \(E(t,\rho )=o(|t|^2\log |1/t|)\). This would mean that in the proof of main results in Sect. 7 we would work on this very small neighborhood and obtain the same range of n and \(\rho \) in the final statements. We find it more convenient to work on the neighborhood \(B_{\delta _0}(0)\) as in the statement above.

Remark 6.5

Let \(q_{\rho }\) be the flight function taking values in \({{\mathbb {R}}}^2\) as opposed to the displacement function \(\kappa _\rho \) taking values in \({{\mathbb {Z}}}^2\). A similar statement holds for the dominant eigenvalue of the perturbed operator \(R_\rho (e^{it q_{\rho }})\). The proof is similar to the one below using that \(|q_{\rho }-\kappa _\rho | \le 1\).

Proposition 6.3

In the notation of Banach spaces of distributions (see, for instance, [21]) for \(h\in C^{q_0}\) we write \(\langle h, \textbf{1}\rangle =\langle \textbf{1}, h\rangle =\int h \textbf{1}\, d m\) and \(\langle m, h\rangle =\int h\, d m\), where \(\textbf{1}\) is both an element of \({{\mathcal {B}}}\) and of \((C^{q_0})'\). Let \(v_{\rho }(t)=\frac{ \Pi _{\rho }(t) 1}{\langle \Pi _{\rho }(t) 1, 1\rangle }\) and recall that \(v_{\rho }(0)=\textbf{1}\). Recall also that for every \(\rho \), \(\lambda _{\rho }(t) v_\rho (t)={\hat{R}}_{\rho }(t) v_{\rho }(t)\) for t small enough, and that \(\lambda _{\rho }(0)=1\). Since \(\langle v_{\rho }(t),\textbf{1}\rangle =1\),

$$\begin{aligned} 1-\lambda _{\rho }(t)&=1-\langle {\hat{R}}_{\rho }(t)v_\rho (t),\textbf{1}\rangle =\mu (1-e^{it\kappa _\rho })+\langle ({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0))(v_{\rho }(t)-\textbf{1}),\textbf{1}\rangle \\&=: \mu (1-e^{it\kappa _\rho })+ V(t,\rho ). \end{aligned}$$

With the meaning of inner product clarified, for ease of notation from here on we will write \(V(t,\rho )=\int _{{{\mathcal {M}}}}(e^{it\kappa _\rho }-1)(v_{\rho }(t)-\textbf{1}) \, d m\). We recall the terminology in Remark 2.2. For \(\xi = (p,q)\) with \(\gcd (p,q) = 1\), we let \(\xi ' = (p',q')\) be the point uniquely determined by \(\xi \) in the sense that \(p'/q'\) is convergent preceding p/q in the continued fraction expansion of p/q; in particular \(|\xi '| \le |\xi |\). Recall that \({\Psi }\) is the set of all such pairs \((\xi ,\xi ')\) with \(|\xi | \le 1/(2\rho )\). With this specified, we write

$$\begin{aligned} \mu (1-e^{it\kappa _\rho })=\sum _{(\xi ,\xi ')\in {\Psi }}\sum _{N=1}^\infty (e^{it( \xi '+ N\xi )}-1) \mu (\{ \kappa _\rho = \xi '+ N\xi \}). \end{aligned}$$

Using the fact that \(\int \kappa _\rho \, d\mu =0\), we compute that

$$\begin{aligned} \mu (1-e^{it\kappa _\rho })&=\sum _{(\xi ,\xi ')\in {\Psi }}\sum _{N=1}^\infty \left( e^{it( \xi '+ N\xi )}-1-it( \xi '+ N\xi )\right) \mu (\{ \kappa _\rho = \xi '+ N\xi \})\\&=\sum _{(\xi ,\xi ')\in {\Psi }}\sum _{N=1}^{1/|t|} \left( e^{it( \xi '+ N\xi )}-1-it( \xi '+ N\xi )\right) \mu (\{ \kappa _\rho = \xi '+ N\xi \}) \\&\quad +O\left( |t|\sum _{(\xi ,\xi ')\in {\Psi }}|\xi |\sum _{N>1/|t|} N\mu (\{ \kappa _\rho = \xi '+ N\xi \}) \right) \\&=\sum _{(\xi ,\xi ')\in {\Psi }}\sum _{N=1}^{1/|t|} \frac{1}{2}\, \langle t, \xi '+ N\xi \rangle ^2 \mu (\{ \kappa _\rho = \xi '+ N\xi \}) +O(|t|^2)\\ {}&:=I(t,\rho )+O(|t|^2), \end{aligned}$$

where the involved constants in the last big O are independent of \(\rho \). Further, using Lemma A.4,

$$\begin{aligned} I(t,\rho )&=\frac{1}{4\pi \rho }\sum _{|\xi | \le 1/(2\rho )}\frac{d_{\rho }(\xi )^2}{|\xi |} \langle t, \xi \rangle ^2\sum _{N=\max \{1,d_\rho (\xi )/(2\rho )\}}^{1/|t|} \frac{1}{N} \\&\qquad +O\left( |t|^2\sum _{(\xi ,\xi ')\in {\Psi }}\frac{1}{4\pi |\xi |\rho } \sum _{N<\max \{1, d_\rho (\xi )/(2\rho )\}}4\rho ^2\, N\,|\xi |\right) \\&=\frac{1}{4\pi \rho }\sum _{|\xi | \le 1/(2\rho )}\frac{d_{\rho }(\xi )^2}{|\xi |}\langle t, \xi \rangle ^2 \sum _{N=\max \{1,d_\rho (\xi )/(2\rho )\}}^{1/|t|} \frac{1}{N} +O\left( |t|^2\rho ^{-1}\right) \\&=\frac{\log (1/|t|)}{4\pi \rho }\sum _{|\xi | \le 1/(2\rho )}\frac{d_{\rho }(\xi )^2}{|\xi |}\langle t, \xi \rangle ^2 + O\left( |t|^2\rho ^{-1}\log (1/\rho )\right) . \end{aligned}$$

Hence, with \({\bar{A}}(t,\rho )\) as in Lemma 6.2,

$$\begin{aligned} \mu (1-e^{it\kappa _\rho })={\bar{A}}(t,\rho ) \frac{\log (1/|t|)}{4\pi \rho }+O\left( |t|^2\rho ^{-1}\log (1/\rho )\right) . \end{aligned}$$

Thus, \(1-\lambda _{\rho }(t)={\bar{A}}(t,\rho ) \frac{\log (1/|t|)}{4\pi \rho }+E(t,\rho )\), where \(E(t,\rho )=O\left( |t|^2\rho ^{-1}\log (1/\rho )\right) + V(t,\rho )\). It remains to estimate \(V(t,\rho )\). Note that

$$\begin{aligned} v_{\rho }(t)-\textbf{1}=\frac{\mu ((\Pi _{\rho }(t)-\Pi _{\rho }(0)) \textbf{1})}{\mu ( \Pi _{\rho }(t) \textbf{1})}\, \Pi _\rho (0)\textbf{1}+\frac{(\Pi _{\rho }(t)-\Pi _{\rho }(0)) \textbf{1}}{\mu _{\rho }( \Pi _{\rho }(t) 1)}. \end{aligned}$$


$$\begin{aligned} V(t,\rho )= & {} \frac{\mu ((\Pi _{\rho }(t)-\Pi _{\rho }(0)) \textbf{1})}{\mu ( \Pi _{\rho }(t) \textbf{1})}\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\, d \mu \\{} & {} +\frac{\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1) (\Pi _{\rho }(t)-\Pi _{\rho }(0)) \textbf{1}\, d m}{\mu ( \Pi _{\rho }(t) \textbf{1})}\\= & {} I_1(t,\rho )+I_2(t,\rho ). \end{aligned}$$

Estimating \( I_1(t,\rho )\). Since \(\int _{{{\mathcal {M}}}_0} \kappa _\rho \, d\mu =0\), we have

$$\begin{aligned} I_1(t,\rho )=\frac{\mu ((\Pi _{\rho }(t)-\Pi _{\rho }(0)) \textbf{1})}{\mu ( \Pi _{\rho }(t) \textbf{1})}\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1-it\kappa _\rho )\, d \mu . \end{aligned}$$

Now, by (35) and Lemma 6.1,

$$\begin{aligned} \int _{{{\mathcal {M}}}} | (\Pi _{\rho }(t)-\Pi _{\rho }(0))\textbf{1}|\, d\mu&=\sum _{(\xi ,\xi ')\in {\Psi }}\sum _{N=1}^\infty \int _{{{\mathcal {M}}}} 1_{\{ \kappa _\rho = \xi '+ N\xi \}} |(\Pi _{\rho }(t)-\Pi _{\rho }(0)) \textbf{1}| \nonumber \\&\le \sum _{(\xi ,\xi ')\in {\Psi }} |\xi |^{-\frac{5}{2}+s_0} \rho ^{-\frac{1}{2}+s_0}\Vert \Pi _{\rho }(t)-\Pi _{\rho }(0)\Vert _s\sum _{N=1}^\infty N^{-\frac{5}{2}} \nonumber \\&\le C \rho ^{-\nu } \gamma _{\rho }^{-2}|t|^{\nu } \end{aligned}$$

for some uniform C. Using also that \(|e^{ix}-1-ix|\le x^y\), for any \(y\in (0,2]\),

$$\begin{aligned} |I_1(t,\rho )|\le C \rho ^{-\nu } \gamma _{\rho }^{-2}|t|^{\nu }|t|^{2-\nu /2}\int _{{{\mathcal {M}}}_0} |\kappa _\rho |^{2-\nu }\, d \mu \le C \rho ^{-\nu -1} \gamma _{\rho }^{-2}|t|^{\nu /2+2}, \end{aligned}$$

where in the last inequality we have used Lemma A.5. Note that for \(|t|\in B_{\delta _0}(0)\) with \(\delta _0\le \gamma _{\rho }^{4/(3\nu -1)}\), as in the statement, \(|t|^{\nu /2}<\gamma _{\rho }^{2\nu /(3\nu -1)}< \gamma _{\rho }^{2}\) for all \(\nu \in (\frac{1}{3},\frac{1}{2})\). Thus, \(|I_1(t,\rho )|\le C \rho ^{-\nu -1} |t|^{2}\).

Estimating \( I_2(t,\rho )\) Recall that (32) holds and that \(\delta \) is chosen so that (34) holds. Using the definition of \(\Pi _{\rho }(t)\) and noting that for every \(\rho \), \((u-{\hat{R}}_{\rho }(0))^{-1}\textbf{1}=(1-u)^{-1}\),

$$\begin{aligned} (\Pi _{\rho }(t)-\Pi _{\rho }(0) 1&=\int _{|u-1|=\delta }(u-{\hat{R}}_{\rho }(t))^{-1}({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0))(u-{\hat{R}}_{\rho }(0))^{-1}\textbf{1}\, du\\&=\int _{|u-1|=\delta }(1-u)^{-1}(u-{\hat{R}}_{\rho }(t)^{-1}({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0))\textbf{1}\, du. \end{aligned}$$


$$\begin{aligned} I_2(t,\rho )&=\int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\int _{|u-1|=\delta }(1-u)^{-1}(u-{\hat{R}}_{\rho }(t))^{-1}({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0))\textbf{1}\, d u\, d m \nonumber \\&:=J_1(t,\rho )+J_2(t,\rho ), \end{aligned}$$


$$\begin{aligned} J_1(t,\rho ):=\int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\int _{|u-1|=\delta }(1-u)^{-1}(u-{\hat{R}}_{\rho }(0))^{-1}({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0))\textbf{1}\, d u\, d m \end{aligned}$$


$$\begin{aligned} J_2(t,\rho )&:= \int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\int _{|u-1|=\delta }(1-u)^{-1}\left( (u-{\hat{R}}_{\rho }(t))^{-1}-(u-{\hat{R}}_{\rho }(0))^{-1}\right) \\&\qquad ({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0))\textbf{1}\, d u\, d m\\&= \int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\int _{|u-1|=\delta } (1-u)^{-1}(u-{\hat{R}}_{\rho }(t))^{-1}({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0)) (u-{\hat{R}}_{\rho }(0))^{-1} \\&\qquad \times ({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0))\textbf{1}\, d u\, d m =: K_1(t,\rho )+K_2(t,\rho ), \end{aligned}$$


$$\begin{aligned} K_1(t,\rho )=\int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\int _{|u-1|=\delta }&(1-u)^{-1}(u-{\hat{R}}_{\rho }(0))^{-1}({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0)) \nonumber \\&\times (u-{\hat{R}}_{\rho }(0))^{-1}({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0))\textbf{1}\, d u\, d m \end{aligned}$$


$$\begin{aligned} \nonumber K_2(t,\rho )=\int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)&\int _{|u-1|=\delta } (1-u)^{-1} \left( (u-{\hat{R}}_{\rho }(t))^{-1}-(u-{\hat{R}}_{\rho }(0))^{-1}\right) \\&\times ({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0)) (u-{\hat{R}}_{\rho }(0))^{-1}({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0))\textbf{1}\, d u\, d m. \end{aligned}$$

We first treat \(K_2(t,\rho )\). Note that for u in the chosen contour, \(\Vert (u-R_\rho (t))^{-1}\Vert _{{{\mathcal {B}}}}\le \gamma _{\rho }^{-1}\). Using (35), for all such u,

$$\begin{aligned}&\left\| \left( (u-{\hat{R}}_{\rho }(t))^{-1}-(u-{\hat{R}}_{\rho }(0))^{-1}\right) ({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0)) (u-{\hat{R}}_{\rho }(0))^{-1}({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0))\right\| _{{{\mathcal {B}}}} \\&\quad \le C \rho ^{-2\nu }|t|^{3\nu }\gamma _{\rho }^{-3}. \end{aligned}$$

This together with Lemma 6.1 gives that

$$\begin{aligned} |K_2(t,\rho )|&\le \sum _{(\xi ,\xi ')\in {\Psi }} \sum _{N=1}^\infty \int _{{{\mathcal {M}}}_0} \int _{|u-1|=\delta }|1-u|^{-1}1_{\{ \kappa _\rho = \xi '+ N\xi \} } |e^{it\kappa _\rho }-1|\\&\quad \times \left| (u-{\hat{R}}_{\rho }(t))^{-1}-(u-{\hat{R}}_{\rho }(0))^{-1} ({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0) (u-{\hat{R}}_{\rho }(0))^{-1}({\hat{R}}_{\rho }(t)-{\hat{R}}_{\rho }(0)) \textbf{1}\right| \, d u\, d m\\&\le |t|^{3\nu } \rho ^{-3\nu } \gamma _{\rho }^{-3} \sum _{(\xi ,\xi ')\in {\Psi }} |\xi |^{-\frac{5}{2}+s_0} \rho ^{-\frac{1}{2}+s_0}\sum _{N=1}^\infty |t| N^{-\frac{3}{2}}\le C \rho ^{-3\nu } \gamma _{\rho }^{-3}|t|^{3\nu +1}. \end{aligned}$$

Hence, \(|K_2(t,\rho )| \le C \rho ^{-1}\gamma _{\rho }^{-3}\,|t|^{2}\, t^{3\nu -1}=C \rho ^{-1}\gamma _{\rho }^{-3}\,|t|^{2}\, \gamma _{\rho }^4\) for all \(|t|\in B_{\delta _0}\) with \(\delta _0<\gamma _{\rho }^{4/(3\nu -1)}\). It follows that \(|K_2(t,\rho )| \le C \rho ^{-1}|t|^{2}\).

Estimating \(J_1(t,\rho )\) in (39) and \(K_1(t,\rho )\) in (40) These terms are in, some sense, independent of the Banach space \({{\mathcal {B}}}\) (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.

We start with \(J_1(t,\rho )\) defined in (39), which is easier using (36). Recall that \({\hat{R}}_{\rho }(0)=R_{\rho }\) and \(\int _{|u-1|=\delta } (1-u)^{-2} \, du=0\) due to Cauchy’s theorem. This gives

$$\begin{aligned} J_1(t,\rho )&=\int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\int _{|u-1|=\delta }(1-u)^{-1}\sum _{j=0}^\infty u^{-j-1} R_{\rho }^j\, R_\rho (e^{it\kappa _\rho }-1)\textbf{1}\, d u\, d m \\&\quad -\left( \int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\, d \mu \right) ^2 \int _{|u-1|=\delta }(1-u)^{-1}\sum _{j=0}^\infty u^{-j-1}\, d u\\&=\int _{|u-1|=\delta }(1-u)^{-1}\sum _{j=0}^\infty u^{-j-1}\int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1) R_\rho ^j\, R_{\rho }(e^{it\kappa _\rho }-1)\textbf{1}\, d m\, d u \\&\quad -\left( \int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\, d \mu \right) ^2 \int _{|u-1|=\delta }(1-u)^{-1}\sum _{j=0}^\infty u^{-j-1}\, d u. \end{aligned}$$

Swapping the order of the integrals is allowed due to (36). The quantity

$$\begin{aligned} \Big (\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1) R_\rho ^{j+1}(e^{it\kappa _\rho }-1)\, d\mu -\int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\, d \mu \Big )^2 \end{aligned}$$

decays exponentially fast. Hence, we can write

$$\begin{aligned} J_1(t,\rho )&=\int _{|u-1|=\delta }(1-u)^{-1}\sum _{j=0}^\infty u^{-j-1}\\&\quad \times \Big (\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,(e^{it\kappa _\rho }-1)\circ T_{\rho }^{j+1} \, d\mu -\left( \int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\, d \mu \right) ^2 \Big )\, du. \end{aligned}$$

Using Lemma A.5 to control the dependence on \(\rho \), \(\left( \int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\, d \mu \right) ^2\le C |t|^2\rho ^{-2}\). Next, recall that (32) holds and that \(\delta < \frac{1}{2} \min \{ \gamma _{\rho }, {\hat{\gamma }}_{\rho } \}\). Note that for \(|u-1|=\delta \), we have \(|u|^{-(j+1)}\ll (1-{\hat{\gamma }}_{\rho }/2)^{-(j+1)}\). This together with (36) gives

$$\begin{aligned} |J_1(t,\rho )|&\le C_\rho \,|t|^2 \int _{|u-1|=\delta }|1-u|^{-1}\sum _{j=0}^\infty |u|^{-j-1}\left( 1-{\hat{\gamma }}_{\rho }\right) ^{j+1}\\&\ll {\hat{C}}_\rho \,|t|^2 \sum _{j=1}^\infty \left( \frac{1-{\hat{\gamma }}_{\rho }}{1-{\hat{\gamma }}_{\rho }/2}\right) ^{j+1}\le 2{\hat{C}}_\rho \,|t|^2\, {\hat{\gamma }}_{\rho }^{-1}. \end{aligned}$$

An argument similar to the one above used in estimating \( J_1(t,\rho )\) with  (37) instead of (36) allows us to deal with \(K_1(t,\rho )\) defined in (40). Compute that

$$\begin{aligned} K_1(t,\rho )=\int _{{{\mathcal {M}}}_0}(e^{it\kappa _\rho }-1)\int _{|u-1|=\delta }&(1-u)^{-1}\sum _{m\ge 1} u^{-m}\sum _{j\ge 1} u^{-j}{\hat{R}}_{\rho }(0)^{j}(e^{it\kappa _\rho }-1)\\&\times {\hat{R}}_{\rho }(0)^{m}(e^{it\kappa _\rho }-1)\, d u\, d m. \end{aligned}$$


$$\begin{aligned} E(t,\rho )&=\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,d\mu \,\int _{|u-1|=\delta }(1-u)^{-1}\\&\qquad \times \sum _{j\ge 1} u^{-j}\sum _{m\ge 1} u^{-m}\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1) R_\rho (0)^{m}(e^{it\kappa _\rho }-1)\,d\mu \, du \\&-\int _{|u-1|=\delta }(1-u)^{-1}\sum _{j\ge 1} u^{-j}\sum _{m\ge 1} u^{-m}\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,d\mu \, \\&\qquad \times \int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\, (e^{it\kappa _\rho }-1)\circ T_{\rho }^{j}\, d\mu \, du\\&-\Big (\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,d\mu \Big )^3\int _{|u-1|=\delta }(1-u)^{-1}\sum _{j\ge 1} u^{-j}\sum _{m\ge 1} u^{-m}\, du\\&=(E_1(t,\rho )-E_2(t,\rho ))\,\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,d\mu -E_3(t,\rho ). \end{aligned}$$

Using (37), we obtain

$$\begin{aligned} \Big |K_1(t,\rho ) -E(t,\rho ) \Big |\le \bar{C}_{\rho }|t|^2 \sum _{m\ge 1} |u|^{-m}\sum _{j\ge 1} |u|^{-j}(1-\bar{\gamma }_{\rho })^{m+j} \le 4 \bar{C}_{\rho }|t|^2 \bar{\gamma }_{\rho }^{-2}, \end{aligned}$$

where in the last inequality we proceeded as in estimating \(J_1\) above.

Finally, we need to argue that E is bounded by \(|t|^2\). First,

$$\begin{aligned}&E_1(t,\rho )=\int _{|u-1|=\delta }(1-u)^{-1}\sum _{j\ge 1} u^{-j}\sum _{m\ge 1} u^{-m}\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1) R_\rho (0)^{m}(e^{it\kappa _\rho }-1)\,d\mu \, du\\&\quad =\int _{|u-1|=\delta }(1-u)^{-2}\sum _{m\ge 1} u^{-m}\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,(e^{it\kappa _\rho }-1)\circ T_\rho ^m\,d\mu \, du\\&\quad =\int _{|u-1|=\delta }(1-u)^{-2}\sum _{m\ge 1} u^{-m} \\&\qquad \times \left( \int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,(e^{it\kappa _\rho }-1)\circ T_\rho ^m\,d\mu \, du - \left( \int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,d\mu \right) ^2\,d\mu \right) \\&\qquad + \left( \int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\,d\mu \right) ^2\int _{|u-1|=\delta }(1-u)^{-2}\sum _{m\ge 1} u^{-m}\, du= E_1^1(t,\rho )+E_1^2(t,\rho ). \end{aligned}$$

Using (36), we have that \(|E_1^1(t,\rho )|\le 2{\hat{C}}_\rho \,|t|^2\, {\hat{\gamma }}_{\rho }^{-1}\).

Also, \( E_2(t,\rho )=\int _{|u-1|=\delta }(1-u)^{-2}\sum _{j\ge 1} u^{-j}\int _{{{\mathcal {M}}}_0} (e^{it\kappa _\rho }-1)\, (e^{it\kappa _\rho }-1)\circ T_{\rho }^{j}\, d\mu \) and again by (36) and Cauchy’s theorem, \(| E_2(t,\rho )|\le 4{\hat{C}}_\rho \,|t|^2\, {\hat{\gamma }}_{\rho }^{-2}\). Finally, \(E_3(t,\rho )=0\). Altogether, \(| K_1(t,\rho )|\le 8\bar{C}_{\rho }\,|t|^2\, \bar{\gamma }_{\rho }^{-2}\). \(\square \)

7 Limit theorems and mixing as \(\rho \rightarrow 0\)

The first result below is the non-standard Gaussian limit law, known to hold when the horizon is infinite. It is a precise version of Theorem A stated in Sect. 1.3.

Our main contribution lies in characterizing the limit paths allowed as \(\rho \rightarrow 0\); this is done up to the unknown \(\gamma _{\rho }\), \(C_\rho \) in (32) and \(\bar{C}_{\rho }\), \(\bar{\gamma }_{\rho }\) as in (37).

Throughout this section, the notation is the same in Sect. 1.1. In particular, \(b_{n,\rho }=\frac{\sqrt{n\log (n/\rho ^2)}}{\sqrt{4\pi }\ \rho }\), and the variance matrix \(\Sigma \) are defined as in (1), in agreement with Lemma 6.2. We recall that \(\implies \) stands for convergence in distribution with respect to the invariant measure \(\mu \).

Theorem 7.1

Let \(\gamma _{\rho }\), \(C_\rho \) be as in (32), let \(\bar{\gamma }_{\rho }\), \(\bar{C}_{\rho }\) be as in (37) and let C be as in Proposition 6.3.

Set \(M(\rho )=\max \{ C_\rho \gamma _{\rho }^{-2},\rho ^2 \bar{C}_{\rho }\bar{\gamma }_{\rho }^{-2}\} + C\). Let \(\rho \rightarrow 0\) and simultaneously \(n\rightarrow \infty \) in such a way that \(M(\rho )= o(\log n)\). Then

$$\begin{aligned} \frac{\kappa _{n,\rho }}{b_{n,\rho }}\implies {{\mathcal {N}}}(0,\Sigma ). \end{aligned}$$

Remark 7.2

A similar statement holds for the flight function \(q_{\rho }\). The only change in the proof is the use of Remark 6.5 instead of Proposition 6.3.


Throughout we let \(\delta <\frac{1}{2} \min \{\gamma _{\rho },{\hat{\gamma }}_{\rho }\}\), so that we can use Proposition 6.3 with \(\delta _0=\delta ^{4/(3\nu -1)}\). By (33), for \(t\in B_{\delta _0}(0)\),

$$\begin{aligned} {{\mathbb {E}}}_\mu (e^{i t\kappa _{n,\rho }} 1)&={{\mathbb {E}}}_\mu ({\hat{R}}_{\rho }(t)^n 1)=\lambda _{\rho }(t)^n\int _{{{\mathcal {M}}}_0}\Pi _{\rho }(t) 1\, d\mu +\int _{{{\mathcal {M}}}_0}Q_{\rho }(t)^n1\, d\mu \\&=\lambda _{\rho }(t)^n\int _{{{\mathcal {M}}}_0}\Pi _{\rho }(t) 1\, d\mu +O(C_\rho \,(1-\gamma _{\rho })^n). \end{aligned}$$

Note that the assumption \(M(\rho )=o(\log n)\) ensures that, for \(\rho \) small enough, \(\frac{t}{b_{n,\rho }}\in B_{\delta _0}(0)\) for all \(t\in {{\mathbb {R}}}^2\). Hence, as \(n\rightarrow \infty \) and given the range of n, equivalently as \(\rho \rightarrow 0\),

$$\begin{aligned} \left| {{\mathbb {E}}}_\mu \left( \exp \left( {it \frac{\kappa _{n,\rho }}{b_{n,\rho }}}\right) \right) - \lambda _{\rho } \left( \frac{t}{b_{n,\rho }}\right) ^n\int _{{{\mathcal {M}}}_0}\Pi _{\rho } \left( \frac{t}{b_{n,\rho }}\right) 1\, d\mu \right| \rightarrow 0, \end{aligned}$$

for all \(t\in {{\mathbb {R}}}^2\).

Also, it follows from (35) that \(\Vert \Pi _{\rho }\left( \frac{t}{b_{n,\rho }}\right) -\Pi _{\rho }(0) \Vert _{{{\mathcal {B}}}}\rightarrow 0\), as \(n\rightarrow \infty \) and given the range of n, equivalently as \(\rho \rightarrow 0\). Thus, a standard argument based on the dominated convergence theorem shows that as \(n\rightarrow \infty \), equivalently as \(\rho \rightarrow 0\),

$$\begin{aligned} \left| {{\mathbb {E}}}_\mu \left( \exp \left( {it \, \frac{\kappa _{n,\rho }}{b_{n,\rho }}}\right) \right) - \lambda _{\rho } \left( \frac{t}{b_{n,\rho }}\right) ^n\right| \rightarrow 0. \end{aligned}$$

It remains to understand \(\lambda _{\rho } \left( \frac{t}{b_{n,\rho }}\right) ^n\) as \(\rho \rightarrow 0\). Since \(\delta _0=\delta ^{4/(3\nu -1)}\), we can apply Proposition 6.3 to obtain

$$\begin{aligned}&n\left( 1-\lambda _{\rho } \left( \frac{t}{b_{n,\rho }}\right) \right) \\ {}&\quad =\frac{n}{8\pi \rho }\,{\bar{A}}\left( \frac{t}{b_{n,\rho }}\,,\,\rho \right) \log (b_{n,\rho }/|t|) +n\, O\left( \left( \bar{C}_{\rho }\bar{\gamma }_{\rho }^{-2} + C\rho ^{-2}\right) \left( \frac{|t|}{b_{n,\rho }}\right) ^2\right) . \end{aligned}$$

By assumption, \(M(\rho )=o(\log n)\). Hence, as \(\rho \rightarrow 0\),

$$\begin{aligned}&n\, \left( \bar{C}_{\rho }\bar{\gamma }_{\rho }^{-2} + C\rho ^{-2}\right) \left( \frac{|t|}{b_{n,\rho }}\right) ^2\\ {}&\quad = \left( \bar{C}_{\rho }\bar{\gamma }_{\rho }^{-2} + C\rho ^{-2}\right) \frac{4\pi |t|^2\rho ^2}{\log (n/\rho ^2)} =O\left( \frac{M(\rho )}{\log n}\right) \cdot |t|^2= o(1)\cdot |t|^2\rightarrow 0. \end{aligned}$$

Now, given that \({\bar{A}}\) is as in Lemma 6.2,

$$\begin{aligned} \frac{n}{4\pi \rho }\,{\bar{A}}\left( \frac{t}{b_{n,\rho }},\,\rho \right) =\frac{1}{\log (n/\rho ^2)}\frac{1}{\rho }\, \rho ^2\bar{A}\left( t,\rho \right) =\frac{\rho {\bar{A}}(t,\rho )}{\log (n/\rho ^2)}. \end{aligned}$$

Also, using Lemma 6.2 and recalling the range of n,

$$\begin{aligned} \lim _{\rho \rightarrow 0}\,&\frac{n}{4\pi \rho }\,\bar{A}\left( \frac{t}{b_{n,\rho }}\, ,\, \rho \right) \log \left( \frac{b_{n,\rho }}{|t|}\right) = \lim _{\rho \rightarrow 0} \frac{\rho {\bar{A}}\left( t,\rho \right) }{\log (n/\rho ^2)}\log \left( \frac{b_{n,\rho }}{|t|}\right) \\&= \lim _{\rho \rightarrow 0}\frac{\rho }{2} \frac{{\bar{A}}(t,\rho )}{\log \left( \frac{\sqrt{n}}{\rho }\right) } \log \left( \frac{\sqrt{n}}{\rho } \frac{\sqrt{\log (n/\rho ^2)} }{ \sqrt{4\pi } |t|} \right) =\langle \Sigma t, t\rangle , \end{aligned}$$

where in the last equality we have used Lemma 6.2 and the uniform convergence theorem for slowly varying functions. Putting the above together,

$$\begin{aligned} \lim _{\rho \rightarrow 0} \lambda _{\rho } \left( \frac{t}{b_{n,\rho }}\right) ^n =\lim _{\rho \rightarrow 0} \exp \left( - n \left( 1-\lambda _{\rho } \left( \frac{t}{b_{n,\rho }}\right) \right) \right) =\exp \left( -\frac{1}{2}\langle \Sigma t, t\rangle \right) , \end{aligned}$$

for any \(t\in {{\mathbb {R}}}^2\). This completes the proof of Theorem 7.1 by Levy’s continuity theorem. \(\square \)

The next result gives a local limit theorem as \(\rho \rightarrow 0\), again up to the unknown \(\gamma _{\rho }\), \(C_\rho \) and \(\bar{\gamma }_{\rho }\), \(\bar{C}_{\rho }\). 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 \(\rho \) 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 \(\Phi _\Sigma \) be the density of a Gaussian random variable distributed according to \({{\mathcal {N}}}(0,\Sigma )\) and recall from Sect. 4.1 that \(C^{p_0}\subset {{\mathcal {B}}}\).

Theorem 7.3

Assume the assumptions and notation of Theorem 7.1; in particular \(M(\rho )\) is defined in the same way. Let \(v\in C^{p_0}({{\mathcal {M}}})\) and \(w\in L^a({{\mathcal {M}}})\), for \(a>1\).

Let \(\rho \rightarrow 0\) and simultaneously \(n\rightarrow \infty \) in such a way that \(M(\rho )=o(\log n)\). Then

$$\begin{aligned} \left| \int _{{{\mathcal {M}}}} v1_{\{\kappa _{n,\rho }=N\}} w\circ T_\rho ^n\, d\mu -\frac{{{\mathbb {E}}}_\mu (v)\,{{\mathbb {E}}}_\mu (w)}{(b_{n,\rho })^2}\, \Phi _\Sigma \left( \frac{N}{b_{n,\rho }}\right) \right| \rightarrow 0. \end{aligned}$$

uniformly in \(N\in {{\mathbb {Z}}}^2\).

Remark 7.4

A similar statement holds for the flight function \(q_{\rho }\). By a similar argument, using Remark 6.5 instead of Proposition 6.3, we obtain \((b_{n,\rho })^2\mu (\{q_{n,\rho }\in V\})\rightarrow \Phi _\Sigma (0)Leb_{{{\mathbb {R}}}^2}(V)\), for any compact neighborhood \(V\in {{\mathbb {R}}}^2\) with \(Leb_{{{\mathbb {R}}}^2}(\partial V)=0\). A uniform LLT for \(q_{\rho }\) 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 \(\rho >0\), \(\kappa _\rho \) is aperiodic, i.e., there exists no non-trivial solution to the equation \(e^{it\kappa _\rho }g\circ T_\rho =g\). The aperiodicity of \(\kappa _\rho \) has been used in [31] (in fact, in [30]) to provide LLT for fixed \(\rho \). Given Proposition 6.3 and the aperiodicity of \(\kappa _\rho \), 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.

Theorem 7.3

Let \(\delta _0=\delta ^{4/(3\nu -1)}\) be so that (34), (32) and Proposition 6.3 hold for all \(|t|\in B_{\delta _0}(0)\). Since \(\kappa _\rho \) is aperiodic, a known argument (see [Lemma 4.3 and Theorem 4.1] [1]) shows that \(\Vert {\hat{R}}_{\rho }(t)^n\Vert _{{{\mathcal {B}}}}\le C_\rho (1-\gamma _{\rho })^n\), for all \(|t|\ge \delta _0\). It follows that \(|{{\mathbb {E}}}_\mu ({\hat{R}}_{\rho }(t)^n 1)|\le \Vert {\hat{R}}_{\rho }(t)^n \Vert _{{{\mathcal {B}}}}\le C_\rho \,(1-\gamma _{\rho })^n\) for every \(|t| \in (\delta _0,\pi )\). Thus, using that \(v\in C^{p_0}\subset {{\mathcal {B}}}\),

$$\begin{aligned} \int _{{{\mathcal {M}}}}&v1_{\{\kappa _{n,\rho }=N\}} w\circ T_\rho ^n\, d\mu =\frac{1}{4\pi ^2}\int _{ [-\pi ,\pi ]^2} e^{-itN} \int _{{{\mathcal {M}}}}{\hat{R}}_{\rho }(t)^n v \, w \, d\mu \, dt \nonumber \\&=\frac{1}{4\pi ^2}\int _{ [-\delta _0,\delta _0]^2} e^{-itN} \int _{{{\mathcal {M}}}}{\hat{R}}_{\rho }(t)^n v \, w \, d\mu \, dt +O\left( C_\rho \,(1-\gamma _{\rho })^n\right) \nonumber \\&=\frac{1}{4\pi ^2}\int _{ [-\delta _0,\delta _0]^2} e^{-itN} \lambda _\rho \left( t\right) ^n\int _{{{\mathcal {M}}}} \Pi _{\rho }(t)v \, w \, d\mu \, dt +O\left( C_\rho \,(1-\gamma _{\rho })^n+{\hat{C}}_\rho \,(1-{\hat{\gamma }}_{\rho })^n\right) \nonumber \\&=\frac{1}{4\pi ^2}I(\rho ,t)+O\left( C_\rho \,(1-\gamma _{\rho })^n\right) . \end{aligned}$$

Recall that \(w\in L^a\), \(a>1\) and set \(b=a/(a-1)\). Using the Hölder inequality,

$$\begin{aligned} I(\rho ,t)= & {} \int _{ [-\delta ,\delta ]^2} e^{-itN} \lambda _\rho \left( t\right) ^n\, dt\, \int _{{{\mathcal {M}}}} v \, d\mu \,\int _{{{\mathcal {M}}}} w \, d\mu \\{} & {} +\int _{ [-\delta ,\delta ]^2} e^{-itN} \lambda _\rho \left( t\right) ^n\int _{{{\mathcal {M}}}} (\Pi _{\rho }(t)-\Pi _{\rho }(0))v \, w \, d\mu \, dt \\= & {} \int _{ [-\delta ,\delta ]^2} e^{-itN} \lambda _\rho \left( t\right) ^n\, dt\, \int _{{{\mathcal {M}}}} v \, d\mu \,\int _{{{\mathcal {M}}}} w \, d\mu \\{} & {} + O\left( \Vert w\Vert _{L^a(\mu )}\int _{ [-\delta ,\delta ]^2} \left| \lambda _\rho \left( t\right) ^n\right| \left( \int _{{{\mathcal {M}}}} |(\Pi _{\rho }(t)-\Pi _{\rho }(0))v|^b\, d\mu \right) ^{1/b}\, dt\right) . \end{aligned}$$

Recall \(v\in {{\mathcal {B}}}\). Using (35), (29) and Lemma 6.1 and proceeding as in equation (38),

$$\begin{aligned} \Big (\int _{{{\mathcal {M}}}} |(\Pi _{\rho }(t)-\Pi _{\rho }(0))v|^b\, d\mu \Big )^{1/b} \le C \rho ^{-\nu }\gamma _{\rho }^{-2} |t|^{\nu }\le C \rho ^{-2}|t|^\varepsilon , \end{aligned}$$

for some uniform C and some \(\varepsilon >0\). In the last inequality we have used that \(|t|<\delta _0\). Thus,

$$\begin{aligned} I(\rho ,t)&=\int _{ [-\delta _0,\delta _0]^2} e^{-itN} \lambda _\rho \left( t\right) ^n\, dt\, \int _{{{\mathcal {M}}}} v \, d\mu \, \int _{{{\mathcal {M}}}} w \, d\mu \\ {}&\quad + O\left( \Vert w\Vert _{L^a(\mu )}\rho ^{-2} \int _{ [-\delta _0,\delta _0]^2} |t|^{\varepsilon } \left| \lambda _\rho \left( t\right) ^n\right| \, dt\right) . \end{aligned}$$

With a change of variables,

$$\begin{aligned} I(\rho ,t)&=\frac{1}{(b_{n,\rho })^2}\int _{ [-\delta _0 b_{n,\rho },\delta _0 b_{n,\rho }]^2} e^{-iu\frac{N}{b_{n,\rho }}} \lambda _{\rho } \left( \frac{u}{b_{n,\rho }}\right) ^n\, du\, \int _{{{\mathcal {M}}}} v \, d\mu \,\int _{{{\mathcal {M}}}} w \, d\mu \nonumber \\&\qquad + O\left( \Vert w\Vert _{L^a(\mu )} \frac{\rho ^{-2}}{(b_{n,\rho })^3}\int _{ [-\delta _0 b_{n,\rho },\delta _0 b_{n,\rho }]^2} |u|^{\varepsilon } \left| \lambda _{\rho } \left( \frac{u}{b_{n,\rho }}\right) ^n\right| du\right) . \end{aligned}$$

Given the range of n in the statement, we use (41) to obtain

$$\begin{aligned} \lim _{\rho \rightarrow 0}\left| 4\pi ^2\,\int _{ [-\delta _0 b_{n,\rho },\delta _0 b_{n,\rho }]^2} e^{-iu\frac{N}{b_{n,\rho }}} \lambda _{\rho } \left( \frac{u}{b_{n,\rho }}\right) ^n\, du-\Phi _\Sigma \left( \frac{N}{b_{n,\rho }}\right) \right| =0. \end{aligned}$$

To deal with the big O term in (43), we use that by (41) there exists a uniform constant C so that

$$\begin{aligned} \frac{\rho ^{-2}}{(b_{n,\rho })^3}\int _{ [-\delta _0 b_{n,\rho },\delta _0 b_{n,\rho }]^2} |u|^{\varepsilon } \left| \lambda _{\rho } \left( \frac{u}{b_{n,\rho }}\right) ^n\right| \,du\le \frac{\rho ^{-2}}{(b_{n,\rho })^{2+\varepsilon }}\int _{ [-\delta _0 b_{n,\rho },\delta _0 b_{n,\rho }]^2} |u|^{\varepsilon } e^{-C|u|^2}\, du. \end{aligned}$$

Since \(M(\rho )=o(\log n)\), we have \(n\gg \exp \left( C\rho ^{-2}\right) \). Thus, \(\frac{\rho ^{-2}}{(b_{n,\rho })^{2+\varepsilon }}\ll \frac{\log n}{(b_{n,\rho })^{2+\varepsilon }}=o\left( \frac{1}{(b_{n,\rho })^2}\right) \) as \(\rho \rightarrow 0\). Putting these together and using (43),

$$\begin{aligned} \lim _{\rho \rightarrow 0}\left| 4\pi ^2\, I(\rho ,t) -\Phi _\Sigma \left( \frac{N}{b_{n,\rho }}\right) \, \int _{{{\mathcal {M}}}} v \, d\mu \,\int _{{{\mathcal {M}}}} w \, d\mu \,\right| =0. \end{aligned}$$

This together with (42) gives that as \(\rho \rightarrow 0\),

$$\begin{aligned}&\left| \int _{{{\mathcal {M}}}} v1_{\{\kappa _{n,\rho }=N\}} w\circ T_\rho ^n\, d\mu -\frac{1}{(b_{n,\rho })^2} \Phi _\Sigma \left( \frac{N}{b_{n,\rho }}\right) \,\int _{{{\mathcal {M}}}} v \, d\mu \,\int _{{{\mathcal {M}}}} w \, d\mu \,\right| \\&\qquad =O\left( (b_{n,\rho })^2\ C_\rho \,(1-\gamma _{\rho })^n \right) =o(1), \end{aligned}$$

where in the last equation we used that \(M(\rho )=o(\log n)\). This concludes the proof. \(\square \)

It is known that the local limit theorem for \(\kappa _\rho \) and the billiard map \(T_{\rho }\) (with \(\rho \) fixed) implies mixing for the planar Lorentz map \({\hat{T}}_{\rho }\) (again \(\rho \) 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 Sect. 1 that the Lorentz map \({\hat{T}}_{\rho }\) defined on \({\widehat{{{\mathcal {M}}}}}={{\mathcal {M}}}\times {{\mathbb {Z}}}^2\) is given by \({\hat{T}}_{\rho }(\theta ,\phi ,\ell ) = ( T_{\rho }(\theta ,\phi ), \ell +\kappa _\rho (\theta ,\phi ))\) for \( (\theta ,\phi ) \in {{\mathcal {M}}},\ \ell \in {{\mathbb {Z}}}^2\). Let \({\hat{\mu }}=\mu \times \text{ Leb}_{{{\mathbb {Z}}}^2}\), where \(\text{ Leb}_{{{\mathbb {Z}}}^2}\) is the counting measure on \({{\mathbb {Z}}}^2\). An immediate consequence of Theorem 7.3 is

Corollary 7.5

Assume the assumptions and notation of Theorem 7.3. Let \(\rho \rightarrow 0\) and simultaneously \(n\rightarrow \infty \) in such a way that \(M(\rho )=o(\log n)\). Then

$$\begin{aligned} \lim _{\rho \rightarrow 0}\left| (b_{n,\rho })^2\int _{{\widehat{{{\mathcal {M}}}}}}v\, w\circ {\hat{T}}_{\rho }\, d{\hat{\mu }}-\int _{{\widehat{{{\mathcal {M}}}}}} v \, d{\hat{\mu }}\,\int _{{\widehat{{{\mathcal {M}}}}}} w \, d{\hat{\mu }}\,\right| =0. \end{aligned}$$

Remark 7.6

The class of functions in Corollary 7.5 is rather restrictive as the functions vw are supported on the cell \({{\mathcal {M}}}\). Given the work [28] (see also [29, Sect. 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 \({\widehat{{{\mathcal {M}}}}}\). Since the involved argument is rather delicate and not a main concern of the present work, we omit this.