Map Lattices Coupled by Collisions: Hitting Time Statistics and Collisions Per Lattice Unit

Abstract

We study map lattices coupled by collision and show how perturbations of transfer operators associated with the spatially periodic approximation of the model can be used to extract information about collisions per lattice unit. More precisely, we study a map on a finite box of L sites with periodic boundary conditions, coupled by collision. We derive, via a non-trivial first-order approximation for the leading eigenvalue of the rare event transfer operator, a formula for the first collision rate and a corresponding first hitting time law. For the former we show that the formula scales at the order of $L\cdot \varepsilon ^2$, where $\varepsilon$ is the coupling strength, and for the latter, by tracking the L dependency in our arguments, we show that the error in the law is of order $O\left( C(L)\frac{L\varepsilon ^2}{\zeta (L)}\cdot \left| \ln \frac{L\varepsilon ^2}{\zeta (L)}\right| \right)$, where $\zeta (L)$ is given in terms of the spectral gap of the rare event transfer operator, and C(L) has an explicit expression. Finally, we derive an explicit formula for the first collision rate per lattice unit.

1 Introduction

Coupled map lattices are discrete-time dynamical systems modeling the interaction between microscopic units organized in a spatial structure given by a lattice. The state of each unit, called a site, evolves according to the interplay between the local dynamics and the effect of the interaction with other sites. Such systems were introduced in the 80s (see [26] and references therein), and since then, they have seen a remarkable amount of research due to their paramount importance in applications and in studying non-equilibrium thermodynamics [5, 31]. From a mathematical viewpoint, the main interest of coupled map lattices is that they provide natural examples of dynamical systems with an infinite-dimensional state space. Ergodic and statistical properties of coupled maps were studied extensively since the seminal work of [8]. Existence and uniqueness of the SRB measure and decay rate of correlations were the questions studied initially, typically in the weak interaction regime. The early results consider smooth expanding or hyperbolic local dynamics with coupling of similar regularity, while piecewise expanding site dynamics with more general coupling schemes (allowing to treat nearest neighbor coupling, for instance) were considered subsequently. See [10, 22] for an extensive list of references. Considering the complete coupling regime, it is natural to expect phase transition-like phenomena. Such results were only proved in case of finite lattices with a small number of sites [25, 33]; however, numerics suggest that the existence of multiple positive Lebesgue measure ergodic components can prevail for large system sizes [14]. Another line of current research studies coupled maps in more complex spatial structures. After the pioneering work of [27], more realistic setups were considered in the form of heterogeneous networks [29].

A natural way to understand a system on an infinite lattice is to consider its spatially periodic approximations, that is, coupled maps defined on finite boxes of sites with periodic boundary conditions. By understanding how various important dynamical quantities, such as entropy [12], escape rates in open systems [4], scale with the size of the system it is possible to meaningfully define the amount of “chaos” per lattice unit¹ in the system [see [34] section 4.2 for a highlight about the importance of this concept in large systems]. A progress in studying the above quantities, entropy and escape rates, has been obtained in [4, 12] for one-dimensional lattices weakly coupled via a convolution operator.

In this work, we study coupled map lattices where the interaction takes place via rare but intense ‘collisions’ and the dynamics on each site is given by a piecewise uniformly expanding map of the interval. Our main goal is to quantify explicitly the amount of chaos per lattice unit in this model. The peculiarity of the model is that this kind of interaction gives rise to a discontinuous coupling function. Many of the technical challenges this poses were tackled in [23], where the existence of a unique Sinai–Ruelle–Bowen measure and exponential decay of correlations within a suitable class of measures was proved via a decoupling technique.² The motivation for this model can be understood through the intimate connections with similar models that are used by mathematicians and physicists in the study of heat transfer and the derivation of an associated Fourier’s law [3, 15].

In this paper, we show how spectral techniques of transfer operators associated with the (finite-dimensional) periodic approximation of the model³ can be used to extract information about collisions per lattice unit. In particular, using this technique, a formula for the first collision rate with respect to Lebesgue measure, i.e., the fraction of orbits that see a first collision at each time step of the dynamics,⁴ and a corresponding hitting time distribution for the system of L sites are derived. For the former we show that the formula scales at the order of $L\cdot \varepsilon ^2$, where $\varepsilon$ is the coupling strength, and for the latter we show that the error in the law is of order $O\left( C(L) \frac{L\varepsilon ^2}{\zeta (L)}\cdot \left| \ln \frac{L\varepsilon ^2}{\zeta (L)}\right| \right)$, where $\zeta (L)$ is given in terms of the speed of mixing, and C(L) is a particular function which is obtained by tracking the L dependency in the abstract perturbation result of [24]. Moreover, we derive an explicit formula for the first collision rate per lattice unit. To the best of our knowledge explicit formulae of this type did not appear before this work in the mathematical literature. We believe that the transfer operator technique employed in this work can be applied to different types of coupled map lattices addressing a related problem, for instance, to derive a precise formula for the escape rate of the open system studied in [4].

In Sect. 2 we introduce the coupled model, state the assumptions on the local dynamics, introduce the corresponding transfer operators and the function space on which the transfer operators act. We finish this section with the statement of our main results, Theorem 2.5 and Theorem 2.8. In Sect. 3 we provide examples to illustrate Theorem 2.5; in particular, we describe in certain low-dimensional cases the structure of the set of collision states, its limit set as the coupling strength goes to zero, and the formula of the ‘derivative’ for the dominant eigenvalue of a rare event transfer operator. Section 4 contains some technical lemmas, their proofs, and the proofs of Theorems 2.5 and 2.8. It also contains a remark discussing the infinite-dimensional limit of the collision rate and its per lattice unit counterpart.

2 Setup

Let I denote the closed unit interval [0, 1]. Let $X=I^{\Lambda }$, where $\Lambda ={\mathbb {Z}}^d/(N{\mathbb {Z}})^d$ for some integers $N,d \ge 1$. As an ease of notation, we are going to write $L=N^d$ from now on. We will refer to $\Lambda$ as the finite lattice of L sites. We denote by $m_{L}$ the L-dimensional Lebesgue measure on X.

Let $\tau :I\rightarrow I$ be the map describing the local dynamics at a single site. Define $T_0:X\rightarrow X$ as the product map $(T_0(x))_{{\mathbf {p}}}=\tau (x_{{{\mathbf {p}}}})$, ${{\mathbf {p}}}\in \Lambda$, i.e., $T_0$ is the uncoupled dynamics on the lattice.

Let $V^+:=\{e_i\}$ be the standard basis of ${{\mathbb {R}}}^d$ and define $V:=V^+ \cup -V^+$. For $\varepsilon >0$, let $\{A_{\varepsilon ,-{{\mathbf {v}}}}\}_{{{\mathbf {v}}}\in V}\subset I$ be a set of disjoint open intervals, each of length $\varepsilon$. Denote by $(a_{{\mathbf {v}}}, a_{-{{\mathbf {v}}}})$ the point that $A_{\varepsilon ,{{\mathbf {v}}}}\times A_{\varepsilon ,-{{\mathbf {v}}}}$ shrinks to as $\varepsilon \rightarrow 0$ and $S:=\{(a_{{\mathbf {v}}}, a_{-{{\mathbf {v}}}})\}_{{{\mathbf {v}}}\in V^+}$ be the set of all such points. Following [23], we consider the coupling⁵

$$\begin{aligned} (\Phi _\varepsilon (x))_{{\mathbf {p}}}={\left\{ \begin{array}{ll} x_{{{\mathbf {p}}}+{{\mathbf {v}}}} \quad \text {if } x_{{\mathbf {p}}}\in A_{\varepsilon ,{{\mathbf {v}}}}\text { and } x_{{{\mathbf {p}}}+{{\mathbf {v}}}}\in A_{\varepsilon ,-{{\mathbf {v}}}} \text { for some } {{\mathbf {v}}}\in V\\ x_{{\mathbf {p}}}\quad \quad \text {otherwise} \end{array}\right. } \end{aligned}$$

(1)

where ${{\mathbf {p}}}+{{\mathbf {v}}}$ is understood modulo N coordinatewise. We define the coupled dynamics $T_\varepsilon :X\rightarrow X$ as the composition $T_\varepsilon :=\Phi _\varepsilon \circ T_0$.

We assume that $\tau :I\rightarrow I$ is piecewise monotone and $C^{1+\beta }$ on the intervals $I_1,\dots ,I_k$ with an extension of similar smoothness to the boundaries and $|\tau '| > 1$. We further assume that $\tau$ has full branches. It is well known that $\tau$ admits a unique absolutely continuous invariant measure $\mu _{\tau }$ whose density h is bounded from below by a positive number on the whole interval I [6], and it is furthermore well known to be (Hölder) continuous. Moreover, to study first-order approximations of collision rates, we assume that $\tau$ is continuous at $a_{{\mathbf {v}}}$ and $a_{-{{\mathbf {v}}}}$ for all ${{\mathbf {v}}}\in V$.

Fig. 1

The Gaspard–Gilbert model [15]. Grey circles represent the obstacles, while the black circles are the moving billiard balls

Remark 2.1

Our primary inspiration to study this this type of coupled map lattice was given by the paper [23]. In addition to this, an analogy to the billiard model of [15] provides further motivation: consider a periodic pattern of smooth scatterers, such as in Fig. 1. Among every four of them a cell is formed that contains a moving billiard ball. Thus, the cells have the structure of a two-dimensional lattice. The radius of the moving ball is large enough so that the ball is confined to its cell, but small enough so that it is able to collide with the moving balls in each of the four neighboring cells. When no such interaction happens, the state of each ball (described by its position and velocity) evolves according to classical billiard dynamics. This can be understood as the local dynamics of the sites. When two moving balls collide, they exchange the components of their velocities that are parallel to the line through their centers at the moment of impact. This is similar in spirit to the coupling (1), where the state of two neighboring sites are interchanged if these states are in an appropriate zone with respect to each other, although in (1) the specific form of the collision is insignificant. As for the local dynamics, following [23] we consider the simpler setting of a piecewise uniformly expanding interval maps. The reason for this simplification is that the coupled system will be of similar regularity, but higher dimensional. Transfer operator techniques for multidimensional piecewise expanding maps are well understood [16, 28, 32], in contrast to high-dimensional hyperbolic systems with singularites,⁶ which would have to be considered in case of the billiard model of [15].

In what follows we will build a theory to study the map on the finite lattice of L sites.

2.1 Transfer Operators, First Hitting Times and First Collision Rates

Denote by $L^1_{{{\mathbb {R}}}^{L}}$ the space of measurable functions $f: {{\mathbb {R}}}^{L} \rightarrow {{\mathbb {R}}}$ for which

$$\begin{aligned} \int _{{{\mathbb {R}}}^{L}} |f| < \infty . \end{aligned}$$

Let $L^\infty$ be the space of essentially bounded measurable functions $f: {{\mathbb {R}}}^{L} \rightarrow {{\mathbb {R}}}$. We will use the notation $\Vert f\Vert _1=\int |f|$ for $f \in L^1_{{{\mathbb {R}}}^{L}}$, and $\Vert f\Vert _\infty :={{\,\mathrm{ess\,sup}\,}}f$. Since $T_0$ is non-singular with respect to $m_L$, we can define the transfer operator associated with $T_0$, denoted by $P_{T_0}$, via the following duality: for $f\in L^1_{{{\mathbb {R}}}^{L}}$ and $g\in L^\infty$

$$\int g\circ T_0\cdot f \mathrm{d}m_L=\int g\cdot P_{T_0} f \mathrm{d}m_L.$$

Now, define ${\mathcal {I}}_{L}$ as the set of rectangles $I_{i_1} \times \cdots \times I_{i_{L}}$, $i_1,\dots , i_{L} \in \{1,\dots ,k\}$ on which $T_0$ is $C^{1+\beta }$. Then, the transfer operator $P_{T_0}$ has the following a.e. pointwise representation:

$$\begin{aligned} P_{T_0}f(x)=\sum _{J \in {\mathcal {I}}_{ L}}\frac{f}{|\det (DT_0)|}\circ (T_0|J)^{-1}(x)1_{T_0(J)}(x). \end{aligned}$$

2.1.1 Rare Events and the Corresponding Transfer Operator

For ${{\mathbf {p}}}\in \Lambda$ let

$$X_{\varepsilon ,{{\mathbf {v}}}}({{\mathbf {p}}})=\{x\in X: x_{{{\mathbf {p}}}}\notin A_{\varepsilon ,{{\mathbf {v}}}}\text { or } x_{{{\mathbf {p}}}+{{\mathbf {v}}}}\notin A_{\varepsilon ,-{{\mathbf {v}}}}\},$$

and set

$$X_{\varepsilon }^0:=\bigcap _{{{\mathbf {p}}}\in \Lambda }\bigcap _{{{\mathbf {v}}}\in V}X_{\varepsilon ,{{\mathbf {v}}}}({{\mathbf {p}}}),$$

and

$$H_\varepsilon :=X\setminus X_\varepsilon ^0.$$

Let

$$X^{n-1}_\varepsilon =\cap _{i=0}^{n-1} T_{0}^{-i}X_{\varepsilon }^0.$$

Notice that $X^{n-1}_\varepsilon$ is the set of points whose orbits did not result in any collision up to time $n-1$. For $n\ge 1$ and a measurable function f, let

$$\begin{aligned} {\hat{P}}^n_{\varepsilon }f= P^n_{T_{0}}(1_{X^{n-1}_{\varepsilon }} f). \end{aligned}$$

(2)

Following the terminology of [19], we are going to call ${\hat{P}}_{\varepsilon }$ the rare event transfer operator.

Note that in case of $\varepsilon =0$ we have $X_0^n=X$ for all $n \in {\mathbb {N}}$, and hence in this case the operator defined by (2) simply reduces to the transfer operator of $T_{0}$.

It is a natural question to study how long a typical trajectory of the coupled system can survive without experiencing a single collision.

Definition 2.2

Let m be a probability measure supported on $X^0_\varepsilon$. Suppose that

$$-\lim _{n \rightarrow \infty }\frac{1}{n}\ln m(X_{\varepsilon }^{n-1}):={{\hat{r}}}$$

exists. Then, the quantity in the above limit measures asymptotically, with respect to m, the fraction of orbits that see a first collision at each time step under the dynamics of $T_\varepsilon$. We call such a quantity the first collision rate under $T_\varepsilon$, with respect to m.

The quantity in Definition 2.2 captures the fact that the proportion of initial conditions that lead to no collision up to time n shrinks exponentially with rate ${\hat{r}}$. The motivation behind this definition is to study how ${{\hat{r}}}$, when it exists, scales with the size of the system to meaningfully define the amount of “chaos” per lattice unit. This is achieved in Theorem 2.8 for the collision models we are interested in.

A natural subsequent question to the fist collision rate is to study how long it takes for a typical trajectory to experience its first collision. We define the first hitting time by

$$t_\varepsilon (x)=\inf \{n\ge 0:\,T^n_\varepsilon (x)\in H_\varepsilon \}$$

, i.e., $t_\varepsilon$ is the first time when we see a collision under the coupled map $T_\varepsilon$. Note that $t_\varepsilon$ can be equivalently defined by

$$t_\varepsilon (x)=\inf \{n\ge 0:\,T^n_0(x)\in H_\varepsilon \}.$$

Remark 2.3

The above defined quantities do not take into consideration how the coupling acts on states that involve a collision, since we study trajectories up until the first collision event occurs. More precisely, no coupling takes place on the trajectories we study, so in fact what we study can be viewed as the product map $T_0$ with a specific hole $H_\varepsilon$.⁷

The main goal of this work is to understand the first collision rates and hitting time statistics in map lattices coupled by collision as introduced by Keller and Liverani [23] and the generalizations of this collision model as described in footnote 1 of [23].

2.2 Quasi-Hölder Spaces

We will study the action of the transfer operators defined above on a suitable⁸ Banach spaces. It turns out that for the problem at hand, the most suitable space is the one introduced in [32]. We start by recalling the function spaces introduced in [32].

Let $0 < \alpha \le 1$ and $\omega _0 >0$ be parameters. Define the oscillation of f on a Borel set $S \subset {{\mathbb {R}}}^{L}$ as

$$\begin{aligned} {{\,\mathrm{osc}\,}}(f,S)={{\,\mathrm{ess\,sup}\,}}_S f - {{\,\mathrm{ess\,inf}\,}}_S f. \end{aligned}$$

Let

$$\begin{aligned} |f|_{\alpha }=\sup _{0 < \omega \le \omega _0} \omega ^{-\alpha }\int _{{{\mathbb {R}}}^{L}}{{\,\mathrm{osc}\,}}(f,B_{\omega }(x))\mathrm{d}x \end{aligned}$$

and

$$\begin{aligned} V_{\alpha }=\left\{ f \in L^1_{{{\mathbb {R}}}^{L}}: |f|_{\alpha } < \infty \right\} . \end{aligned}$$

Set $\Vert f\Vert _{\alpha }=|f|_{\alpha }+\Vert f\Vert _{1}$ for $f \in V_{\alpha }$. Then, by [ [32], Proposition 3.3] $(V_{\alpha },\Vert \cdot \Vert _{\alpha })$ is a Banach space, and the intersection of the unit ball of $V_{\alpha }$ with the set of functions supported on X is compact in $L^1_{{{\mathbb {R}}}^{L}}$. Moreover, by [ [32], Proposition 3.4] $V_{\alpha }$ is continuously injected in $L^\infty$; in particular, $\forall f\in V_\alpha$

$$\begin{aligned} \Vert f\Vert _\infty \le \frac{\max (1,\omega _0^\alpha )}{\gamma _{ L}\omega _0^{L}}\Vert f\Vert _\alpha , \end{aligned}$$

(3)

where $\gamma _{L}$ denotes the volume of the unit ball of ${{\mathbb {R}}}^{L}$. As this constant will be important to us on multiple occasions, we introduce the notation

$$\begin{aligned} C(L,\omega _0)=\frac{\max (1,\omega _0^\alpha )}{\gamma _{ L}\omega _0^{L}}. \end{aligned}$$

(4)

2.3 Notation, Collision States and Their Limit Sets

Now we introduce some notation that are needed in the statements of our main results. All the notation introduced in this subsection is illustrated through simple examples in Sect. 3. Recall that $H_{\varepsilon }=X\backslash X_{\varepsilon }^0$ is the set of points whose state results in a collision. Let

$$\begin{aligned} H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}}=\{ x \in X: x_{{{\mathbf {p}}}} \in A_{\varepsilon ,{{\mathbf {v}}}}, x_{{{\mathbf {p}}}+{{\mathbf {v}}}} \in A_{\varepsilon ,-{{\mathbf {v}}}}\} \end{aligned}$$

(5)

for ${{\mathbf {p}}}\in \Lambda , {{\mathbf {v}}}\in V^+$. This set denotes the set of points whose state results in a collision at site ${{\mathbf {p}}}$ in the direction ${{\mathbf {v}}}$. Then,

$$\begin{aligned} H_{\varepsilon }=\cup _{{{\mathbf {p}}}\in \Lambda }\cup _{{{\mathbf {v}}}\in V^+}H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}}. \end{aligned}$$

Note that this is a finite, but, in general, not a disjoint union as multiple collisions can happen at the same time.

Note that $H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}}$ are L-dimensional boxes with a base of area $\varepsilon ^2$. Their limiting sets, as $\varepsilon \rightarrow 0$, will play a crucial role in the statement and proof of Theorem 2.5. Denote these limiting sets by $I_{{{\mathbf {p}}},{{\mathbf {v}}}}=\lim _{\varepsilon \rightarrow 0}H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}}$. Explicitly they are given as

$$\begin{aligned} I_{{{\mathbf {p}}},{{\mathbf {v}}}}=\{ x \in X: x_{{{\mathbf {p}}}}=a_{{\mathbf {v}}}, x_{{{\mathbf {p}}}+{{\mathbf {v}}}}= a_{-{{\mathbf {v}}}}\}. \end{aligned}$$

Remark 2.4

It is easy to see that $\lim _{\varepsilon \rightarrow 0}m_{L}(H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}})=m_{ L}(I_{{{\mathbf {p}}},{{\mathbf {v}}}})=0$. This implies that $m_{L}(H_{\varepsilon }) \rightarrow 0$ as $\varepsilon \rightarrow 0$.

Recall that for each ${{\mathbf {v}}}\in V^+$, $(a_{{\mathbf {v}}}, a_{-{{\mathbf {v}}}})$ denotes the point that $A_{\varepsilon ,{{\mathbf {v}}}}\times A_{\varepsilon ,-{{\mathbf {v}}}}$ shrinks to as $\varepsilon \rightarrow 0$ and let $S:=\{(a_{{\mathbf {v}}}, a_{-{{\mathbf {v}}}})\}_{{{\mathbf {v}}}\in V^+}$ be the set of all such points. Let $S^{per} \subset S$ be the set of periodic points in S with respect to the map $T_{2,0}=\tau \times \tau$. For each $(a_{{{\mathbf {v}}}},a_{-{{\mathbf {v}}}}) \in S^{per}$ define $k({{\mathbf {v}}})$ as the smallest integer k such that $T_{2,0}^{k+1}(a_{{{\mathbf {v}}}},a_{-{{\mathbf {v}}}}) \in S$. Let $V_k^+ \subset V^+$ denote the set of directions for which $k({{\mathbf {v}}})=k$ and $K=\{k({{\mathbf {v}}})\}_{{{\mathbf {v}}}\in V^+}$. Note that the case when $S^{per}\not =\emptyset$ is interesting and important to study since, as the calculation below shows, the collision rate will be higher when the collision zones are centered around periodic points with higher minimal period.⁹

2.4 Statements of the Main Results

In the following theorem L is fixed and we understand $o(1) \rightarrow 0$ as $\varepsilon \rightarrow 0$.

Theorem 2.5

Fix $L \!\in \! {\mathbb {N}}$. For $\varepsilon \ge 0$ sufficiently small, the operator ${\hat{P}}_\varepsilon : V_\alpha \!\rightarrow \!V_\alpha$ admits a spectral gap. In particular $T_0$ admits an absolutely continuous invariant measure $\mu _0=\rho _0\cdot m_L$, with $\rho _0(x)=\Pi _{i=1}^{L} h(x_i)$. Moreover, for $\varepsilon >0$,

• 1. ${\hat{P}}_\varepsilon$ has an eigenfunction $\rho _\varepsilon >0$, $\int \rho _\varepsilon \mathrm{d}x=1$, corresponding to a simple dominant eigenvalue $\lambda _\varepsilon \in (0,1)$, with

  $$\lambda _{\varepsilon }=1-\mu _0(H_\varepsilon )\cdot \theta (1+o(1)),$$

  where

  $$\mu _0(H_\varepsilon )= L\Xi _{\varepsilon }(1+o(1)) \sim \varepsilon ^2 L d$$

  such that

  $$\begin{aligned} \Xi _{\varepsilon }=\sum _{{{\mathbf {v}}}\in V^+}\int _{A_{\varepsilon ,{{\mathbf {v}}}}}\int _{A_{\varepsilon ,-{{\mathbf {v}}}}}h(x_1)h(x_2)\mathrm{d}x_1\mathrm{d}x_2, \end{aligned}$$

  and $\theta$ is a constant. In particular,

  $\bullet$ if $S^{per} \ne \emptyset$ then¹⁰

  $$\begin{aligned} \theta =1-\frac{1}{\sum _{{{\mathbf {v}}}\in V^+}h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})} \sum _{k \in K}\sum _{{{\mathbf {v}}}\in V_k^+}\frac{h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})}{|(\tau ^{k+1})'(a_{{{\mathbf {v}}}})(\tau ^{k+1})'(a_{-{{\mathbf {v}}}})|}; \end{aligned}$$

  (6)

  $\bullet$ otherwise $\theta =1$.

• 2. The first collision rate of $T_\varepsilon$ with respect to $m_L$

  $${{\hat{r}}}_L:=-\lim _{n\rightarrow \infty }\frac{1}{n}\ln m_L(X_\varepsilon ^{n-1})$$

  exists and

  $${{\hat{r}}}_L=\mu _0(H_\varepsilon ) \cdot \theta (1+o(1)).$$

• 3. In addition, there exists $C>0$, $\xi _\varepsilon >0$, with $\lim _{\varepsilon \rightarrow 0}\xi _\varepsilon =\theta$, such that for all $t>0$

  $$\left| \mu _0\Big \{t_\varepsilon \ge \frac{t}{\xi _\varepsilon \mu _0(H_\varepsilon )}\Big \}-e^{-t}\right| \le C(t\vee 1)e^{-t} C(L)\frac{L\mathrm{d} \varepsilon ^2}{\zeta (L)}\left| \ln \left( \frac{L\mathrm{d}\varepsilon ^2}{\zeta (L)}\right) \right| $$

  where $\zeta (L)$ is given in terms of the spectral gap of the rare event transfer operator (see (35)) and $C(L)=C(L,\bar{c}(\tau ) L^{-1/2})$ for $C(\cdot ,\cdot )$ defined by (4).

Remark 2.6

The formula (6) simplifies considerably when $d=1$. In that case $V^+=\{1\}$ and $S=\{(a_1, a_{-1})\}$. Supposing that k is the smallest integer such that $T_{2,0}^{k+1}(a_1, a_{-1})=(a_1, a_{-1})$, we get

$$\begin{aligned} \theta =1-\frac{1}{|(\tau ^{k+1})'(a_1)(\tau ^{k+1})'( a_{-1})|}, \end{aligned}$$

independently of the invariant density h of $\tau$.

Remark 2.7

In the third item of Theorem 2.5 we provide a bound on the first hitting time law that depends on the time scale t. We follow the spectral approach of [19] (inspired by [1]) and compute the precise L dependence of the error term. We note that the particular time-dependent bound we obtain improves as $t \rightarrow \infty$, so it is meaningful for large values t: our goal with this statement is to give a bound on the decay of the tail of the hitting time distribution. We note that for small values of t, a uniform bound such as the one given in [18] could be more appropriate.

The proof of Theorem 2.5 follows closely the abstract argument of [24]. However, since we are interested in how dynamical quantities scale with the lattice size L, we carefully track the L dependency of various quantities and error terms. Furthermore, we provide a precise formula for $\theta$ in this specific system (sometimes referred to in the literature as the extremal index), a result which is interesting on its own right. Tracking the L dependency in the argument of [24] is useful in particular in identifying explicitly the constants $\zeta (L)$ and C(L) in the third item of Theorem 2.5. Moreover, such understanding is essential to progress in the direction of computing the infinite-dimensional limit of the first collision rate per lattice unit. See Remark 4.5.

As the second item of Theorem 2.5 states, we have

$$\begin{aligned} {{\hat{r}}}_L= \mu _0(H_\varepsilon ) \cdot \theta (1+o(1)). \end{aligned}$$

(7)

On the other hand the proof of Theorem 2.5 also shows that

$$\begin{aligned} \mu _0(H_{\varepsilon })={ L}\Xi _{\varepsilon }(1+o(1))\sim \varepsilon ^2 { L}d \end{aligned}$$

(8)

and consequently ${{\hat{r}}}_L$, scales linearly in L. Thus, to obtain a first collision rate per lattice unit equation (8) suggests that a normalization by L is needed. This is made precise in our second main theorem.

Theorem 2.8

For any $L\ge 2$, the first collision rate per lattice unit is given by

$$\frac{1}{L} {{\hat{r}}}_L= \Xi _{\varepsilon } \cdot \theta (1+o(1))$$

as $\varepsilon \rightarrow 0$, where $\Xi _\varepsilon$ and $\theta$ are as in Theorem 2.5.

Remark 2.9

Note that the little o notation in Theorem 2.8 depends on L, so taking the limit $L \rightarrow \infty$ is not an obvious task. Again, see Remark 4.5.

3 Examples

In this section we provide examples to illustrate Theorem 2.5. In particular, we describe the structure of the set $H_{\varepsilon }$, its limit when $\varepsilon \rightarrow 0$ and the formula of $\theta$ (6) for some low-dimensional finite lattices.

3.1 One-Dimensional Lattice

Let $d=1$. We naturally identify the sites ${{\mathbf {p}}}_{1},\dots ,{{\mathbf {p}}}_{L}$ with the numbers $1\dots , L$. Now $V^+=\{1\}$ and $S=(a_{1},a_{-1})$, so to simplify notation we will denote $H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}}$ as $H_{\varepsilon }^{i}$ and $I_{{{\mathbf {p}}}_i,{{\mathbf {v}}}}$ as $I_{i}$, as it causes no confusion.

Fig. 2

The state space when $L=2$: the system with two interacting sites

Fig. 3

Interactions in case of $L=3$

Suppose $(a_1,a_{-1})$ is a $k+1$ periodic point of $T_{2,0}$. We now study lattices of some small number of sites.

• $L=2$: $H_{\varepsilon }^{1}=A_{\varepsilon ,1}\times A_{\varepsilon ,-1}$ and $H_{\varepsilon }^{2}=A_{\varepsilon ,-1}\times A_{\varepsilon ,1}$ are two disjoint squares, while $I_{1}=(a_{1},a_{-1})$ and $I_{2}=(a_{-1},a_{1})$. Both squares represent the single possible interaction between the two sites. See Fig. 2 for an illustration.

• $L=3$: $H_{\varepsilon }^{1}=A_{\varepsilon ,1}\times A_{\varepsilon ,-1}\times I$, $H_{\varepsilon }^{2}=I \times A_{\varepsilon ,1}\times A_{\varepsilon ,-1}$ and $H_{\varepsilon }^{3}=A_{\varepsilon ,-1} \times I \times A_{\varepsilon ,1}$, while their limits are the intervals $I_{1}=a_{1}\times a_{-1}\times I$, $I_{2}=I \times a_{1}\times a_{-1}$ and $I_{2}=a_{-1} \times I \times a_{1}$. Notice that since $A_{\varepsilon ,1}$ and $A_{\varepsilon ,-1}$ are disjoint, the sets $H_{\varepsilon }^i$, $i=1,2,3$ are pairwise disjoint. This translates to the fact that we cannot have multiple interactions simultaneously. The set $H_{\varepsilon }^1$ corresponds to the interaction between sites 1 and 2, $H_{\varepsilon }^2$ to 2 and 3 and $H_{\varepsilon }^3$ to 3 and 1. See Fig. 3 for an illustration.

• ${L}=4$: now $H_{\varepsilon }^{1}=A_{\varepsilon ,1}\times A_{\varepsilon ,-1}\times I \times I$, $H_{\varepsilon }^{2}=I \times A_{\varepsilon ,1}\times A_{\varepsilon ,-1}\times I$, $H_{\varepsilon }^{3}=I \times I \times A_{\varepsilon ,1}\times A_{\varepsilon ,-1}$ and $H_{\varepsilon }^{4}=A_{\varepsilon ,-1} \times I \times I \times A_{\varepsilon ,1}$. Notice that $H_{\varepsilon }^{1}$ and $H_{\varepsilon }^{3}$ have a non-empty intersection, and so does $H_{\varepsilon }^{2}$ and $H_{\varepsilon }^{4}$:

  $$\begin{aligned} H_{\varepsilon }^{1} \cap H_{\varepsilon }^{3}&=A_{\varepsilon ,1}\times A_{\varepsilon ,-1}\times A_{\varepsilon ,1}\times A_{\varepsilon ,-1} \\ H_{\varepsilon }^{2} \cap H_{\varepsilon }^{4}&=A_{\varepsilon ,-1}\times A_{\varepsilon ,1}\times A_{\varepsilon ,-1}\times A_{\varepsilon ,1}. \end{aligned}$$

  In terms of the coupled system, this means that sites 1 and 3 can simultaneously interact with their neighbors to the right, and so can sites 2 and 4. See Fig. 4.

According to Remark 2.6, we obtain in all three cases studied above

$$\begin{aligned} \theta =1-\frac{1}{|(\tau ^{k+1})'(a_1)(\tau ^{k+1})'(a_{-1})|}, \end{aligned}$$

independently of h. This is easy to prove when the sets $H_{\varepsilon }^i$ are pairwise disjoint, as in the case of $L=2$ and $L=3$.

Fig. 4

Possible simultaneous interactions when $d=1$, ${L}=4$

However, we have to be more careful when the intersections $H_{\varepsilon }^i \cap H_{\varepsilon }^j$, $i \ne j$ are not necessarily empty, although we eventually obtain the same formula. Explaining the reason for this through the example of $L=4$, the key fact the proof will depend on is that the limit sets $I_{1}=a_{1}\times a_{-1}\times I \times I$, $I_{2}=I \times a_{1}\times a_{-1}\times I$, $I_{3}=I \times I \times a_{1}\times a_{-1}$ and $I_{4}=a_{-1} \times I \times I \times a_{1}$ are two-dimensional sets, but the intersections

$$\begin{aligned} I_{1} \cap I_{3}&=a_{1}\times a_{-1}\times a_{1}\times a_{-1} \\ I_{2} \cap I_{4}&=a_{-1}\times a_{1}\times a_{-1}\times a_{1} \end{aligned}$$

are two points with zero two-dimensional Lebesgue measure,¹¹ which will prove to be enough for them to be negligible in the computation of $\theta$.

3.2 Two-Dimensional Lattice

Let $d=2$ and $L=4$. Denote the sites by ${{\mathbf {p}}}_1=(0,0)$, ${{\mathbf {p}}}_2=(1,0)$, ${{\mathbf {p}}}_3=(0,1)$ and ${{\mathbf {p}}}_4=(1,1)$. Then, $V^+=\{{{\mathbf {v}}},{{\mathbf {h}}}\}$, where ${{\mathbf {v}}}=(0,1)$ and ${{\mathbf {h}}}=(1,0)$. The set S has two elements: $(a_{{{\mathbf {v}}}},a_{-{{\mathbf {v}}}})$ and $(a_{{{\mathbf {h}}}},a_{-{{\mathbf {h}}}})$. Suppose the minimal period of $(a_{{{\mathbf {v}}}},a_{-{{\mathbf {v}}}})$ is $k+1$, while the minimal period of $(a_{{{\mathbf {h}}}},a_{-{{\mathbf {h}}}})$ is $\ell +1$. If we study the sets $I_{{{\mathbf {p}}}_i,{{\mathbf {v}}}_j}$, we find that no three can intersect, but there are eight pairs that can intersect. Namely, $I_{{{\mathbf {p}}}_1,{{\mathbf {h}}}}$ and $I_{{{\mathbf {p}}}_3,{{\mathbf {h}}}}$; $I_{{{\mathbf {p}}}_2,{{\mathbf {h}}}}$ and $I_{{{\mathbf {p}}}_4,{{\mathbf {h}}}}$; $I_{{{\mathbf {p}}}_1,{{\mathbf {v}}}}$ and $I_{{{\mathbf {p}}}_2,{{\mathbf {v}}}}$; $I_{{{\mathbf {p}}}_3,{{\mathbf {v}}}}$ and $I_{{{\mathbf {p}}}_4,{{\mathbf {v}}}}$; $I_{{{\mathbf {p}}}_2,{{\mathbf {h}}}}$ and $I_{{{\mathbf {p}}}_3,{{\mathbf {h}}}}$; $I_{{{\mathbf {p}}}_1,{{\mathbf {h}}}}$ and $I_{{{\mathbf {p}}}_4,{{\mathbf {h}}}}$; $I_{{{\mathbf {p}}}_1,{{\mathbf {v}}}}$ and $I_{{{\mathbf {p}}}_4,{{\mathbf {v}}}}$ and $I_{{{\mathbf {p}}}_2,{{\mathbf {v}}}}$ and $I_{{{\mathbf {p}}}_3,{{\mathbf {v}}}}$ (Fig. 5).

Fig. 5

Possible simultaneous interactions when $d=2$, ${L}=4$. Left: corresponds to the intersection of $H_{\varepsilon }^{{{\mathbf {p}}}_1,{{\mathbf {h}}}}$ and $H_{\varepsilon }^{{{\mathbf {p}}}_3,{{\mathbf {h}}}}$, $H_{\varepsilon }^{{{\mathbf {p}}}_2,{{\mathbf {h}}}}$ and $H_{\varepsilon }^{{{\mathbf {p}}}_4,{{\mathbf {h}}}}$, $H_{\varepsilon }^{{{\mathbf {p}}}_1,{{\mathbf {h}}}}$ and $H_{\varepsilon }^{{{\mathbf {p}}}_4,{{\mathbf {h}}}}$ and $H_{\varepsilon }^{{{\mathbf {p}}}_2,{{\mathbf {h}}}}$ and $H_{\varepsilon }^{{{\mathbf {p}}}_3,{{\mathbf {h}}}}$; right: corresponds to the intersection of $H_{\varepsilon }^{{{\mathbf {p}}}_1,{{\mathbf {v}}}}$ and $H_{\varepsilon }^{{{\mathbf {p}}}_2,{{\mathbf {v}}}}$, $H_{\varepsilon }^{{{\mathbf {p}}}_3,{{\mathbf {v}}}}$ and $H_{\varepsilon }^{{{\mathbf {p}}}_4,{{\mathbf {v}}}}$, $H_{\varepsilon }^{{{\mathbf {p}}}_1,{{\mathbf {v}}}}$ and $H_{\varepsilon }^{{{\mathbf {p}}}_4,{{\mathbf {v}}}}$ and $H_{\varepsilon }^{{{\mathbf {p}}}_2,{{\mathbf {v}}}}$ and $H_{\varepsilon }^{{{\mathbf {p}}}_3,{{\mathbf {v}}}}$

By Theorem 2.5 we obtain

$$\begin{aligned} \theta =1&-\frac{1}{h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})+h(a_{{{\mathbf {h}}}})h(a_{-{{\mathbf {h}}}})} \cdot \frac{h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})}{|(\tau ^{k+1})'(a_{{{\mathbf {v}}}})(\tau ^{k+1})'(a_{-{{\mathbf {v}}}})|} \\&+\frac{1}{h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})+h(a_{{{\mathbf {h}}}})h(a_{-{{\mathbf {h}}}})} \cdot \frac{h(a_{{{\mathbf {h}}}})h(a_{-{{\mathbf {h}}}})}{|(\tau ^{\ell +1})'(a_{{{\mathbf {h}}}})(\tau ^{\ell +1})'(a_{-{{\mathbf {h}}}})|}. \end{aligned}$$

So unless there is a close relation between the values at the points $a_{\pm {{\mathbf {v}}}}$ and $a_{\pm {{\mathbf {h}}}}$ of h and the derivative of $\tau$, this expression depends non-trivially on the density of the site dynamics $\tau$.

4 Proofs

We first compute a formula for the measure of $H_{\varepsilon }$ with respect to the $T_0$-invariant measure $\mu _0=\Pi _{i=1}^{L} h \cdot m$.

Lemma 4.1

$$\begin{aligned} \mu _0(H_{\varepsilon })= L\Xi _{\varepsilon }(1+o(1)) \end{aligned}$$

where

$$\begin{aligned} \Xi _{\varepsilon }=\sum _{{{\mathbf {v}}}\in V^+}\int _{A_{\varepsilon ,{{\mathbf {v}}}}}\int _{A_{\varepsilon ,-{{\mathbf {v}}}}}h(x)h(y)\mathrm{d}x\mathrm{d}y \end{aligned}$$

and $o(1) \rightarrow 0$ as $L\varepsilon ^2 \rightarrow 0$.

Proof

We start by making some observations that will considerably simplify the formula for $\mu _0(H_{\epsilon })$. First notice that for each ${{\mathbf {p}}}\in \Lambda$ we must have $H_\varepsilon ^{{{\mathbf {p}}},{{\mathbf {v}}}_i} \cap H_\varepsilon ^{{{\mathbf {p}}},{{\mathbf {v}}}_j} =\emptyset$ for ${{\mathbf {v}}}_i \ne {{\mathbf {v}}}_j$, as the state of site ${{\mathbf {p}}}$ cannot be simultaneously in $A_{\varepsilon ,{{\mathbf {v}}}_i}$ and $A_{\varepsilon ,{{\mathbf {v}}}_j}$. Notice also that at most L/2 sets $H_\varepsilon ^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}$ can intersect: indeed, $x \in H_\varepsilon ^{{{\mathbf {p}}},{{\mathbf {v}}}}$ implies that $x_{{\mathbf {p}}}\in A_{\varepsilon ,{{\mathbf {v}}}}$ for some ${{\mathbf {v}}}\in V^{\pm }$ and $x_{{{\mathbf {p}}}+{{\mathbf {v}}}}$ is in $A_{\varepsilon ,-{{\mathbf {v}}}}$ where $-{{\mathbf {v}}}\in V^{\mp }$. This shows that $x \in H_\varepsilon ^{{{\mathbf {p}}},{{\mathbf {v}}}}$ implies that $x \notin H_\varepsilon ^{{{\mathbf {p}}}+{{\mathbf {v}}},{{\mathbf {v}}}'}$ for any ${{\mathbf {v}}}\in V^{\pm }$. So we can have x in at most L/2 sets $H_\varepsilon ^{{{\mathbf {p}}},{{\mathbf {v}}}}$ simultaneously.

Let $\Lambda ^k=\{({{\mathbf {p}}}_1,\dots ,{{\mathbf {p}}}_k): {{\mathbf {p}}}_i \in \Lambda ,\, i=1,\dots ,k\}$ and $(V^+)^k\!=\!\{({{\mathbf {v}}}_1,\dots ,{{\mathbf {v}}}_k): {{\mathbf {v}}}_i \in V^+,\, i=1,\dots ,k\}$. Fix an ordering on the elements of $\Lambda$ and denote it by <. Then, according to de Moivre’s inclusion–exclusion principle and using the above observations we can write

$$\begin{aligned} \mu _0(H_{\varepsilon })&=\sum _{{{\mathbf {p}}}\in \Lambda }\sum _{{{\mathbf {v}}}\in V^+}\mu _0(H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}})-\sum _{\begin{array}{c} {{\mathbf {p}}}_1,{{\mathbf {p}}}_2 \in \Lambda \\ {{\mathbf {p}}}_1< {{\mathbf {p}}}_2 \end{array}}\sum _{{{\mathbf {v}}}_1,{{\mathbf {v}}}_2 \in V^+}\mu _0(H_{\varepsilon }^{{{\mathbf {p}}}_1,{{\mathbf {v}}}_1} \cap H_{\varepsilon }^{{{\mathbf {p}}}_2,{{\mathbf {v}}}_2}) \nonumber \\&\quad +\sum _{\begin{array}{c} {{\mathbf {p}}}_1,{{\mathbf {p}}}_2,{{\mathbf {p}}}_3 \in \Lambda \\ {{\mathbf {p}}}_1< {{\mathbf {p}}}_2< {{\mathbf {p}}}_3 \end{array}}\sum _{{{\mathbf {v}}}_1,{{\mathbf {v}}}_2,{{\mathbf {v}}}_3 \in V^+}\mu _0(H_{\varepsilon }^{{{\mathbf {p}}}_1,{{\mathbf {v}}}_1} \cap H_{\varepsilon }^{{{\mathbf {p}}}_2,{{\mathbf {v}}}_2}\cap H_{\varepsilon }^{{{\mathbf {p}}}_3,{{\mathbf {v}}}_3})-\dots \nonumber \\&=\sum _{k=1}^{\lfloor {L}/2 \rfloor }(-1)^{k+1}\sum _{\begin{array}{c} \{{{\mathbf {p}}}_i\}_{i=1}^{k} \in \Lambda ^k \\ {{\mathbf {p}}}_i < {{\mathbf {p}}}_{i+1} \end{array}}\sum _{\{{{\mathbf {v}}}_i\}_{i=1}^{k} \in (V^+)^k}\mu _0\left( \bigcap _{i=1}^k H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}\right) . \end{aligned}$$

(9)

Note that some intersections might be empty, but selecting the non-empty ones would make the formula unnecessarily complicated and would provide little benefit at this point. We also note here that the principle applied to a finite measure of density F with respect to the Lebesgue measure implies that

$$\begin{aligned} \int _{H_{\varepsilon }} F(x) \mathrm{d}x&=\sum _{\ell =1}^{\lfloor {L}/2 \rfloor }(-1)^{\ell +1}\sum _{\begin{array}{c} \{{{\mathbf {p}}}_i\}_{i=1}^{\ell } \in \Lambda ^{\ell } \\ {{\mathbf {p}}}_i < {{\mathbf {p}}}_{i+1} \end{array}}\sum _{\{{{\mathbf {v}}}_i\}_{i=1}^{\ell } \in (V^+)^{\ell }}\int _{\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}}F(x)\mathrm{d}x. \end{aligned}$$

(10)

For $k=1$ we get

$$\begin{aligned} \sum _{{{\mathbf {p}}}\in \Lambda }\sum _{{{\mathbf {v}}}\in V^+}\mu _0(H_\varepsilon ^{{{\mathbf {p}}},{{\mathbf {v}}}})={L}\sum _{{{\mathbf {v}}}\in V^+}\int _{A_{\varepsilon ,{{\mathbf {v}}}}}\int _{A_{\varepsilon ,-{{\mathbf {v}}}}}h(x_1)h(x_2)\mathrm{d}x_1\mathrm{d}x_2={L}\Xi _{\varepsilon }. \end{aligned}$$

(11)

where

$$\begin{aligned} \Xi _{\varepsilon }=\sum _{{{\mathbf {v}}}\in V^+}\int _{A_{\varepsilon ,{{\mathbf {v}}}}}\int _{A_{\varepsilon ,-{{\mathbf {v}}}}}h(x_1)h(x_2)\mathrm{d}x_1\mathrm{d}x_2. \end{aligned}$$

Now let $k > 1$. We remind the reader that we denoted the density of the site dynamics $\tau$ by h. Since h is of bounded variation, there exists ${\overline{C}} > 0$ such that $h \le {\overline{C}}$. If the intersection $\cap _{i=1}^k H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}$ is non-empty we can write

$$\begin{aligned}&\mu _0\left( \bigcap _{i=1}^k H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}\right) \\&\quad =\int _{A_{\varepsilon ,{{\mathbf {v}}}_1}}\int _{A_{\varepsilon ,-{{\mathbf {v}}}_1}}\dots \int _{A_{\varepsilon ,{{\mathbf {v}}}_k}}\int _{A_{\varepsilon ,-{{\mathbf {v}}}_k}}\prod _{i=1}^{2k}h(x_i)\mathrm{d}x_1\dots \mathrm{d}x_{2k} \\&\quad =\mu _0(H_{\varepsilon }^{{{\mathbf {p}}}_1,{{\mathbf {v}}}_1})\int _{A_{\varepsilon ,{{\mathbf {v}}}_2}}\int _{A_{\varepsilon ,-{{\mathbf {v}}}_2}}\dots \int _{A_{\varepsilon ,{{\mathbf {v}}}_k}}\int _{A_{\varepsilon ,-{{\mathbf {v}}}_k}}\prod _{i=3}^{2k}h(x_i)\mathrm{d}x_3\dots \mathrm{d}x_{2k} \end{aligned}$$

which can be bounded by

$$\begin{aligned} \mu _0\left( \bigcap _{i=1}^k H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}\right) \le {\overline{C}}^{2(k-1)} \varepsilon ^{2(k-1)}\mu _0(H_{\varepsilon }^{{{\mathbf {p}}}_1,{{\mathbf {v}}}_1}) \end{aligned}$$

(12)

Now first notice that

$$\begin{aligned} \mu _0(H_{\varepsilon })=\mu _0\left( \bigcup _{{{\mathbf {p}}}\in \Lambda }\bigcup _{{{\mathbf {v}}}\in V^+}H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}}\right) \le \sum _{{{\mathbf {p}}}\in \Lambda } \sum _{{{\mathbf {v}}}\in V^+}\mu _0(H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}}) = {L}\Xi _{\varepsilon } \end{aligned}$$

by (11).

For the lower bound we use (11) for $k=1$ and (12) for each $k > 1$:

$$\begin{aligned}&\sum _{k=1}^{\lfloor {L}/2 \rfloor }(-1)^{k+1}\sum _{\begin{array}{c} \{{{\mathbf {p}}}_i\}_{i=1}^{k} \in \Lambda ^k \\ {{\mathbf {p}}}_i< {{\mathbf {p}}}_{i+1} \end{array}}\sum _{\{{{\mathbf {v}}}_i\}_{i=1}^{k} \in (V^+)^k}\mu _0\left( \bigcap _{i=1}^k H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}\right) \\&\quad ={L}\Xi _{\varepsilon }+\sum _{k=2}^{\lfloor {L}/2 \rfloor }(-1)^{k+1}\sum _{\begin{array}{c} \{{{\mathbf {p}}}_i\}_{i=1}^{k} \in \Lambda ^k \\ {{\mathbf {p}}}_i< {{\mathbf {p}}}_j \end{array}}\sum _{\{{{\mathbf {v}}}_i\}_{i=1}^{k} \in (V^+)^k}\mu _0\left( \bigcap _{i=1}^k H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}\right) \\&\quad \ge {L}\Xi _{\varepsilon }-\left( \sum _{k=2}^{\lfloor {L}/2 \rfloor }\sum _{\begin{array}{c} \{{{\mathbf {p}}}_i\}_{i=1}^{k} \in \Lambda ^k \\ {{\mathbf {p}}}_i< {{\mathbf {p}}}_{i+1} \end{array}}\sum _{\{{{\mathbf {v}}}_i\}_{i=1}^{k} \in (V^+)^k}\mu _0\left( \bigcap _{i=1}^k H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}\right) \right) \\&\quad \ge {L}\Xi _{\varepsilon }-\sum _{k=2}^{\lfloor {L}/2 \rfloor }\sum _{\begin{array}{c} \{{{\mathbf {p}}}_i\}_{i=1}^{k} \in \Lambda ^k \\ {{\mathbf {p}}}_i < {{\mathbf {p}}}_{i+1} \end{array}}\sum _{\{{{\mathbf {v}}}_i\}_{i=1}^{k} \in (V^+)^k}{\overline{C}}^{2(k-1)} \varepsilon ^{2(k-1)} \mu _0(H_{\varepsilon }^{{{\mathbf {p}}}_1,{{\mathbf {v}}}_1}) \\&\quad = {L}\Xi _{\varepsilon }-\sum _{k=2}^{\lfloor {L}/2 \rfloor }\left( {\begin{array}{c}{L}\\ k-1\end{array}}\right) d^{k-1}\sum _{{{\mathbf {p}}}_1 \in \Lambda }\sum _{{{\mathbf {v}}}_1 \in V^+}{\overline{C}}^{2(k-1)} \varepsilon ^{2(k-1)}\mu _0(H_{\varepsilon }^{{{\mathbf {p}}}_1,{{\mathbf {v}}}_1}) \\&\quad = {L}\Xi _{\varepsilon }-\left( \sum _{{{\mathbf {p}}}_1 \in \Lambda }\sum _{{{\mathbf {v}}}_1 \in V^+} \mu _0(H_{\varepsilon }^{{{\mathbf {p}}}_1,{{\mathbf {v}}}_1}) \right) \sum _{k=2}^{\lfloor {L}/2 \rfloor }\left( {\begin{array}{c}{L}\\ k-1\end{array}}\right) d^{k-1}{\overline{C}}^{2(k-1)}\varepsilon ^{2(k-1)}, \end{aligned}$$

which can be written as

$$\begin{aligned} \mu _0(H_{\varepsilon }) \ge {L}\Xi _{\varepsilon }\left( 1-\sum _{k=2}^{\lfloor {L}/2 \rfloor }\left( {\begin{array}{c}{L}\\ k-1\end{array}}\right) d^{k-1}{\overline{C}}^{2(k-1)} \varepsilon ^{2(k-1)} \right) . \end{aligned}$$

The first term in the sum $\sum _{k=2}^{\lfloor {L}/2 \rfloor }\left( {\begin{array}{c}{L}\\ k-1\end{array}}\right) d^{k-1}{\overline{C}}^{2(k-1)}\varepsilon ^{2(k-1)}$ is ${L}d{\overline{C}}^2\varepsilon ^2$. This is the leading term assuming ${L}\varepsilon ^2$ is sufficiently small, so

$$\begin{aligned} \mu _0(H_{\varepsilon }) \ge {L}\Xi _{\varepsilon }(1-C{L}\mathrm{d}\varepsilon ^2)={L}\Xi _{\varepsilon }(1+o(1)). \end{aligned}$$

$\square$

We now turn to the study of the transfer operators ${{\hat{P}}}_{\varepsilon }$. Observe that $T_0=\tau \times \cdots \times \tau$ satisfies conditions (PE1)-(PE4) of [32] for any $\alpha \le \beta$. Indeed, the sets $U_i$ are given by the rectangles $I_{{\underline{i}}}=I_{i_1}\times \cdots \times I_{i_{ L}}$ where ${\underline{i}}=(i_1,\dots ,i_{L}) \in \{1,\dots ,k\}^L$. We further observe that since $\tau$ has full branches, that is, $\tau (I_{j})=I$ for each $j=1,\dots ,k$, we have $T_0(I_{{\underline{i}}})=I^L$ for all ${\underline{i}} \in \{1,\dots ,k\}^L$. We are going to use the notation $T_{0,{\underline{i}}}^{-1}=(T_0|_{I_{{\underline{i}}}})^{-1}$ for the inverse branches, defined on $I^L$. On the sets $I_{{\underline{i}}}$, the map $T_0$ is $C^{1+\beta }$ and can be extended to the boundary as a function of the same smoothness. This means that the first requirement of (PE2) about the piecewise smoothness of $T_0$ is satisfied, while the second requirement asking that for all ${\underline{i}}$, $\omega \le \omega _0$, $z \in I^L$ and $x,y \in B_{\omega }(z) \cap I^L$

$$\begin{aligned} |\det D_x T_{0,{\underline{i}}}^{-1}-\det D_y T_{0,{\underline{i}}}^{-1}| \le c_L|\det D_z T_{0,{\underline{i}}}^{-1}|\omega ^{\alpha }, \end{aligned}$$

(13)

also holds, and we can choose $c_L=c(\tau )\sqrt{L}$. Finally, (PE4) requiring that for all $x,y \in I^L$

$$\begin{aligned} d(T_{0,{\underline{i}}}^{-1}(x),T_{0,{\underline{i}}}^{-1}(y)) \le sd(x,y) \end{aligned}$$

(14)

holds with $s:=\frac{1}{\min |\tau '|}$.

Lemma 4.2

Let $n \in {\mathbb {N}}$ and $\varepsilon \ge 0$. There exist constants $\sigma \in (0,1)$, $D(L,\varepsilon , \omega _0) > 0$ such that

$$\begin{aligned} \Vert {\hat{P}}_{\varepsilon }^nf\Vert _1&\le \Vert f\Vert _1 \end{aligned}$$

(15)

$$\begin{aligned} \Vert {\hat{P}}_{\varepsilon }^nf\Vert _{\alpha }&\le \sigma ^n \Vert f\Vert _{\alpha }+ D(L,\varepsilon , \omega _0)\Vert f\Vert _1. \end{aligned}$$

(16)

Proof

Since

$$\begin{aligned} \Vert {\hat{P}}_{\varepsilon }f\Vert _1=\int _X |P_{T_0}(1_{X_{\varepsilon }^0}f)| \le \int _X |1_{X_{\varepsilon }^0}f| \le \int _X |f|, \end{aligned}$$

we have $\Vert {\hat{P}}_{\varepsilon }^nf\Vert _1 \le \Vert f\Vert _1$ for all $n \in {\mathbb {N}}$ and $\varepsilon \ge 0$.

To prove (16) we show the following:

$$\begin{aligned} \Vert P_{T_0}f\Vert _{\alpha }&\le \sigma \Vert f\Vert _{\alpha }+K\Vert f\Vert _1 ; \end{aligned}$$

(17)

$$\begin{aligned} \Vert 1_{X_{\varepsilon }^0}f\Vert _{\alpha }&\le \Vert f\Vert _{\alpha }+ 8 Ld\varepsilon \omega _0^{1-\alpha }\Vert f\Vert _1, \end{aligned}$$

(18)

Writing $B(L,\varepsilon ,\omega _0)=\sigma \cdot 8 Ld\varepsilon \omega _0^{1-\alpha }+K$, the inequalities (17) and (18) imply that

$$\begin{aligned} \Vert {\hat{P}}_{\varepsilon }f\Vert _{\alpha } \le \sigma \Vert f\Vert _{\alpha }+ B(L,\varepsilon ,\omega _0)\Vert f\Vert _1, \end{aligned}$$

(19)

which proves (16) by iteration with $D(L,\varepsilon ,\omega _0)=B(L,\varepsilon ,\omega _0)\sum _{k=0}^{\infty }\sigma ^k$.

We first prove (17), arguing similarly as in Lemma 4.1 of [32], but in a simplified setting since $T_0$ is piecewise onto.¹²

$$\begin{aligned} {{\,\mathrm{osc}\,}}(P_{T_0}f,B_{\omega }(x))&\le \sum _{{\underline{i}}}{{\,\mathrm{osc}\,}}\left( \left( \frac{f}{|\det DT_0|} \right) \circ T_{0,{\underline{i}}}^{-1} \cdot {\mathbf {1}}_{T_0 (I_{{\underline{i}}})},B_{\omega }(x) \right) \\&=\sum _{{\underline{i}}}{{\,\mathrm{osc}\,}}\left( \left( \frac{f}{|\det DT_0|} \right) \circ T_{0,{\underline{i}}}^{-1},B_{\omega }(x) \right) \\&\le \sum _{{\underline{i}}}{{\,\mathrm{osc}\,}}\left( \frac{f}{|\det DT_0|} ,I_{{\underline{i}}} \cap T_0^{-1}B_{\omega }(x) \right) \\&\le \sum _{{\underline{i}}}{{\,\mathrm{osc}\,}}\left( \frac{f}{|\det DT_0|} ,I_{{\underline{i}}} \cap B_{s\omega }(y_{{\underline{i}}}) \right) , \end{aligned}$$

setting $y_{{\underline{i}}}=T_{0,{\underline{i}}}^{-1}(x)$ and remembering that s was defined as the reciprocal of the minimum expansion of $\tau$ (and consequently $T_0$). It is easy to prove that for $g_1, g_2 \in L^{\infty }$, $g_2>0$ and some S Borel set

$$\begin{aligned} {{\,\mathrm{osc}\,}}(g_1g_2,S) \le {{\,\mathrm{osc}\,}}(g_1,S){{\,\mathrm{ess\,sup}\,}}_S g_2 + {{\,\mathrm{osc}\,}}(g_2,S){{\,\mathrm{ess\,inf}\,}}_S |g_1| \end{aligned}$$

holds. (The reader is invited to check Proposition 3.2 (iii) in [32].) This implies that

$$\begin{aligned} {{\,\mathrm{osc}\,}}(P_{T_0}f,B_{\omega }(x))&\le \sum _{{\underline{i}}}{{\,\mathrm{osc}\,}}\left( f,B_{s\omega }(y_{{\underline{i}}}) \right) {{\,\mathrm{ess\,sup}\,}}_{I_{{\underline{i}}}\cap B_{s\omega }(y_{{\underline{i}}})}\frac{1}{|\det DT_0|}\\&\quad +{{\,\mathrm{osc}\,}}\left( \frac{1}{|\det DT_0|},I_{{\underline{i}}}\cap B_{s\omega }(y_{{\underline{i}}}) \right) {{\,\mathrm{ess\,inf}\,}}_{ B_{s\omega }(y_{{\underline{i}}})}|f| \\&\le \sum _{{\underline{i}}}(1+ c_Ls^{\alpha }\omega ^{\alpha }){{\,\mathrm{osc}\,}}\left( f,B_{s\omega }(y_{{\underline{i}}}) \right) \frac{1}{|\det DT_0|}(y_{{\underline{i}}})\\&\quad + c_Ls^{\alpha }\omega ^{\alpha }\frac{1}{|\det DT_0|}(y_{{\underline{i}}}) |f|(y_{{\underline{i}}}) \\&\le (1+ c_Ls^{\alpha }\omega ^{\alpha })P_{T_0}({{\,\mathrm{osc}\,}}(f,B_{s\omega }( x)))+ c_Ls^{\alpha }\omega ^{\alpha }P_{T_0}|f| (x), \end{aligned}$$

where $c_L$ and s are given by (13) and (14), respectively. Integrating this on ${{\mathbb {R}}}^L$ we get

$$\begin{aligned} \int _{{{\mathbb {R}}}^L} {{\,\mathrm{osc}\,}}(P_{T_0}f,B_{\omega }(\cdot )) \le (1+ c_Ls^{\alpha }\omega ^{\alpha })\int _{{{\mathbb {R}}}^L}{{\,\mathrm{osc}\,}}(f,B_{s\omega }(\cdot ))+ c_Ls^{\alpha }\omega ^{\alpha }\int _{{{\mathbb {R}}}^L}|f|, \end{aligned}$$

giving

$$\begin{aligned} |P_{T_0}f|_{\alpha } \le (1+c_Ls^{\alpha }\omega _0^{\alpha })s^{\alpha }|f|_{\alpha }+ c_Ls^{\alpha }\Vert f\Vert _1 \end{aligned}$$

which gives (17) provided that $\omega _0$ is small enough. We note that choosing $\omega _0^{\alpha } \sim L^{-1/2}$ allows to choose $c_Ls^{\alpha }\omega _0^{\alpha }$, thus $\sigma$ and K uniformly in L. Assume from now on that we fix a choice of ${{\bar{c}}}$ and let $\omega _0$ such that $\omega _0^{\alpha }= \bar{c}L^{-1/2}$, $c_Ls^{\alpha }\omega _0^{\alpha }=\delta$ for some fixed $\delta >0$ sufficiently small so that $(1+\delta )s^{\alpha }=:\sigma < 1$.

To prove (18), we are going to use that for $f,g \in V_{\alpha }$

$$\begin{aligned} \Vert fg\Vert _{\alpha } \le |f|_{\alpha }\Vert g\Vert _{\infty }+\Vert f\Vert _{1}|g|_{\alpha }+\Vert f\Vert _{1}\Vert g\Vert _{\infty }, \end{aligned}$$

(20)

as argued in the proof of Proposition 3.4, [32]. Let $f \in V_{\alpha }$ and $g=1_{X_{\varepsilon }^0}$. Recall that $X_{\varepsilon }^0=X\backslash H_{\varepsilon }$. Notice that ${{\,\mathrm{osc}\,}}(g,B_{\omega }(x)) = 1$ if $x \in B_{\omega }(\partial X_{\varepsilon }^0)=B_{\omega }(\partial H_{\varepsilon })$. Recall also that $H_{\varepsilon }=\cup _{{{\mathbf {v}}}\in V^+}\cup _{{{\mathbf {p}}}\in \Lambda } H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}}$, where the union is not necessarily disjoint. Nevertheless, ${{\,\mathrm{osc}\,}}(g,B_{\omega }(x))$ is either 1 or 0 for $x \in B_{\omega }(\partial H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}})$ for each ${{\mathbf {p}}}\in \Lambda$ and ${{\mathbf {v}}}\in V^+$, and 0 elsewhere. This implies that

$$\begin{aligned} \int _{{{\mathbb {R}}}^{ L}}{{\,\mathrm{osc}\,}}(g,B_{\omega }(x))\mathrm{d}x&\le \sum _{{{\mathbf {v}}}\in V^+}\sum _{{{\mathbf {p}}}\in \Lambda } m_L(B_{\omega }(\partial H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}})) \\&\le Ld((\varepsilon +2\omega )^2-{(\varepsilon -2\omega )^2})= 8Ld\varepsilon \omega , \end{aligned}$$

and thus

$$\begin{aligned} \sup _{0 < \omega \le \omega _0}\omega ^{-\alpha }\int _{{{\mathbb {R}}}^{ L}}{{\,\mathrm{osc}\,}}(g,B_{\omega }(x))\mathrm{d}x \le 8 Ld\varepsilon \omega _0^{1-\alpha }. \end{aligned}$$

So $g \in V_{\alpha }$ indeed, and by (20) we obtain

$$\begin{aligned} \Vert f1_{X_{\varepsilon }^0}\Vert _{\alpha }&\le |f|_{\alpha }\Vert 1_{X_{\varepsilon }^0}\Vert _{\infty }+\Vert f\Vert _1 |1_{X_{\varepsilon }^0}|_{\alpha }+\Vert f\Vert _1 \Vert 1_{X_{\varepsilon }^0}\Vert _{\infty } \\&\le |f|_{\alpha } + \Vert f\Vert _1 + \Vert f\Vert _1 |1_{X_{\varepsilon }^0}|_{\alpha } \\&\le \Vert f\Vert _{\alpha }+8 Ld\varepsilon \omega _0^{1-\alpha }\Vert f\Vert _{1}. \end{aligned}$$

$\square$

Our choice of $\omega _0$ implied that $\omega _0^{\alpha }L^{1/2} \sim 1$. In turn, $L\omega _0^{1-\alpha } \sim L^{\frac{3}{2}-\frac{1}{2\alpha }}<1$ for sufficiently small $\alpha$. We now fix the value of $\alpha$ such that $\frac{3}{2}-\frac{1}{2\alpha } < 0$, that is, $\alpha \le \min \{\beta ,1/3\}$. Then, $B(L,\varepsilon ,\omega _0) \le 2\sigma +O(\varepsilon )+K$, in particular

$$\begin{aligned} B(L,\varepsilon ,\omega _0) \le C \quad \text {for }\varepsilon \text { sufficiently small.} \end{aligned}$$

(21)

Remark 4.3

Note that $f\ge 0$ with $\int _Xf\mathrm{d}x=1$, there exists an $n\ge 0$, such that

$$\Vert {\hat{P}}^n_{\varepsilon }f\Vert _1 =\int _{X}P^n_{T_{0}}(1_{X^{n-1}_{\varepsilon }} f)\mathrm{d}x=\int _{X}1_{X^{n-1}_{\varepsilon }} f\mathrm{d}x<\int _{X} f\mathrm{d}x=\Vert f\Vert _1.$$

Lemma 4.4

Let $C(L,\omega _0)$ be the constant defined by (4). Then,

$$\sup _{\Vert f\Vert _{\alpha }\le 1} \Vert ({\hat{P}}_{\varepsilon }-P_{T_0})f\Vert _1\le C(L,\omega _0)\cdot L\mathrm{d}\varepsilon ^2.$$

Proof

Using (3) we get:

$$\begin{aligned} \begin{aligned} \Vert ({\hat{P}}_{\varepsilon }-P_{T_0})f\Vert _1&= \Vert P_{T_0}(1_{X_\varepsilon }f)-P_{T_0}f\Vert _1\le ||1_{ H_\varepsilon }f||_1\\&\le | H_\varepsilon |\cdot \Vert f\Vert _\infty \le C(L,\omega _0) | H_\varepsilon |\cdot \Vert f\Vert _{\alpha }. \end{aligned} \end{aligned}$$

The proof is then completed by noting that $| H_\varepsilon |= \left| \cup _{{{\mathbf {v}}}\in V^+}\cup _{{{\mathbf {p}}}\in \Lambda } H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}}\right| \le L\mathrm{d}\varepsilon ^2$. $\square$

Proof of Theorem 2.5

By Lemma 4.2 and the fact that the unit ball of $(V_{\alpha },\Vert \cdot \Vert _{\alpha })$ is compact in the weak norm $\Vert \cdot \Vert _1$, the result of [17] implies the residual spectrum of each operator ${\hat{P}}_{\varepsilon }$, $\varepsilon \ge 0$ is contained in $\{z \in {\mathbb {C}}: |z| \le \sigma \}$. The map $T_0$ admits an invariant density $\rho _0(x)=\Pi _{i=1}^{L} h(x_i)$. Since h is mixing, $\rho _0$ is mixing. This with the quasi-compactness of $P_0$ implies $P_0$ has a spectral gap when acting on $V_\alpha$. By Lemmas 4.2 and 4.4, and the abstract perturbation result of [20] the operator ${\hat{P}}_{\varepsilon }:V_\alpha \rightarrow V_\alpha$, $\varepsilon >0$ has a spectral gap with a leading simple eigenvalue $\lambda _{\varepsilon }$ and by Remark 4.3$\lambda _\varepsilon \in (\sigma ,1)$ holds. Moreover, there is $\rho _{\varepsilon }\in V_\alpha$, a probability Borel measure $\nu _{\varepsilon }$ and linear operators $Q_{\varepsilon }:V_\alpha \rightarrow V_\alpha$ such that

$$\begin{aligned} \lambda _{\varepsilon }^{-1}{\hat{P}}_{\varepsilon }=\rho _{\varepsilon }\otimes \nu _{\varepsilon }+ Q_{\varepsilon } \end{aligned}$$

(22)

with

• ${\hat{P}}_{\varepsilon } \rho _{\varepsilon }=\lambda _{\varepsilon }\rho _{\varepsilon },\, \nu _{\varepsilon }{\hat{P}}_{\varepsilon }=\lambda _{\varepsilon }\nu _{\varepsilon },\, Q_{\varepsilon }\rho _{\varepsilon }=0,\, \nu _{\varepsilon }Q_{\varepsilon }=0$;

• $\sum _{n=0}^{\infty }\sup _{\varepsilon }\Vert Q_{\varepsilon }^n\Vert _{\alpha }<\infty$;

• $\int \rho _{\varepsilon }\mathrm{d}x=1$ and $\sup _{\varepsilon }\Vert \rho _{\varepsilon }\Vert _{\alpha }<\infty$.

We assume the normalization $\nu _{0}(\rho _{\varepsilon })=1$ for all $\varepsilon \ge 0$.

We now state some more specific bounds for the above defined quantities. Observing the second item, the spectral radius of $Q_{\varepsilon }$ is some $K_\varepsilon (L)\in (0,1)$, so we have

$$\begin{aligned} \sum _{k=0}^{\infty }\Vert Q_{\varepsilon }^k\Vert _\alpha \le \frac{1}{1-K_\varepsilon (L)}\le \frac{1}{1-K(L)} \end{aligned}$$

(23)

for some K(L) that is independent of $\varepsilon .$ Moreover, for the last item, since $\sigma$ is a uniform upper bound on the essential spectral radii of ${{\hat{P}}}_\varepsilon$, we can fix $r\in (\sigma ,1)$ and $\delta _0\in (0,1-r)$, both r and $\delta _0$ are independent of L, so that B(0, r) contains the essential spectrum of ${{\hat{P}}}_\varepsilon$ and $\lambda _\varepsilon \in B(1,\delta _0)$. Then, using the fact that $\rho _\varepsilon$ is the nonnegative eigenfunction corresponding to the dominant real eigenvalue $\lambda _\varepsilon$, and $\int \rho _\varepsilon =1$, by Lemma 4.2, we have

$$\lambda _\varepsilon \Vert \rho _\varepsilon \Vert _\alpha =\Vert {{\hat{P}}}_\varepsilon \rho _\varepsilon \Vert _\alpha \le \sigma \Vert \rho _\varepsilon \Vert _\alpha + B(L, \varepsilon ,\omega _0).$$

Therefore,

$$\begin{aligned} \Vert \rho _\varepsilon \Vert _\alpha \le \frac{ B(L, \varepsilon ,\omega _0)}{\lambda _\varepsilon -\sigma }\le \frac{ C}{r-\sigma } \quad \text {for }\varepsilon \text { sufficiently small} \end{aligned}$$

(24)

by (21).

To obtain the first-order approximation of $\lambda _\varepsilon$ we proceed as in [24]. Notice that

$$\begin{aligned} \int _X({\hat{P}}_{\varepsilon }-P_{T_0})f \mathrm{d}x=\int _X 1_{ H_\varepsilon }f \mathrm{d}x. \end{aligned}$$

Consequently,

$$\begin{aligned} \eta _{\varepsilon }:=\sup _{\Vert f\Vert _{\alpha }\le 1}|\int _X({\hat{P}}_{\varepsilon }-P_{T_0})f \mathrm{d}x|\le C(L,\omega _0)\int _X 1_{ H_\varepsilon }\mathrm{d}x \rightarrow 0\text { as }\varepsilon \rightarrow 0, \end{aligned}$$

where we used that $\Vert f\Vert _{\infty } \le C(L,\omega _0)\Vert f\Vert _{\alpha }$ by inequality (3). In particular we obtained that

$$\begin{aligned} \eta _{\varepsilon } =O(C(L,\omega _0)L\varepsilon ^2) \end{aligned}$$

(25)

As $X^0_{\varepsilon }\cup H_{\varepsilon } = X$, it holds that $|1_{H_{\varepsilon }}|_{\alpha }=|1_{X^0_{\varepsilon }}|_{\alpha }$. Thus, we obtain the analogue of (18)

$$\begin{aligned} \Vert 1_{H_{\varepsilon }}f\Vert _{\alpha }\le \Vert f\Vert _{\alpha }+ 8L\mathrm{d}\varepsilon \omega _0^{1-\alpha }\Vert f\Vert _1, \end{aligned}$$

and in particular

$$\begin{aligned} \Vert 1_{H_{\varepsilon }}\rho _0\Vert _{\alpha }\le \Vert \rho _0\Vert _{\alpha }+ 8L\mathrm{d}\varepsilon \omega _0^{1-\alpha }. \end{aligned}$$

Using this and (17) we obtain

$$\begin{aligned} \begin{aligned} \Vert P_{T_0}(1_{ H_\varepsilon }\rho _0)\Vert _{\alpha }&\le \sigma \Vert 1_{ H_\varepsilon }\rho _0\Vert _\alpha +K\\&\le \sigma \left( \Vert \rho _0\Vert _\alpha +8L\mathrm{d}\varepsilon \omega _0^{1-\alpha }\right) + K\\&= \sigma \Vert \rho _0\Vert _\alpha + B(L,\varepsilon ,\omega _0)\\&\le \frac{C\sigma }{r-\sigma } + C=\frac{Cr}{r-\sigma }=:C_1 \end{aligned} \end{aligned}$$

Recall further that ${\underline{C}}:=\inf _x h(x)>0$. Then,

$$\begin{aligned}&\eta _{\varepsilon }\cdot \Vert ({\hat{P}}_{\varepsilon }-P_{T_0})\rho _0\Vert _{\alpha }\\&\quad \le C(L,\omega _0)\int _X 1_{H_\varepsilon }\mathrm{d}x\cdot \Vert P_{T_0}(1_{ H_\varepsilon }\rho _0)\Vert _{\alpha } \le C_1C(L,\omega _0) \int _X 1_{ H_\varepsilon }\mathrm{d}x \\&\quad \le \frac{ C_1C(L,\omega _0)}{{\underline{C}}}\int _X 1_{ H_\varepsilon }\rho _0\mathrm{d}x=\frac{ C_1C(L,\omega _0)}{{\underline{C}}}\cdot \int _X({\hat{P}}_{\varepsilon }-P_{T_0})\rho _0 \mathrm{d}x \end{aligned}$$

Thus, we obtained

$$\begin{aligned} \eta _{\varepsilon }\cdot \Vert ({\hat{P}}_{\varepsilon }-P_{T_0})\rho _0\Vert _{\alpha } = O(C(L,\omega _0)) \int _X({\hat{P}}_{\varepsilon }-P_{T_0})\rho _0 \mathrm{d}x. \end{aligned}$$

(26)

We now give a bound on $|1-\nu _{\varepsilon }(\rho _0)|$ following the proof of [24, Lemma 6.1 (a)]. We will repeatedly use the fact that $(1-\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^k(\rho _0)=(1-\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })Q_{\varepsilon }^k(\rho _0)$ for all $k \ge 0$.

Recall the normalization $\nu _0(\rho _{\varepsilon })=1$ for all $\varepsilon \ge 0$, and that $\lambda _{\varepsilon }{\hat{P}}_{\varepsilon }=\rho _{\varepsilon } \otimes \nu _{\varepsilon }+Q_{\varepsilon }$. Then, for all $n \in {\mathbb {N}}$

$$\begin{aligned} \nu _{\varepsilon }(\rho _0)\rho _{\varepsilon }=(\lambda _{\varepsilon }{\hat{P}}_{\varepsilon })^n (\rho _0)-Q^n_{\varepsilon }(\rho _0), \end{aligned}$$

which implies

$$\begin{aligned} \nu _{\varepsilon }(\rho _0)=\nu _0((\lambda _{\varepsilon }{\hat{P}}_{\varepsilon })^n (\rho _0)-Q^n_{\varepsilon }(\rho _0)). \end{aligned}$$

Thus, we can write

$$\begin{aligned} |1-\nu _{\varepsilon }(\rho _0)|&=\lim _{n \rightarrow \infty }|\nu _0(\rho _0-(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^n(\rho _0))| \\&= \left| \sum _{k=0}^{n-1} \nu _0([(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^{k}-(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^{k+1}](\rho _0)) \right| \\&\le \sum _{k=0}^{n-1}| \nu _0([(1-\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^k](\rho _0))| \\&\le \sum _{k=0}^{\infty }|\nu _0((1-\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })Q_{\varepsilon }^k(\rho _0))| \end{aligned}$$

as $(1-\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^k=(1-\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })Q_{\varepsilon }^k$. Thus,

$$\begin{aligned} |1-\nu _{\varepsilon }(\rho _0)|&\le \sum _{k=0}^{\infty }|\nu _0((1-\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })Q_{\varepsilon }^k(\rho _0))| \\&\le \sum _{k=0}^{\infty }|\nu _0((P_{T_0}-{{\hat{P}}}_{\varepsilon })Q_{\varepsilon }^k(\rho _0))|+\sum _{k=0}^{\infty }|\nu _0(({\hat{P}}_{\varepsilon }-\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })Q_{\varepsilon }^k(\rho _0))| \\&\le \sum _{k=0}^{\infty }|\nu _0((P_{T_0}-{{\hat{P}}}_{\varepsilon })Q_{\varepsilon }^k(\rho _0))|+|\lambda _{\varepsilon }-1|\sum _{k=0}^{\infty }|\nu _0((\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })Q_{\varepsilon }^k(\rho _0))| \\&= \sum _{k=0}^{\infty }|\nu _0((P_{T_0}-{{\hat{P}}}_{\varepsilon })Q_{\varepsilon }^k(\rho _0))|+|\lambda _{\varepsilon }-1|\sum _{k=0}^{\infty }|\nu _0(Q_{\varepsilon }^{k+1}(\rho _0))| \\&\le \eta _{\varepsilon } \sum _{k=0}^{\infty }\Vert Q_{\varepsilon }^k\Vert _\alpha \cdot \Vert \rho _0\Vert _\alpha + |1-\lambda _{\varepsilon }|\Vert \nu _0\Vert _{\alpha } \sum _{k=1}^{\infty }\Vert Q_{\varepsilon }^k\Vert _\alpha \cdot \Vert \rho _0\Vert _\alpha \\&\le \frac{C}{1-K(L)}\left( \eta _{\varepsilon } + |1-\lambda _{\varepsilon }|\right) \end{aligned}$$

Since $|1-\lambda _{\varepsilon }|=|\nu _0((P_{T_0} -{{\hat{P}}}_{\varepsilon })(\rho _{\varepsilon }))|\le \Vert \rho _{\varepsilon }\Vert _{\alpha }\eta _{\varepsilon } \le C\eta _{\varepsilon }$ (by (24)) we obtain that

$$\begin{aligned} |1-\nu _{\varepsilon }(\rho _0)| \le \frac{C\eta _{\varepsilon }}{1-K(L)}. \end{aligned}$$

Finally, by (23) and (24) we have

$$\begin{aligned} |1-\nu _{\varepsilon }(\rho _0)|=O\left( \frac{C(L,\omega _0)L\varepsilon ^2}{1-K(L)}\right) \end{aligned}$$

(27)

by (25).

In what follows, we bound $\Vert Q_{\varepsilon }^N(\rho _0)\Vert _{\alpha }$, $N \ge 0$ with the help of the tail-sum $\kappa _N=\sum _{n=N}^{\infty }\sup _{\varepsilon }\Vert Q_{\varepsilon }^n\Vert _{\alpha }$. The argument is analogous to [24, Lemma 6.1 (b)].

$$\begin{aligned}&\Vert Q_{\varepsilon }^N(\rho _0)\Vert _{1} \\&\quad \le {{\overline{{\mathrm{lim}}}}}_{n \rightarrow \infty }\Vert Q_{\varepsilon }^N(\rho _0-(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^n(\rho _0))\Vert _{1}+{{\overline{{\mathrm{lim}}}}}_{n \rightarrow \infty }\Vert Q_{\varepsilon }^{N+n}(\rho _0)\Vert _{\alpha } \\&\quad \le {{\overline{{\mathrm{lim}}}}}_{n \rightarrow \infty }\left\| Q_{\varepsilon }^N\left( \sum _{k=0}^{n-1}(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^k\rho _0-(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^{k+1}(\rho _0)\right) \right\| _{1}+{{\overline{{\mathrm{lim}}}}}_{n \rightarrow \infty }\kappa _{N+n}\Vert \rho _0\Vert _{\alpha } \\&\quad \le \sum _{k=0}^{\infty } \Vert Q_{\varepsilon }^N(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^k(1-\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })(\rho _0)\Vert _{1}+{{\overline{{\mathrm{lim}}}}}_{n \rightarrow \infty }\kappa _{N+n}\Vert \rho _0\Vert _{\alpha } \\&\quad \le \sum _{k=0}^{\infty } \Vert Q_{\varepsilon }^{N+k}(P_{T_0}-{{\hat{P}}}_{\varepsilon }+(1-\lambda _{\varepsilon }^{-1}){{\hat{P}}}_{\varepsilon })(\rho _0)\Vert _{1} \\&\quad \le \sum _{k=0}^{\infty } (\Vert Q_{\varepsilon }^{N+k}(P_{T_0}-{{\hat{P}}}_{\varepsilon })(\rho _0)\Vert _1+|1-\lambda _{\varepsilon }^{-1}|\cdot \Vert Q_{\varepsilon }^{N+k+1}(\rho _0)\Vert _{1}) \\&\quad \le \sum _{k=0}^{\infty }\Vert Q_{\varepsilon }^{N+k}\Vert _{\alpha }(\Vert (P_{T_0}-{{\hat{P}}}_{\varepsilon })(\rho _0)\Vert _{\alpha }+|1-\lambda _{\varepsilon }|\cdot \Vert \rho _0\Vert _{\alpha }), \end{aligned}$$

thus

$$\begin{aligned} \Vert Q_{\varepsilon }^N(\rho _0)\Vert _{1} = O(\kappa _N) (\Vert (P_{T_0}-{{\hat{P}}}_{\varepsilon })(\rho _0)\Vert _{\alpha }+|1-\lambda _{\varepsilon }|) \end{aligned}$$

(28)

by using (24).

We now prove the formula for $\lambda _{\varepsilon }$ as stated in item 1 of Theorem 2.5. We follow the proof of [24, Theorem 2.1] while tracking the L dependence of the constants. First notice that $\nu _0((P_{T_0} - {{\hat{P}}}_{\varepsilon })(\rho _0))=\mu _{0}(H_\varepsilon )$.

$$\begin{aligned}&\nu _{\varepsilon }(\rho _0)(1-\lambda _{\varepsilon }) \\&\quad =(1-\nu _\varepsilon (\rho _0))\nu _0((P_{T_0} - {{\hat{P}}}_{\varepsilon })(\rho _\varepsilon )) \\&\quad =\mu _{0}(H_\varepsilon )-\nu _0((P_{T_0} - {{\hat{P}}}_{\varepsilon })(1-(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^n)(\rho _0))-\nu _0((P_{T_0} - {{\hat{P}}}_{\varepsilon })Q_{\varepsilon }^n(\rho _0)) \\&\quad = \mu _{0}(H_\varepsilon )-\sum _{k=0}^{n-1}\nu _0((P_{T_0} - {{\hat{P}}}_{\varepsilon })(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^k(1-\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })(\rho _0))+O(\eta _{\varepsilon }\Vert Q_{\varepsilon }^n\rho _0\Vert _{1}) \\&\quad = \mu _{0}(H_\varepsilon )-\sum _{k=0}^{n-1}\nu _0((P_{T_0} - {{\hat{P}}}_{\varepsilon })(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^k(P_{T_0}-{{\hat{P}}}_{\varepsilon })(\rho _0))\\&\qquad +(1-\lambda _{\varepsilon })\sum _{k=0}^{n}\nu _0((P_{T_0} - {{\hat{P}}}_{\varepsilon })(\lambda _{\varepsilon }^{-1}{{\hat{P}}}_{\varepsilon })^k(\rho _0))\\&\qquad +O(\kappa _n)(O(C(L,\omega _0))\mu _{0}(H_\varepsilon )+\eta _{\varepsilon }|1-\lambda _{\varepsilon }|) \end{aligned}$$

where for the error term we used (28), then (26). Continuing,

$$\begin{aligned} \nu _{\varepsilon }(\rho _0)(1-\lambda _{\varepsilon })&=\mu _{0}(H_\varepsilon )\left( 1-\sum _{k=0}^{n-1}\lambda _{\varepsilon }^{-k}q_{k,\varepsilon } \right) \\&\quad +O(\eta _{\varepsilon })|1-\lambda _{\varepsilon }|\sum _{k=0}^{n-1}(|\nu _{\varepsilon }(\rho _0)| \cdot \Vert \rho _{\varepsilon }\Vert _1+\Vert Q_{\varepsilon }^k\rho _0\Vert _1)\\&\quad +O(\kappa _n)(O(C(L,\omega _0))\mu _{0}(H_\varepsilon )+\eta _{\varepsilon }|1-\lambda _{\varepsilon }|) \end{aligned}$$

where we used the notation

$$\begin{aligned} q_{k,\varepsilon }=\frac{\nu _0((P_{T_0} - {{\hat{P}}}_{\varepsilon }){{\hat{P}}}_{\varepsilon }^k(P_{T_0}-{{\hat{P}}}_{\varepsilon })(\rho _0))}{\nu _0((P_{T_0} - {{\hat{P}}}_{\varepsilon })(\rho _0))}. \end{aligned}$$

Using (27), (24) and (23), we can write this as

$$\begin{aligned}&\left( 1+O\left( \frac{C\left( L,\omega _0\right) L\varepsilon ^2}{1-K(L)}\right) \right) (1-\lambda _{\varepsilon })(1+nO(\eta _{\varepsilon }))\nonumber \\&\quad =\mu _{0}(H_\varepsilon )\left( 1-\sum _{k=0}^{n-1}\lambda _{\varepsilon }^{-k}q_{k,\varepsilon } \right) +O(\kappa _n)C(L,\omega _0)\mu _{0}(H_\varepsilon ) \nonumber \\&\left( 1+O\left( \frac{C\left( L,\omega _0\right) L\varepsilon ^2}{1-K(L)}\right) \right) (1-\lambda _{\varepsilon })(1+nO(C(L,\omega _0)L\varepsilon ^2))\nonumber \\&\quad =\mu _{0}(H_\varepsilon )\left( 1-\sum _{k=0}^{n-1}\lambda _{\varepsilon }^{-k}q_{k,\varepsilon } \right) +O(\kappa _n)C(L,\omega _0)\mu _{0}(H_\varepsilon ) \end{aligned}$$

(29)

Assuming the limits $\lim _{\varepsilon \rightarrow 0}q_{k,\varepsilon }=:q_k$ exist, this implies that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{1-\lambda _{\varepsilon }}{\mu _{0}(H_\varepsilon )}=1-\sum _{k=0}^{n-1}q_{k}+O(\kappa _n), \end{aligned}$$

and finally taking $n \rightarrow \infty$ gives the formula

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{1-\lambda _{\varepsilon }}{\mu _{0}(H_\varepsilon )}=\theta \end{aligned}$$

(30)

by using the notation $\theta =1-\sum _{k=0}^{\infty }q_{k}$.

We now compute the values of $q_{k}$, $k \ge 0$ to obtain an explicit formula for $\theta$. To this end, we recall from [19] that $q_{k,\varepsilon }$ can be computed as

$$\begin{aligned} q_{k,\varepsilon }&=\frac{1}{\mu _0(H_{\varepsilon })}\int _{H_{\varepsilon }}{\hat{P}}_{\varepsilon }^kP_{T_0}(1_{H_{\varepsilon }}\rho _0)\mathrm{d}x \nonumber \\&=\frac{\mu _0(H_{\varepsilon }\cap T_0^{-1}H_{\varepsilon }^c\cap \cdots T_0^{-k}H_{\varepsilon }^c \cap T_0^{-(k+1)}H_{\varepsilon })}{\mu _0(H_{\varepsilon }) } \end{aligned}$$

(31)

Recall further that $H_{\varepsilon }=\cup _{{{\mathbf {v}}}\in V^+}\cup _{{{\mathbf {p}}}\in \Lambda }H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}}$ where

$$ H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}}=\{x_{{{\mathbf {p}}}} \in A_{\varepsilon ,{{\mathbf {v}}}}, x_{{{\mathbf {p}}}+{{\mathbf {v}}}} \in A_{\varepsilon ,-{{\mathbf {v}}}}, x_{{{\mathbf {q}}}} \in I \text { for } {{\mathbf {q}}}\ne {{\mathbf {p}}}, {{\mathbf {p}}}+{{\mathbf {v}}}\}, $$

and $I_{{{\mathbf {p}}},{{\mathbf {v}}}}=\lim _{\varepsilon \rightarrow 0}H_{\varepsilon }^{{{\mathbf {p}}},{{\mathbf {v}}}}$, i.e.,

$$\begin{aligned} I_{{{\mathbf {p}}},{{\mathbf {v}}}}=\{x_{{{\mathbf {p}}}}=a_{{\mathbf {v}}}, x_{{{\mathbf {p}}}+{{\mathbf {v}}}}= a_{-{{\mathbf {v}}}}, x_{{{\mathbf {q}}}} \in I \text { for } {{\mathbf {q}}}\ne {{\mathbf {p}}}, {{\mathbf {p}}}+{{\mathbf {v}}}\}, \end{aligned}$$

where $a_{{{\mathbf {v}}}}=\lim _{\varepsilon \rightarrow 0}A_{\varepsilon ,{{\mathbf {v}}}}$ and $a_{-{{\mathbf {v}}}}=\lim _{\varepsilon \rightarrow 0}A_{\varepsilon ,-{{\mathbf {v}}}}$.

Consider first the case when none of the elements of $S=\{(a_{{{\mathbf {v}}}}, a_{-{{\mathbf {v}}}})\}_{{{\mathbf {v}}}\in V^+}$ are periodic points, that is, $S^{per}=\emptyset$. Then, by the expression (31) it is evident that in the $\varepsilon \rightarrow 0$ limit we get $q_k=0$ for all $k \ge 0$. As a consequence, $\theta =1$.

For each $(a_{{{\mathbf {v}}}},a_{-{{\mathbf {v}}}}) \in S^{per}$ recall that $k({{\mathbf {v}}})$ is the smallest integer such that $T_{2,0}^{k+1}(a_{{{\mathbf {v}}}},a_{-{{\mathbf {v}}}}) \in S$, and $V_k^+ \subset V^+$ is the set of directions for which $k({{\mathbf {v}}})=k$ and $K=\{k({{\mathbf {v}}})\}_{{{\mathbf {v}}}\in V^+}$. Notice that S can be divided into periodic orbits, where the first preimage of each point intersecting S is well-defined. We will denote the said preimage of $(a_{{{\mathbf {v}}}},a_{-{{\mathbf {v}}}})$ by $(\tau _*^{-(k'+1)}(a_{{{\mathbf {v}}}}),\tau _*^{-(k'+1)}(a_{-{{\mathbf {v}}}}))$. Consider now some $k \in K$. Computing $q_{k,\varepsilon }$, we get

$$\begin{aligned} q_{k,\varepsilon }&=\frac{1}{\mu _0(H_{\varepsilon })}\int _{H_{\varepsilon }}P_{\varepsilon }^kP_{T_0}(1_{H_{\varepsilon }}\rho _0)\mathrm{d}x \\&=\frac{1}{\mu _0(H_{\varepsilon })}\int _{H_{\varepsilon }^k}\sum _{y \in T_0^{-(k+1)}(x)}\frac{\rho _0 (y)}{\Vert D T_0^{k+1}y\Vert }\mathrm{d}x \\&=\frac{1}{\mu _0(H_{\varepsilon })}\int _{H_{\varepsilon }^k}\sum _{y \in T_0^{-(k+1)}(x)}\frac{\prod _{{{\mathbf {p}}}\in \Lambda }h(y_{{{\mathbf {p}}}})}{\prod _{{{\mathbf {p}}}\in \Lambda }|(\tau ^{k+1})'(y_{{{\mathbf {p}}}})|}\mathrm{d}x \end{aligned}$$

where $H_{\varepsilon }^k=H_{\varepsilon }\cap T_0(H_{\varepsilon }^c)\cap \cdots \cap T_0^{k}(H_{\varepsilon }^c)\cap T_0^{k+1}(H_{\varepsilon })$.

As a notational ease write

$$F(x)=\sum _{y \in T_0^{-(k+1)}(x)}\frac{\prod _{{{\mathbf {p}}}\in \Lambda }h(y_{{{\mathbf {p}}}})}{\prod _{{{\mathbf {p}}}\in \Lambda }|(\tau ^{k+1})'(y_{{{\mathbf {p}}}})|}.$$

We remark that $F=P_{T_0^{ k+1}}\rho _0=\rho _0$, but we prefer to keep this expression for reasons that will become clear in what follows.

Recall that we defined $\Lambda ^{\ell }=\{({{\mathbf {p}}}_1,\dots ,{{\mathbf {p}}}_k): {{\mathbf {p}}}_i \in \Lambda ,\, i=1,\dots ,\ell \}$, $(V^+)^{\ell }=\{({{\mathbf {v}}}_1,\dots ,{{\mathbf {v}}}_{\ell }): {{\mathbf {v}}}_i \in V^+,\, i=1,\dots ,\ell \}$ and let $(V_k^+)^{\ell }=\{({{\mathbf {v}}}_1,\dots ,{{\mathbf {v}}}_{\ell }): {{\mathbf {v}}}_i \in V_k^+,\, i=1,\dots ,\ell \}$. Recalling (10) we can write

$$\begin{aligned} \int _{H_{\varepsilon }^k} F(x) \mathrm{d}x&=\sum _{\ell =1}^{\lfloor {L}/2 \rfloor }(-1)^{\ell +1}\sum _{\begin{array}{c} \{{{\mathbf {p}}}_i\}_{i=1}^{\ell } \in \Lambda ^{\ell } \\ {{\mathbf {p}}}_i < {{\mathbf {p}}}_{i+1} \end{array}}\sum _{\{{{\mathbf {v}}}_i\}_{i=1}^{\ell } \in (V^+)^{\ell }}\int _{(\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i})\cap H_{\varepsilon }^k}F(x)\mathrm{d}x. \end{aligned}$$

(32)

Let us introduce the notation $x_{\ne \{{{\mathbf {p}}}_i,{{\mathbf {p}}}_i+{{\mathbf {v}}}_i\}_{i=1}^{\ell }}$ for the $(L-2\ell )$-dimensional vector that we get from $\{x_{{{\mathbf {q}}}}\}_{{{\mathbf {q}}}\in \Lambda }$ by discarding the coordinates $x_{\{{{\mathbf {p}}}_i,{{\mathbf {p}}}_i+{{\mathbf {v}}}_i\}_{i=1}^{\ell }}$. Assume $\{{{\mathbf {v}}}_i\}_{i=1}^{\ell } \in (V_k^+)^{\ell }$. Then, $(\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}) \cap H_{\varepsilon }^k=\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}$ for small enough $\varepsilon$. We now compute the limit

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{\mu _0(\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i})}{\mu _0(H_{\varepsilon })} \cdot \frac{1}{\mu _0(\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i})}\int _{\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}}F(x)\mathrm{d}x. \end{aligned}$$

Since F is continuous at $x \in S$ (based on the continuity assumptions and $\tau$ and $\tau '$, implying the continuity of h), we obtain

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0} \frac{1}{\mu _0(\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i})}\int _{\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}}F(x)\mathrm{d}x \\&\quad =\prod _{i=1}^{\ell }\frac{h(\tau _*^{-(k+1)}(a_{{{\mathbf {v}}}_i}))h(\tau _*^{-(k+1)}(a_{-{{\mathbf {v}}}_i}))}{|(\tau ^{k+1})'(\tau _*^{-(k+1)}(a_{{{\mathbf {v}}}_i}))(\tau ^{k+1})'(\tau _*^{-(k+1)}(a_{-{{\mathbf {v}}}_i}))|}\cdot \frac{1}{h(a_{{{\mathbf {v}}}_i})h(a_{-{{\mathbf {v}}}_i})} \\&\quad \times \int \limits _{I^{{L}-{2}\ell }}\sum _{y \in T_0^{-(k+1)}(x)}\prod _{\begin{array}{c} {{\mathbf {q}}}\in \Lambda \\ {{\mathbf {q}}}\notin \{{{\mathbf {p}}}_i,{{\mathbf {p}}}_i+{{\mathbf {v}}}_i\}_{i=1}^{\ell } \end{array}}\frac{h(y_{{\mathbf {q}}})}{|(\tau ^{k+1})'(y_{{{\mathbf {q}}}})|}\mathrm{d}x_{\ne \{{{\mathbf {p}}}_i,{{\mathbf {p}}}_i+{{\mathbf {v}}}_i\}_{i=1}^{\ell }} \\&\quad \times \chi (\cap _{i=1}^{\ell } I_{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}) \end{aligned}$$

where $\chi (A)=1$ if A is non-empty and $\chi (A)=0$ otherwise. Notice that

$$\begin{aligned} \int \limits _{I^{{L}-{ 2}\ell }}\sum _{y \in T_0^{-(k+1)}(x)}\prod _{\begin{array}{c} {{\mathbf {q}}}\in \Lambda \\ {{\mathbf {q}}}\notin \{{{\mathbf {p}}}_i,{{\mathbf {p}}}_i+{{\mathbf {v}}}_i\}_{i=1}^{\ell } \end{array}}\frac{h(y_{{\mathbf {q}}})}{|(\tau ^{k+1})'(y_{{{\mathbf {q}}}})|}\mathrm{d}x_{\ne \{{{\mathbf {p}}}_i,{{\mathbf {p}}}_i+{{\mathbf {v}}}_i\}_{i=1}^{\ell }}=1 \end{aligned}$$

by integrating successively according to each variable $x_{{{\mathbf {q}}}}$ and using that $P_{\tau }h=h$ and $\int _I h=1$. Then, we can write

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0} \frac{1}{\mu _0(\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i})}\int _{\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}}F(x)\mathrm{d}x \\&\quad =\prod _{i=1}^{\ell }\frac{h(\tau _*^{-(k+1)}(a_{{{\mathbf {v}}}_i}))h(\tau _*^{-(k+1)}(a_{-{{\mathbf {v}}}_i}))}{|(\tau ^{k+1})'(\tau _*^{-(k+1)}(a_{{{\mathbf {v}}}_i}))(\tau ^{k+1})'(\tau _*^{-(k+1)}(a_{-{{\mathbf {v}}}_i}))|}\cdot \frac{1}{h(a_{{{\mathbf {v}}}_i})h(a_{-{{\mathbf {v}}}_i})} \\&\qquad \times \chi (\cap _{i=1}^{\ell } I_{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}). \end{aligned}$$

Notice that $(\tau _*^{-(k+1)}(a_{{{\mathbf {v}}}_i}),\tau _*^{-(k+1)}(a_{-{{\mathbf {v}}}_i}))=(a_{{{\mathbf {v}}}_j},a_{-{{\mathbf {v}}}_j})$ for some ${{\mathbf {v}}}_j \in V_k^+$.

On the other hand, using (9) we obtain

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0}\frac{\mu _0(\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i})}{\mu _0(H_{\varepsilon })}\\&\quad =\lim _{\varepsilon \rightarrow 0}\frac{\mu _0(\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i})}{\sum _{k=1}^{\lfloor {L}/2 \rfloor }(-1)^{k+1}\sum _{\begin{array}{c} \{{{\mathbf {p}}}_i\}_{i=1}^{k} \in \Lambda ^k \\ {{{\mathbf {p}}}_i < {{\mathbf {p}}}_{i+1}} \end{array}}\sum _{\{{{\mathbf {v}}}_i\}_{i=1}^{k} \in (V^+)^k}\mu _0\left( \bigcap _{i=1}^k H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}\right) } \\&\quad ={\left\{ \begin{array}{ll} \frac{h(a_{{{\mathbf {v}}}_1})h(a_{-{{\mathbf {v}}}_{1}})}{\sum _{{{\mathbf {p}}}\in \Lambda }\sum _{{{\mathbf {v}}}\in V^+}h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})} &{} \quad \text {if }\ell =1, \\ 0 &{} \quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

The reason we get zero in the limit for $\ell > 1$ is that $m_{L-2}(\cap _{i=1}^{\ell } I_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i})=0$ while $m_{L-2}( I_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}) > 0$ (we have $m_{L-2}( I_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i})=1$ in fact.) This implies

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0}\frac{\mu _0(\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i})/\varepsilon ^2}{\sum \limits _{k=1}^{\lfloor {L}/2 \rfloor }(-1)^{k+1}\sum \limits _{\begin{array}{c} \{{{\mathbf {p}}}_i\}_{i=1}^{k} \in \Lambda ^k \\ { {{\mathbf {p}}}_i< {{\mathbf {p}}}_{i+1}} \end{array}}\sum \limits _{\{{{\mathbf {v}}}_i\}_{i=1}^{k} \in (V^+)^k}\mu _0\left( \bigcap \limits _{i=1}^k H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}\right) /\varepsilon ^2} \\&\quad =\lim _{\varepsilon \rightarrow 0}\frac{\mu _0(\cap _{i=1}^{\ell } H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i})/\varepsilon ^{2^{\ell }}\cdot \varepsilon ^{2(\ell -1)}}{\sum \limits _{k=1}^{\lfloor {L}/2 \rfloor }(-1)^{k+1}\sum \limits _{\begin{array}{c} \{{{\mathbf {p}}}_i\}_{i=1}^{k} \in \Lambda ^k \\ {{{\mathbf {p}}}_i< {{\mathbf {p}}}_{i+1}} \end{array}}\sum \limits _{\{{{\mathbf {v}}}_i\}_{i=1}^{k} \in (V^+)^k}\mu _0\left( \bigcap \limits _{i=1}^k H_{\varepsilon }^{{{\mathbf {p}}}_i,{{\mathbf {v}}}_i}\right) /\varepsilon ^{2^{k}}\cdot \varepsilon ^{2(k-1)}} \\&\quad =\lim _{\varepsilon \rightarrow 0}\frac{\prod \limits _{i=1}^{\ell }h(a_{{{\mathbf {v}}}_i})h(a_{-{{\mathbf {v}}}_i}) \varepsilon ^{2(\ell -1)}}{\sum \limits _{k=1}^{\lfloor {L}/2 \rfloor }(-1)^{k+1}\sum \limits _{\begin{array}{c} \{{{\mathbf {p}}}_i\}_{i=1}^{k} \in \Lambda ^k \\ {{{\mathbf {p}}}_i < {{\mathbf {p}}}_{i+1}} \end{array}}\sum \limits _{\{{{\mathbf {v}}}_i\}_{i=1}^{k} \in (V^+)^k}\prod \limits _{i=1}^{k}h(a_{{{\mathbf {v}}}_i})h(a_{-{{\mathbf {v}}}_i}) \varepsilon ^{2(k-1)}} \\&\quad =\frac{0}{\sum _{{{\mathbf {p}}}\in \Lambda }\sum _{{{\mathbf {v}}}\in V^+}h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})+0}=0. \end{aligned}$$

To finish the proof, consider (32), keep only the term $\ell =1$ and repeating this computation for each ${{\mathbf {p}}}\in \Lambda$ and ${{\mathbf {v}}}\in V_k^+$, we obtain

$$\begin{aligned} q_k&=\frac{1}{\sum _{{{\mathbf {p}}}\in \Lambda }\sum _{{{\mathbf {v}}}\in V^+}h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})}\sum _{{{\mathbf {p}}}\in \Lambda } \sum _{{{\mathbf {v}}}\in V_k^+}\frac{h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})}{|(\tau ^{k+1})'(a_{{{\mathbf {v}}}})(\tau ^{k+1})'(a_{-{{\mathbf {v}}}})|} \\&=\frac{1}{\sum _{{{\mathbf {v}}}\in V^+}h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})} \sum _{{{\mathbf {v}}}\in V_{k}^+}\frac{h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})}{|(\tau ^{k+1})'(a_{{{\mathbf {v}}}})(\tau ^{k+1})'(a_{-{{\mathbf {v}}}})|}. \end{aligned}$$

Finally,

$$\begin{aligned} \theta&=1-\sum _{k \in K} q_k \nonumber \\&=1-\frac{1}{\sum _{{{\mathbf {v}}}\in V^+}h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})} \sum _{k \in K}\sum _{{{\mathbf {v}}}\in V_{k}^+}\frac{h(a_{{{\mathbf {v}}}})h(a_{-{{\mathbf {v}}}})}{|(\tau ^{k+1})'(a_{{{\mathbf {v}}}})(\tau ^{k+1})'(a_{-{{\mathbf {v}}}})|} . \end{aligned}$$

(33)

Lemma 4.1 readily implies that $\mu _0(H_\varepsilon )={L}\Xi _{\varepsilon }(1+o(1)) \sim \varepsilon ^2 {L} d$ as

$$\begin{aligned} d{\underline{C}}^2\varepsilon ^2 \le \sum _{{{\mathbf {v}}}\in V^+}\int _{A_{\varepsilon ,{{\mathbf {v}}}}}\int _{A_{\varepsilon ,-{{\mathbf {v}}}}}h(x_1)h(x_2)\mathrm{d}x_1\mathrm{d}x_2 \le d{\overline{C}}^2\varepsilon ^2 \end{aligned}$$

where ${\overline{C}},{\underline{C}}$ are the respective upper and lower bounds on h noted previously. The o(1) error term is understood in the limit $L\varepsilon ^2 \rightarrow 0$, which for fixed L is equivalent to $\varepsilon \rightarrow 0$.

To prove the second item, first notice that

$$\begin{aligned} m_{L}(X_{\varepsilon }^{n-1})&=\int _X 1_{X_{\varepsilon }^{n-1}}(x)\mathrm{d}x=\int _X P_{T_0}^{n}(1_{X_{\varepsilon }^{n-1}}(x))\mathrm{d}x=\int _X{\hat{P}}_{\varepsilon }^{n}1(x)\mathrm{d}x\\&=\lambda ^n_\varepsilon \left[ \int _X\rho _\varepsilon \otimes \nu _{\varepsilon }(1) \mathrm{d}x + \int _X Q^n_\varepsilon 1(x) \mathrm{d}x\right] , \end{aligned}$$

where in the last step we have used the spectral decomposition (22). Consequently,

$$ \frac{1}{n}\ln \left( m_{L}(X_{\varepsilon }^{n-1})\right) -\ln \lambda _\varepsilon =\frac{1}{n}\ln \left[ \int _X\rho _\varepsilon \otimes \nu _{\varepsilon }(1) \mathrm{d}x + \int _X Q^n_\varepsilon 1(x) \mathrm{d}x\right] . $$

Since $\Vert Q_\varepsilon \Vert _1\le \Vert Q_\varepsilon \Vert _{\alpha }<1$, $\exists C>0$ such that

$$ \left| \frac{1}{n}\ln \left( m_{L}(X_{\varepsilon }^{n-1})\right) -\ln \lambda _\varepsilon \right| \le \frac{C}{n}. $$

Hence,

$$\lim _{n\rightarrow \infty }\frac{1}{n}\ln \left( m_{L}(X_{\varepsilon }^{n-1})\right) =\ln \lambda _\varepsilon $$

and the first collision rate, with respect to $m_L$ in the L-dimensional system is given by ${{\hat{r}}}_L=-\ln \lambda _\varepsilon$. Using the asymptotics of $\lambda _\varepsilon$, we have

$$\begin{aligned} \begin{aligned} \lambda _{\varepsilon }&=1-\mu _0(H_\varepsilon )\cdot \theta (1+o(1))\\&=e^{-\mu _0(H_\varepsilon )\cdot \theta (1+ o(1))} \end{aligned} \end{aligned}$$

This implies

$$\begin{aligned} {{\hat{r}}}_L= \mu _0(H_\varepsilon ) \cdot \theta (1+o(1)). \end{aligned}$$

(34)

The proof of the third item of the theorem follows from item 1 along the lines of Proposition 2 of [19].

We choose N such that $N\eta _{\varepsilon }+O(C(L,\omega _0)\kappa _N)$ is minimal. Denote

$$\begin{aligned} \zeta (L)=|\log (1-K(L))| \end{aligned}$$

(35)

where K(L) was defined by (23). Thus, $N=O\left( \frac{1}{\zeta (L)} \cdot \left| \log \left( \frac{\eta _{\varepsilon }}{C(L,\omega _0)\zeta (L)}\right) \right| \right)$. Set $\delta _{\varepsilon }=O\left( \frac{\eta _{\varepsilon }}{\zeta (L)}\left| \log \frac{\eta _{\varepsilon }}{C(L,\omega _0)\zeta (L)}\right| \right)$.

Let $\xi _{\varepsilon }=\theta _{N,\varepsilon }+O(C(L,\omega _0)\kappa _N)$ where $\theta _{N,\varepsilon }=\sum _{k=0}^{N-1} \lambda _{\varepsilon }^{-k}q_{k,\varepsilon }$. By (29) we obtain that

$$\begin{aligned} \lambda _{\varepsilon }&=1-\mu _0(H_{\varepsilon })\xi _{\varepsilon } (1+O(N\eta _{\varepsilon })) \\&=exp[-\mu _0(H_{\varepsilon })\xi _{\varepsilon } (1+O(\delta _{\varepsilon }))] \end{aligned}$$

As $\lambda _{\varepsilon }=1-\mu _0(H_{\varepsilon })(\theta +o(1))$ by (30) we also have $\lim _{\varepsilon \rightarrow 0}\xi _{\varepsilon }=\theta > 0$, in particular $\xi _{\varepsilon } > 0$ for $\varepsilon$ small enough.

Let $n=\lfloor t/(\xi _{\varepsilon }\mu _0(H_{\varepsilon })) \rfloor$. First assume that $n > \left| \frac{\log \eta _{\varepsilon }}{\zeta (L)} \right|$. Then, as argued in [19, Equation (26)], we can write

$$\begin{aligned} \mu _0\Big \{t_\varepsilon \ge \frac{t}{\xi _\varepsilon \mu _0(H_\varepsilon )}\Big \}&=\lambda _{\varepsilon }^{n+q}(\nu _{\varepsilon }(\rho _0)+O(\Vert Q_\varepsilon ^n\Vert _{\alpha })) \\&=exp[-t(1+O(\delta _{\varepsilon }))]\cdot \left( 1+O\left( \frac{C\left( L,\omega _0\right) L\varepsilon ^2}{1-K(L)}\right) \right) \end{aligned}$$

where in the last line we used (27). So for $\varepsilon$ small enough we have

$$\begin{aligned} \left| \mu _0\Big \{t_\varepsilon \ge \frac{t}{\xi _\varepsilon \mu _0(H_\varepsilon )}\Big \}-e^{-t}\right|&\le C(t\vee 1)e^{-t} \delta _{\varepsilon } \\&\le C(t\vee 1)e^{-t} \frac{\eta _{\varepsilon }}{\zeta (L)}\log \frac{\eta _{\varepsilon }}{C(L,\omega _0)\zeta (L)} \\&\le C(t\vee 1)e^{-t} C(L,\omega _0) \frac{L\varepsilon ^2}{\zeta (L)}\log \frac{L\varepsilon ^2}{\zeta (L)}. \end{aligned}$$

The case of $n \le \left| \frac{\log \eta _{\varepsilon }}{\zeta (L)} \right|$ can be argued in the same way as in the proof of [19, Proposition 2]. $\square$

Proof of Theorem 2.8

To derive the expression in the theorem, we consider

$$ \frac{{{\hat{r}}}_L}{\mu _0(H_\varepsilon )}\cdot \frac{\mu _0(H_\varepsilon )}{{L}\Xi _\varepsilon } $$

as $\varepsilon \rightarrow 0$. By equation (7)

$$\begin{aligned} \frac{{{\hat{r}}}_L}{\mu _0(H_\varepsilon )}= \theta (1+o(1)) \end{aligned}$$

in the limit as $\varepsilon \rightarrow 0$. Moreover, equation (8) implies

$$\frac{\mu _0(H_{\varepsilon })}{{L}\Xi _{\varepsilon }}=1+o(1)$$

as $\varepsilon \rightarrow 0$. This completes the proof. $\square$

We finish the paper by a remark on considering the infinite-dimensional limit of the collision rate and its per lattice unit counterpart.

Remark 4.5

If we want to study collision rates in the infinite limit of the system, i.e., as ${L}\rightarrow \infty$, we need an asymptotic formula for $\lambda _{\varepsilon }$ in some kind of joint limit where $L \rightarrow \infty$ and $\varepsilon \rightarrow 0$. To this end, recall (29) :

$$\begin{aligned} \left( 1+O\left( \frac{C\left( L,\omega _0\right) L\varepsilon ^2}{1-K(L)}\right) \right)&\frac{1-\lambda _{\varepsilon }}{\mu _{0}(H_\varepsilon )}(1+nO(C(L,\omega _0)L\varepsilon ^2))\nonumber \\&=1-\sum _{k=0}^{n-1}\lambda _{\varepsilon }^{-k}q_{k,\varepsilon } +O(\kappa _n)C(L,\omega _0) \end{aligned}$$

The term $O(\kappa _n)C(L,\omega _0)$ causes serious difficulties in taking any kind of limit in which $L \rightarrow \infty$, as $C(L,\omega _0)$, defined by (4) diverges as $L \rightarrow \infty$. We note that the constant $C(L,\omega _0)$ appears in that term due to (26), the analogue of (A6) in [24]. In addition, the term K(L) (a bound on the modulus of the second largest eigenvalue of ${\hat{P}}_{\varepsilon }$) is also difficult to control.