Rare Events Statistics for ℤd Map Lattices Coupled by Collision

TL;DR

This paper studies collision statistics in a lattice model of gas particles to better understand rare collision events.

Contribution

The paper introduces a novel approach using transfer operators to analyze collision rates and times in infinite-dimensional systems.

Findings

A first-order approximation for the collision rate at a lattice site is derived.

The first collision time converges to an exponential distribution with a sharp error term.

The number of collisions converges to a compound Poisson distribution.

Abstract

Understanding the statistics of collisions among locally confined gas particles poses a major challenge. In this work we investigate \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z}^d$$\end{document}Zd-map lattices coupled by collision with simplified local dynamics that offer significant insights for the above challenging problem. We obtain a first order approximation for the first collision rate at a site \documentclass[12pt]{minimal}
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\begin{document}$${\textbf{p}}^*\in \mathbb {Z}^d$$\end{document}p∗∈Zd and we prove a distributional convergence for the first collision time to an exponential, with sharp error term. Moreover, we prove that the number of collisions at site \documentclass[12pt]{minimal}
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\begin{document}$${\textbf{p}}^*$$\end{document}p∗ converge in distribution to a compound Poisson distributed random variable. Key to our analysis in this infinite dimensional setting is the use of transfer operators associated with the decoupled map lattice at site \documentclass[12pt]{minimal}
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Full text

Introduction

The study of rare events in dynamical systems is one of the most active branches of research in ergodic theory. An example of a rare event in dynamical systems appears in the study of ‘open’ systems where orbits infrequently escape from the domain, typically by falling into a small ‘hole’ in the phase space, see [2, 10, 30], the recent article [9] and references therein. Another important example of a rare event in dynamical systems appears in the analysis of extreme value statistics, see for instance [25], the recent work [13, 16] and references therein. In parallel the study of large, possibly infinite-dimensional, systems is of paramount importance in many fields, including mathematics and physics, see for example [12, 29, 31, 32] and references therein. Coupled map lattices are basic models of large dynamical systems. They were investigated by many authors [8], starting with Kaneko [17], and were first put in a rigorous setting by Bunimovich and Sina? [7]. The models studied in [7, 8, 17] cover large coupled systems with weak interaction among its units. However, in several important coupled systems, such as those of locally confined gas particles [14], the interaction among its neighboring units is rare but strong. Proving statistical properties in models such as that of [14] is currently untractable.1 In this work we investigate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^d$$\end{document} -map lattices coupled by collision with simplified local dynamics, introduced by Keller and Liverani [22], where we prove statistical properties that offer significant insights for models such as that of [14].

The system that we consider has two distinctive features: the rarity of the event that we study appears naturally in the model, and the dynamical system that describes the model is infinite dimensional. More precisely, the infinite dynamical system is defined by a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^d$$\end{document} -coupled map lattice where the site dynamic is a chaotic map of the interval, while the coupling is defined via rare interaction between two neighbours. But when two neighbouring sites interact their states change drastically: when the dynamics of two neighbouring sites in the lattice fall simultaneously in their ‘collision’ zones, the states of the neighbouring sites get interchanged. By a loose analogy with classical mechanics, this type of coupling is called ‘coupling by collision’. Indeed, as we mentioned earlier, the model introduced in [22] is reminiscent to collision models studied by mathematicians [5] and physicists [14].

In this work, in the framework of the infinite dimensional model of [22], we obtain a first order approximation for the first collision rate at a site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*\in \mathbb {Z}^d$$\end{document} and we obtain a law for the corresponding first hitting time, with sharp error term. Moreover, we prove that the number of collisions at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} converge in distribution to a compound Poisson distributed random variable. Our techniques are based on analysing the spectrum of rare event transfer operators, which to the best of our knowledge have not been explored before in an infinite dimensional setting. A key idea is to observe that such operators can be associated with the decoupled2 map lattice at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} .

For a finite dimensional version of the model of [22] a different rare event3 was considered in [4]. In the work of [4], the expression of the coupling map did not matter, and basically the rare event in that work can be seen as a rare event of a finite dimensional product map (a rare event for completely decoupled and finite dimensional system). Unlike the work of [4] we deal with an infinite dimensional system. Moreover, the rare event that we study gets really influenced by the coupling map (Eq. (1) below) of the system. At the technical level, it is worth noting that when dealing with an infinite dimensional phase space, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^d$$\end{document} in our case, standard spectral techniques are not available and hence delicate verification of the assumptions of the framework of [18, 19, 23] is needed to obtain limit laws for the rare event. In addition, the computation of the so-called ‘extremal index’, which is needed in the derivation of the first hitting time law, is quite involved in our setting.

The paper is organised as follows. In Sect. 2 we introduce the collision model, the rare event and we state our main results, Theorem 2.1, Theorem 2.2 and Theorem 2.5. We end Sect. 2 by presenting an example that satisfies the main results where we compute the corresponding probabilistic quantities in a concrete situation. Section 3 contains the proof of Theorem 2.1, while Sects. 4 and 5 contain the proofs of Theorems 2.1 and 2.5 respectively. Section 6 contains the proof of a uniform Lasota-Yorke inequality for an auxiliary operator, which is used on several occasions for different specific operators in previous sections of the paper.

Setting and Statement of the Main Result

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I=[0,1]$$\end{document} and m be Lebesgue measure on it. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau : I\rightarrow I$$\end{document} be a piecewise \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} and piecewise onto, uniformly expanding map with expanding factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >1$$\end{document} ; i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists $$\end{document} a finite partition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0=\xi _0<\dots <\xi _M=1$$\end{document} such that, denoting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_i:=(\xi _{i-1},\xi _{i})$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{|I_i}$$\end{document} is monotone and onto, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} , with a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} extension to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{I}}_i$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\tau '(x)|\ge \alpha $$\end{document} . It is well known that such maps admit an absolutely continuous invariant measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _\tau $$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\mu _\tau /dm=\rho _\tau $$\end{document} ; moreover \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\inf _{x\in I}\rho _\tau (x)>0$$\end{document} , the system is mixing [6] and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\tau }\in C^1(I)$$\end{document} , see for instance Lecture 1 of [24].4

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}:=(x_{{\textbf {p}}})_{{\textbf {p}}\in \mathbb {Z}^d} \in I^{\mathbb {Z}^d}$$\end{document} let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_0: I^{\mathbb {Z}^d}\rightarrow I^{\mathbb {Z}^d} $$\end{document} be the product map:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( T_0({\textbf {x}})\right) _{{\textbf {p}}}=\tau (x_{{\textbf {p}}}). \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V^+:=\{e_i, 1\le i\le d\}$$\end{document} be the standard basis of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^d$$\end{document} and define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V:=V^+ \cup -V^+$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{A_{\varepsilon ,-{\textbf{v}}}\}_{{\textbf{v}}\in V}\subset I$$\end{document} be a set of disjoint open intervals, each of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} . Following [22], we consider the coupling

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\Phi _\varepsilon ({\textbf {x}}))_{\textbf{p}}={\left\{ \begin{array}{ll} x_{{\textbf{p}}+{\textbf{v}}} \quad \text {if } x_{\textbf{p}}\in A_{\varepsilon ,{\textbf{v}}}\text { and } x_{{\textbf{p}}+{\textbf{v}}}\in A_{\varepsilon ,-{\textbf{v}}} \text { for some } {\textbf{v}}\in V\\ x_{\textbf{p}}\quad \quad \text {otherwise.} \end{array}\right. } \end{aligned}$$\end{document}

We thus define the coupled dynamics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_\varepsilon :X\rightarrow X$$\end{document} as the composition

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} T_\varepsilon :=T_0\circ \Phi _\varepsilon . \end{aligned}$$\end{document}

Banach spaces and notation

Before setting the problem and stating the main results, we define Banach spaces and fix some notation which will be needed in the sequel.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}$$\end{document} be the space of functions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{\mathbb {Z}^d}$$\end{document} that depend only on a finite number of coordinates and are locally differentiable. Let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\mathcal {D}_1:=\{\varphi \in \mathcal {D}: |\varphi |_\infty \le 1\}.\end{aligned}$$\end{document}

For any Borel complex measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{\mathbb {Z}^d}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{\mathbb {Z}^d}$$\end{document} is equipped with the product topology, we define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\mu |&=\sup _{\varphi \in \mathcal {D}_1}\mu (\varphi ) \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \mu \Vert&=\sup _{{\textbf{p}}\in \mathbb {Z}^d}\sup _{\varphi \in \mathcal {D}_1}\mu (\partial _{{\textbf{p}}}\varphi ) \end{aligned}$$\end{document}

that is, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert $$\end{document} is the bounded variation norm and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\cdot |$$\end{document} is the total variation norm. Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}:=\{\Vert \mu \Vert <\infty \}$$\end{document} is a Banach space as mentioned [20]; moreover, elements in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} consist of measures whose finite dimensional marginals are absolutely continuous with respect to Lebesgue and the corresponding density is a function of bounded variation. Throughout the paper we use the following notation:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \subset \mathbb {Z}^d$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Lambda }: I^{\mathbb {Z}^d} \rightarrow I^{\Lambda }$$\end{document} be the canonical projection. When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda :=\{{\textbf{q}}\}$$\end{document} , by a slight abuse of notation, we will write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\textbf{q}}$$\end{document} instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\{{\textbf{q}}\}}$$\end{document} .

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} By \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Lambda |$$\end{document} we denote the cardinality of the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} .

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\mathbb {Z}^d}:=Leb^{\otimes \mathbb {Z}^d}$$\end{document} is the Lebesgue measure on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{\mathbb {Z}^d}$$\end{document} whereas \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\Lambda :=\pi _\Lambda *m_{\mathbb {Z}^d}$$\end{document} is the Lebesgue measure on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{\Lambda }$$\end{document} .

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _\Lambda *\nu $$\end{document} is the image measure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} through \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _\Lambda $$\end{document} . When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \in \mathcal {B}$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu _{\Lambda }$$\end{document} is the density of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _\Lambda *\nu $$\end{document} with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\Lambda }$$\end{document} .

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} Finally, we use \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|f|_{BV}$$\end{document} to denote the usual bounded variation norm for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:I\rightarrow \mathbb {R}$$\end{document} .

Rare events at one site

In this work we study rare events at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*\in \mathbb {Z}^d$$\end{document} . For this purpose we study the asymptotics as the size of the collision intervals involved with site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*\in \mathbb {Z}^d$$\end{document} shrinks to 0. To do this we use \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \in (0,\varepsilon ]$$\end{document} to label such intervals, i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\delta ,{\textbf{v}}}$$\end{document} and assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(A_{\delta ,{\textbf{v}}})=\delta $$\end{document} . All other collision intervals that are not involved with site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} will be denoted, as before, by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\varepsilon ,{\textbf{v}}}$$\end{document} and their size is independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} . Consequently, form now on we start to denote the fully coupled system by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,\delta }$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{v}}\in V$$\end{document} , let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}X^{{\textbf {v}}}_{\delta }({\textbf {p}}^*):=\{{\textbf {x}}\in I^{\mathbb {Z}^d}: x_{{\textbf {p}}^*}\notin A_{\delta ,{\textbf {v}}} \text { or } x_{{\textbf {p}}^*+{\textbf {v}}}\notin A_{\delta ,-{\textbf {v}}}\}\end{aligned}$$\end{document}

and define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & X_{0,\delta }({\textbf {p}}^*):=\bigcap _{{\textbf {v}}\in V}X^{{\textbf {v}}}_{\delta }({\textbf {p}}^*),\nonumber \\ & H_\delta ({\textbf {p}}^*):=I^{\mathbb {Z}^d}\setminus X_{0,\delta }({\textbf {p}}^*). \end{aligned}$$\end{document}

Through the paper, for a set5E and a subset \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \subset E$$\end{document} we will use the notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^cA:= E\backslash A$$\end{document} to refer to the complement of a set A in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{\mathbb {Z}^d}$$\end{document} . In that regard, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{0,\delta }({\textbf {p}}^*)=^cH_{\delta }({\textbf{p}}^*)$$\end{document} . Moreover, let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X_{n,\delta }({\textbf {p}}^*)=\cap _{i=0}^{n-1} T_{\varepsilon ,\delta }^{-i}X_{0,\delta }({\textbf {p}}^*) \end{aligned}$$\end{document}

and notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{n,\delta }({\textbf {p}}^*)$$\end{document} is the set of points whose orbits under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,\delta }$$\end{document} did not see a collision at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*$$\end{document} up to time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-1$$\end{document} . It is also called the survival set up to time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-1$$\end{document} .

We endow \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{\mathbb {Z}^d}$$\end{document} with the product topology and we recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\mathbb {Z}^d}$$\end{document} denotes Lebesgue measure on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{\mathbb {Z}^d}$$\end{document} . Suppose that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\lim _{n \rightarrow \infty }\frac{1}{n}\ln m_{\mathbb {Z}^d}(X_{n,\delta }({\textbf {p}}^*)) \end{aligned}$$\end{document}

exists. Then the quantity in the Eq. (7) measures asymptotically, with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\mathbb {Z}^d}$$\end{document} , the fraction of orbits that did not see a first collision at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*$$\end{document} . We call such a quantity the first collision rate under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_\varepsilon $$\end{document} at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*$$\end{document} with respect to the measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\mathbb {Z}^d}$$\end{document} . We now introduce a ‘decoupling’ operator at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} , which will play a crucial role in our work. Let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\Phi _{\varepsilon ,{\textbf{p}}^*}({\textbf {x}}))_{\textbf{q}}={\left\{ \begin{array}{ll} (\Phi _{\varepsilon }({\textbf {x}}))_{\textbf{q}}\quad \text {if } {\textbf{q}}-{\textbf{p}}^*\notin V\cup \{0\}\\ (\Phi _{\varepsilon }({\textbf {x}}))_{\textbf{q}}\quad \text {if } {\textbf{v}}={\textbf{q}}-{\textbf{p}}^* \text { and } x_{{\textbf{q}}}\notin A_{\varepsilon ,-{\textbf{v}}}\\ x_{\textbf{q}}\quad \quad \text {otherwise.} \end{array}\right. } \end{aligned}$$\end{document}

Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{\varepsilon ,{\textbf{p}}^*}$$\end{document} differs from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{\varepsilon }$$\end{document} , which was defined in (1), only in the coordinate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{{\textbf{p}}^*}$$\end{document} : under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{\varepsilon ,{\textbf{p}}^*}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{{\textbf{p}}^*}$$\end{document} is independent of the other coordinates. Now define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}T_{\varepsilon ,{\textbf {p}}^*}:=T_0\circ \Phi _{\varepsilon ,{\textbf{p}}^*}\end{aligned}$$\end{document}

and notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{n,\delta }({\textbf {p}}^*)$$\end{document} , the survival set defined in (6), is also given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X_{n,\delta }({\textbf {p}}^*)=\cap _{i=0}^{n-1} T_{\varepsilon , {\textbf{p}}^*}^{-i}X_{0,\delta }({\textbf {p}}^*); \end{aligned}$$\end{document}

that it is the survival set can be defined via the decoupled dynamics at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*\in \mathbb {Z}^d$$\end{document} . This is a very important idea in this work for the following reasons: firstly, the decoupled dynamics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,{\textbf {p}}^*}$$\end{document} does not depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} , consequently if we prove that it admits a physical measure, then the measure will depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} but not on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} and hence this will make this measure suitable to study asymptotics in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} for the rare event (see Theorem 2.1 below). More importantly, if we prove a spectral gap, on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} , for the transfer operator associated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,{\textbf {p}}^*}$$\end{document} , all its spectral data are independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} . The latter will be an important point to obtain a spectral gap for the corresponding rare event transfer operator, via a perturbation argument.

We define the transfer operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon , {\textbf {p}}^*}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathcal {L}}}_{\varepsilon ,\delta , {\textbf {p}}^*}$$\end{document} associated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon , {\textbf {p}}^*}$$\end{document} and the rare event respectively as follows. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in \mathcal {D}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} a Borel complex measure

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\int \varphi d\mathcal {L}_{\varepsilon , {\textbf {p}}^*}\mu =\int \varphi \circ T_{\varepsilon ,{\textbf{p}}^*}d\mu \end{aligned}$$\end{document}

and let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int \varphi d\hat{\mathcal {L}}_{\varepsilon , \delta , {\textbf {p}}^*}(\mu )=\int \varphi \circ T_{\varepsilon ,{\textbf {p}}^*} 1_{X_{0,\delta }({\textbf{p}}^*)} d\mu . \end{aligned}$$\end{document}

Note that by definition

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}|\mathcal {L}_{\varepsilon ,\delta }\mu |\le |\mu |\quad { and }\quad |{\hat{\mathcal {L}}}_{\varepsilon ,\delta ,{\textbf{p}}^*}\mu |\le |\mu |.\end{aligned}$$\end{document}

The following theorem is the first main result of the paper:

Theorem 2.1

For sufficiently small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document}

1. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon , {\textbf {p}}^*}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathcal {L}}}_{\varepsilon ,\delta , {\textbf {p}}^*,}$$\end{document} admit a spectral gap on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} ;
2. the first collision rate with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\mathbb {Z}^d}$$\end{document} exists and satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}-\lim _{n \rightarrow \infty }\frac{1}{n}\ln m_{\mathbb {Z}^d}(X_{n,\delta }({\textbf {p}}^*))=-\ln \lambda _{\varepsilon ,\delta },\end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{\varepsilon ,\delta }\in (0,1)$$\end{document} is the dominant simple eigenvalue of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathcal {L}}}_{\varepsilon , {\textbf {p}}^*,\delta }$$\end{document} .

In the next theorem, we study the asymptotic as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \rightarrow 0$$\end{document} , but keeping \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} fixed small enough as in Theorem 2.1. Denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{\textbf{v}}, a_{-{\textbf{v}}})$$\end{document} the point that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\delta ,{\textbf{v}}}\times A_{\delta ,-{\textbf{v}}}$$\end{document} shrinks to as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \rightarrow 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S:=\{(a_{\textbf{v}}, a_{-{\textbf{v}}})\}_{{\textbf{v}}\in V^+}$$\end{document} be the set of all such points. We assume that each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{\textbf{v}}\in I_i$$\end{document} , for some monotonicity interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_i$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} as defined at the beginning of Sect. 2 and we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{\textbf{v}}$$\end{document} is the midpoint of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\delta ,{\textbf{v}}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{v}}\in V$$\end{document} . Let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {N}_{{\textbf {p}}^*}=\{({\textbf{q}},{\textbf{v}}):\, {\textbf{q}}\in {\textbf {p}}^*+V, {\textbf{v}}={\textbf {p}}^*-{\textbf{q}}\}\cup \{{\textbf {p}}^*\}\times V \end{aligned}$$\end{document}

and introduce for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}A_{\delta ,{\textbf{v}}}({\textbf{q}})=\{{\textbf {x}}\in I^{\mathbb {Z}^d}\, \text { and } x_{{\textbf{q}}}\in A_{\delta , {\textbf{v}}}\}.\end{aligned}$$\end{document}

Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{A_{\delta ,{\textbf{v}}}({\textbf{q}})\}_{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}$$\end{document} consists only of collision sets that lead to a collision with the dynamics at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*$$\end{document} . We introduce the subset \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}\subset S$$\end{document} that consists of recurrent points of S, under the dynamics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,\delta }$$\end{document} , in the following sense:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} S^{rec}:=\,&\{(a_{\textbf{v}}, a_{-{\textbf{v}}}) \in S, \exists k \in \mathbb {N}, \exists {\textbf {x}}\in I^{\mathbb {Z}^d}, (x_{{\textbf{p}}^*},x_{{\textbf{p}}^*+{\textbf{v}}})=(a_{\textbf{v}}, a_{-{\textbf{v}}}) \text { and }\\&\exists ({\textbf{q}},{\textbf{v}}')\in \mathcal {N}_{{\textbf{p}}^*}\text { so that } (\tau ^ka_{{\textbf{v}}}, \tau ^ka_{-{\textbf{v}}})=(a_{{\textbf{v}}'}, a_{-{\textbf{v}}'})\\ &\text { and } \Psi _k^{{\textbf{p}}^*+{\textbf{v}}}({\textbf {x}})={\textbf{q}}\}, \end{aligned} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _k^{{\textbf {p}}}: I^{\mathbb {Z}^d} \rightarrow \mathbb {Z}^d$$\end{document} is an index map such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(T_{\varepsilon ,{\textbf{p}}^*}^k {\textbf {x}})_{\Psi _k^{{\textbf {p}}}({\textbf {x}})}=\tau ^k(x_{{\textbf{p}}})$$\end{document} . It is defined recursively as follows:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{r l c} \Psi _1^{{\textbf {p}}}({\textbf {x}})& ={\textbf{p}}+{\textbf{v}}\text { if } x_{\textbf{p}}\in A_{\varepsilon ,{\textbf{v}}}\text { and } x_{{\textbf{p}}+{\textbf{v}}}\in A_{\varepsilon ,-{\textbf{v}}} \\ & \qquad \text { for some } {\textbf{v}}\in V \text { whenever } {\textbf{p}}^* \notin \{{\textbf{p}},{\textbf{p}}+{\textbf{v}}\} \\ \Psi _1^{{\textbf {p}}}({\textbf {x}})& = {\textbf{p}}\text { otherwise}\\ \Psi _k^{{\textbf {p}}}({\textbf {x}})& =\Psi _{k-1}^{{\textbf {p}}}({\textbf {x}})+\Psi _1^{\Psi _{k-1}^{{\textbf {p}}}({\textbf {x}})}(T_{\varepsilon ,{\textbf {p}}^*}^{k-1}{\textbf {x}}) \text { if }k\ne 0. \end{array} \right. \end{aligned}$$\end{document}

Note that the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}$$\end{document} can strictly contain the set of periodic points of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,\delta }$$\end{document} in S. Indeed, some point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}$$\end{document} can return to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}$$\end{document} without being periodic of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,\delta }$$\end{document} . For each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{{\textbf{v}}},a_{-{\textbf{v}}})\in S^{rec}$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}({\textbf{v}},{\textbf{v}}')$$\end{document} be the set of integers k such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau ^{k+1}a_{{\textbf{v}}},\tau ^{k+1}a_{-{\textbf{v}}})=(a_{{\textbf{v}}'},a_{-{\textbf{v}}'})$$\end{document} . To each such k we associate the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_k$$\end{document} composed of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_1\le \dots \le j_r\le k$$\end{document} , the recurrent time into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}$$\end{document} up to time k (i.e, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau ^{j_i}(a_{{\textbf{v}}}),\tau ^{j_i}(a_{-{\textbf{v}}}))=(a_{{\textbf {w}}_{j_i}},a_{-{\textbf {w}}_{j_i}})$$\end{document} ).

The following notation will appear below in the statement of the second main result of the paper. Using the result of Theorem 2.1, we denote the eigenmeasure corresponding to the eigenvalue 1 for the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon , {\textbf {p}}^*}$$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon , {\textbf{p}}^*}\in \mathcal {B}$$\end{document} . Fix the box

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\Lambda :={\textbf {p}}^* + \{-(k+1),\dots ,(k+1)\}^d\end{aligned}$$\end{document}

around the vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,\Lambda }$$\end{document} be the density of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Lambda }*\mu _{\varepsilon ,{\textbf {p}}^*}$$\end{document} .

The first hitting/collision time at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*$$\end{document} is defined by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}t_\delta ({\textbf {x}})=\inf \{n\ge 0:\,T^n_{\varepsilon ,{\textbf {p}}^*}({\textbf {x}})\in H_\delta ({\textbf {p}}^*)\}.\end{aligned}$$\end{document}

Theorem 2.2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} be small enough as in Theorem 2.1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \in (0,\varepsilon ]$$\end{document} be the size of intervals in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{A_{\delta ,{\textbf{v}}}({\textbf{q}})\}_{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\ln \lambda _{\varepsilon ,\delta }$$\end{document} denote the corresponding first collision rate. Then

1. as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \rightarrow 0$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\lambda _{\varepsilon ,\delta }=1-\mu _{\varepsilon ,{\textbf {p}}^*}(H_\delta ({\textbf {p}}^*))\cdot \theta (1+o(1)),\end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in (0,1]$$\end{document} . In particular, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}= \emptyset $$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =1$$\end{document} ; otherwise \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in (0,1)$$\end{document} and is given by:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta =1-\sum _{{\textbf{v}},{\textbf{v}}'\in V}\sum _{k\in {\mathbb {K}({\textbf{v}},{\textbf{v}}')}}q_{k}({\textbf{v}},{\textbf{v}}'), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{k}({\textbf{v}},{\textbf{v}}')$$\end{document} is non zero whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau ^{k+1}(a_{{\textbf{v}}}),\tau ^{k+1}(a_{-{\textbf{v}}}))=(a_{{\textbf{v}}'},a_{-{\textbf{v}}'})$$\end{document} and is defined as follows:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&q_{k}({\textbf{v}},{\textbf{v}}')=\frac{1}{\sum _{(q,{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\rho _{\tau }(a_{\textbf{v}})\frac{\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^+)+\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^-)}{2}}\\&\quad \times \frac{\rho _{\tau }(a_{{\textbf{v}}})}{|(\tau ^{k+1})'(a_{{\textbf{v}}})|}\left( \frac{1}{2|(\tau ^{k+1})'(a_{-{\textbf{v}}})|}\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^+)+\frac{1}{2|(\tau ^{k+1})'(a_{-{\textbf{v}}})|}\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^-)\right) , \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{\rho }_{\varepsilon ,\Lambda ,k}=\mathbb {E}(1_{ (\Psi _k^{{\textbf{q}}}={\textbf{p}}^*+{\textbf{v}}') \cap \bigcap _{i=1}^r \,^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})}\rho _{\varepsilon ,\Lambda }(.)|I^{\{q\}}). \end{aligned}$$\end{document}

2. Moreover, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _\delta >0$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0}\xi _\delta =\theta $$\end{document} , such that for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\left| \mu _{\varepsilon ,{\textbf {p}}^*}\Big \{t_\delta \ge \frac{t}{\xi _\delta \mu _{\varepsilon ,{\textbf {p}}^*}(H_\delta ({\textbf {p}}^*))}\Big \}-e^{-t}\right| \le (t\vee 1)e^{-t} C\delta ^2\left| \ln (\delta ^2)\right| .\end{aligned}$$\end{document}

The key quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} explicitly stated in the above theorem is the extremal index associated with the event of having a collision at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} . Thus, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}\ne \emptyset $$\end{document} indicates clustering of collisions at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} .

Next, we are also interested in studying the distribution of number of collisions at a particular site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} . For this purpose, we study the following process

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Z_\delta (t):=\sum _{k=1}^{\left\lfloor \frac{t}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))}\right\rfloor }1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^k, \end{aligned}$$\end{document}

where, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,\delta }$$\end{document} is the fully coupled map lattice defined in (2), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \in (0,\varepsilon ]$$\end{document} being the size of collision zones involved with site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} .

Remark 2.3

Note that as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \rightarrow 0$$\end{document} the fully coupled map lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,\delta }$$\end{document} converges to the decoupled map, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon , {\textbf{p}}^*}$$\end{document} . Therefore, we consider in (13) the scaling \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lfloor \frac{t}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))}\rfloor $$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*}$$\end{document} is the measure associated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon , {\textbf{p}}^*}$$\end{document} .

Remark 2.4

Generally in classical dynamical systems literature, see for instance [11, 15], when studying the statistics of number of visits to a set the dynamics does not depend on the parameter of the target set. However, since we want to count the collisions of the fully coupled system that take place at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} , it is natural in our model to consider the process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_\delta (t)$$\end{document} (13) where both the target set and the dynamics depend on the size, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} , of the collision zones associated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} . The dependence on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} in the dynamics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,\delta }$$\end{document} leads to a ‘non-standard’ characteristic function of the iid sequence associated with the limiting compounded Poisson process in Theorem 2.5 below.

To state our next result, in addition to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}$$\end{document} defined above in (14), we need to introduce the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{S}^{rec}$$\end{document} as follows:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \tilde{S}^{rec}:=&\{(a_{\textbf{v}}, a_{-{\textbf{v}}}) \in S, \exists k \in \mathbb {N}, {\textbf{v}}'\in V, (\tau ^k a_{{\textbf{v}}},\tau ^ka_{-{\textbf{v}}})=(a_{{\textbf{v}}'},a_{-{\textbf{v}}'}) \\&\text { s.t } \exists {\textbf {x}}\in I^{\mathbb {Z}^d}, (x_{{\textbf{p}}^*},x_{{\textbf{p}}^*+{\textbf{v}}})=( a_{-{\textbf{v}}}, a_{\textbf{v}}) \text { and }\Psi _k^{{\textbf{p}}^*+{\textbf{v}}}({\textbf {x}})={\textbf{p}}^*-{\textbf{v}}'\}. \end{aligned} \end{aligned}$$\end{document}

Note that the main difference between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{S}^{rec}$$\end{document} is that in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}$$\end{document} we consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{{\textbf{p}}^*},x_{{\textbf{p}}^*+{\textbf{v}}})=( a_{{\textbf{v}}}, a_{-{\textbf{v}}})$$\end{document} , while in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{S}^{rec}$$\end{document} , we consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{{\textbf{p}}^*},x_{{\textbf{p}}^*+{\textbf{v}}})=( a_{-{\textbf{v}}}, a_{\textbf{v}})$$\end{document} , there is an intervertion coming from the fact we consider recurrence linked to the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\epsilon ,\delta }$$\end{document} instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\epsilon ,{\textbf{p}}^*}$$\end{document} .

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{{\textbf{v}}},a_{-{\textbf{v}}})\in S^{rec}\cup \tilde{S}^{rec}$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathbb {K}}({\textbf{v}},{\textbf{v}}')$$\end{document} be the set of integers k such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau ^{k+1}a_{{\textbf{v}}},\tau ^{k+1}a_{-{\textbf{v}}})=(a_{{\textbf{v}}'},a_{-{\textbf{v}}'})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_k$$\end{document} is the set composed of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_1\le \dots \le j_r\le k$$\end{document} , the recurrent time into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}\cup \tilde{S}^{rec}$$\end{document} up to time k. Before stating the next theorem, we recall the definition of a compound Poisson random process:

A stochastic process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(t): I^{\mathbb {Z}^d} \rightarrow \mathbb {N}$$\end{document} is compound Poisson distributed if there exists a Poisson process variable N(t) and a sequence of iid random variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_k: I^{\mathbb {Z}^d}\rightarrow \mathbb {N}$$\end{document} , which is also independent of N(t), such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}Z(t)=\sum _{k=1}^{N(t)}X_i.\end{aligned}$$\end{document}

Recall the process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_\delta (t)$$\end{document} which was defined in (13), which counts the number of collisions at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} from time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1$$\end{document} up to time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\lfloor \frac{t}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))}\right\rfloor $$\end{document} . We prove the following theorem:

Theorem 2.5

The process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_\delta (t)$$\end{document} converges in law (for any probability measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document} ; in particular \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*}$$\end{document} ) to a compound Poisson process

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}Z(t)=\sum _{i=1}^{N(t)}X_i,\end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N(t))_t$$\end{document} is a Poisson process of intensity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta Leb_{|[0,\infty )}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in (0,1]$$\end{document} is as in Theorem 2.2, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X_i)_{i\in \mathbb {N}}$$\end{document} is an iid sequence whose characteristic function is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _X(s)=\frac{\tilde{\theta }(s)(e^{is}-1)}{\theta }+1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\theta }:\mathbb {R}\rightarrow \mathbb {C}$$\end{document} . In particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\theta }(s) =1$$\end{document} when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}\cup \tilde{S}^{rec}=\emptyset $$\end{document} ; otherwise,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\tilde{\theta }(s)= 1-\sum _{{\textbf{v}},{\textbf{v}}'\in V}\sum _{k\in \tilde{\mathbb {K}}({\textbf{v}},{\textbf{v}}')}\sum _{j=0}^ke^{isj}\left( \beta _k^{(1)}(j,{\textbf{v}},{\textbf{v}}')-e^{is}\beta _k^{(2)}(j,{\textbf{v}},{\textbf{v}}')\right) \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\lim _{\delta \rightarrow 0}\frac{\mu _{\varepsilon ,{\textbf {p}}^*}\left( H_\delta ({\textbf {p}}^*)\cap T_{\varepsilon ,{\textbf{p}}^*}^{-1}\left( T_{\varepsilon ,\delta }^{-k}H_\delta ({\textbf {p}}^*)\cap \left( \sum _{i=0}^{k-1}1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^i=j\right) \right) \right) }{\mu _{\varepsilon ,{\textbf {p}}^*}\left( H_\delta ({\textbf {p}}^*)\right) }\\&\quad =:\beta _k^{(1)}(j,{\textbf{v}},{\textbf{v}}') \end{aligned} \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\lim _{\delta \rightarrow 0}\frac{\mu _{\varepsilon ,{\textbf {p}}^*}\left( H_\delta ({\textbf {p}}^*)\cap T_{\varepsilon ,\delta }^{-(k+1)}H_\delta ({\textbf {p}}^*)\cap \left( \sum _{i=1}^k1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^i=j\right) \right) }{\mu _{\varepsilon ,{\textbf {p}}^*}\left( H_\delta ({\textbf {p}}^*)\right) }\\&\quad =:\beta _{k}^{(2)}(j,{\textbf{v}},{\textbf{v}}'). \end{aligned} \end{aligned}$$\end{document}

Remark 2.6

Notice that in general, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}\not \subset \tilde{S}^{rec}$$\end{document} , indeed one may consider the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document} with periodic points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{{\textbf{v}}},a_{-{\textbf{v}}})$$\end{document} in the sense that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau a_{{\textbf{v}}},\tau a_{-{\textbf{v}}})=(a_{{\textbf{v}}},a_{-{\textbf{v}}})$$\end{document} , in that case, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{{\textbf{v}}},a_{-{\textbf{v}}}) \in S^{rec}$$\end{document} but \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{{\textbf{v}}},a_{-{\textbf{v}}}) \notin \tilde{S}^{rec}$$\end{document} . We also have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{S}^{rec}\not \subset S^{rec}$$\end{document} . One can take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{\mathbb {Z}^2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{v}}\ne {\textbf{v}}'$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{{\textbf{v}}},a_{-{\textbf{v}}})\in \tilde{S}^{rec}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau ^3 a_{{\textbf{v}}},\tau ^3 a_{-{\textbf{v}}})=(a_{-{\textbf{v}}'},a_{{\textbf{v}}'})$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau a_{{\textbf{v}}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^2 a_{{\textbf{v}}'}$$\end{document} belong respectively to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\varepsilon ,-{\textbf{v}}'} \backslash a_{-{\textbf{v}}'}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\varepsilon ,-{\textbf{v}}} \backslash a_{-{\textbf{v}}}$$\end{document} and suppose that the orbit of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau ^i a_{{\textbf{v}}})_{i\in \mathbb {N}}$$\end{document} crosses no collision interval except for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^3a_{{\textbf{v}}}$$\end{document} . Then by construction, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$( a_{{\textbf{v}}}, a_{-{\textbf{v}}})\in \tilde{S}^{rec}\backslash S^{rec}$$\end{document} .

Remark 2.7

By Remark 2.6, in general, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{k}^{(2)}(j,{\textbf{v}},{\textbf{v}}')\not =\beta _{k}^{(1)}(j,{\textbf{v}},{\textbf{v}}')$$\end{document} . Indeed, as a by-product of the proof of Theorem 2.5 in Sect. 5, we obtain that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{k}^{(2)}(j,{\textbf{v}},{\textbf{v}}')=0$$\end{document} whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{S}^{rec}=\emptyset $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{k}^{(1)}(j,{\textbf{v}},{\textbf{v}}')=0$$\end{document} whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}=\emptyset $$\end{document} . Finally, notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _k^{(1)}(0,{\textbf{v}},{\textbf{v}}')=q_k({\textbf{v}},{\textbf{v}}')$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_k({\textbf{v}},{\textbf{v}}')$$\end{document} as given in Theorem 2.2.

Remark 2.8

Starting with the work of [27], see also [28, 33], spatio-temporal Poisson processes were obtained by recording not only the successive times of visits to a set, but also the positions. It would be interesting to investigate if similar results can be obtained in infinite dimensional collision models as the one studied in this current work.

An example

In this subsection we present an example that satisfies the above results and show how to compute \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} of Theorem 2.2 and the law for the random variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X_i)_{i\in \mathbb {N}}$$\end{document} of Theorem 2.5 in a concrete situation. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau :[0,1] \rightarrow [0,1]$$\end{document} be given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\mapsto 5 x \,\mod 1$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document} , we only have two collision sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\delta ,{\textbf{v}}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{v}}\in V:=\{-1,1\}$$\end{document} . Assume that the collision sets are centered around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{{\textbf{v}}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{v}}\in V$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1=\frac{1}{2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{-1}=\frac{1}{4}$$\end{document} . Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1,a_{-1}$$\end{document} are fixed points for the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau a_{{\textbf{v}}}=a_{\textbf{v}}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{v}}\in V$$\end{document} ). One can check that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}=\{(a_1,a_{-1}),(a_{-1},a_1)\}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{S}^{rec}=\emptyset $$\end{document} 6. We can now apply Theorem 2.1 and Theorem 2.2: the first collision rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\ln (\lambda _{\varepsilon ,{\textbf{p}}^*})$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\lambda _{\varepsilon ,\delta }=1-\mu _{\varepsilon ,{\textbf {p}}^*}(H_\delta ({\textbf {p}}^*))\cdot \theta (1+o(1)),\end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta&=1-\sum _{{\textbf{v}},{\textbf{v}}'\in V}\sum _{k\in {\mathbb {K}({\textbf{v}},{\textbf{v}}')}}q_{k}({\textbf{v}},{\textbf{v}}')\\&=1- \sum _{k\ge 1}q_k(1,-1)-\sum _{k\ge 1}q_k(-1,1) \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&q_{k}({\textbf{v}},{\textbf{v}}')= \frac{1}{\sum _{(q,{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\rho _{\tau }(a_{\textbf{v}})\frac{\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^+)+\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^-)}{2}}\\&\qquad \times \frac{\rho _{\tau }(a_{{\textbf{v}}})}{|(\tau ^{k+1})'(a_{{\textbf{v}}})|}\left( \frac{1}{2|(\tau ^{k+1})'(a_{-{\textbf{v}}})|}\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^+)+\frac{1}{2|(\tau ^{k+1})'(a_{-{\textbf{v}}})|}\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^-)\right) \\&\quad =\frac{1}{\rho _{\tau }(a_1)\left( \frac{\rho _{\varepsilon ,{\textbf{p}}^*+1}(a_{-1}^+)+\rho _{\varepsilon ,{\textbf{p}}^*+1}(a_{-1}^-)}{2}\right) +\rho _{\tau }(a_{-1})\left( \frac{\rho _{\varepsilon ,{\textbf{p}}^*+1}(a_{1}^+)+\rho _{\varepsilon ,{\textbf{p}}^*+1}(a_{1}^-)}{2}\right) }\\&\qquad \times \frac{\rho _{\tau }(a_{1})}{|(\tau ^{k+1})'(a_{1})|}\left( \frac{1}{2|(\tau ^{k+1})'(a_{-1})|}\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-1}^+)+\frac{1}{2|(\tau ^{k+1})'(a_{-1})|}\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-1}^-)\right) . \end{aligned}$$\end{document}

We now compute \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^\pm )$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 1$$\end{document} . We consider the case when the direction is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{v}}=+1$$\end{document} . The case when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{v}}=-1$$\end{document} follows the same reasoning. To estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^\pm )$$\end{document} notice that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\textbf{q}}({\textbf {x}})$$\end{document} belongs to a little neighborhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{\textbf{v}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\textbf{q}}(T_{\epsilon ,{\textbf{p}}^*}^j{\textbf {x}})$$\end{document} remains close to the periodic point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{\textbf{v}}$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\le k$$\end{document} . Thus, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\le k$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _j^{\textbf{q}}({\textbf{p}}^*+ 1)={\textbf{p}}^*+1$$\end{document} and thus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1_{(\Psi _k^{{\textbf{q}}}={\textbf{p}}^*+1) \cap \bigcap _{i=1}^r \,^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})}({\textbf {x}})=0$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1_{(\Psi _k^{{\textbf{q}}}={\textbf{p}}^*+1)}({\textbf {x}})=1$$\end{document} . Then,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{\rho }_{\varepsilon ,\Lambda ,k}&=\mathbb {E}(1_{ (\Psi _k^{{\textbf{q}}}={\textbf{p}}^*+{\textbf{v}}') \cap \bigcap _{i=1}^r \,^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})}\rho _{\varepsilon ,\Lambda }(.)|I^{\{q\}}). \end{aligned}$$\end{document}

Consequently, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\hat{\rho }_{\varepsilon ,\Lambda ,k}=0\end{aligned}$$\end{document}

while

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\hat{\rho }_{\varepsilon ,\Lambda ,1}=\rho _{\varepsilon ,{\textbf{q}}}.\end{aligned}$$\end{document}

Thus,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} q_{1}(1,-1)+q_{1}(-1,1)&=\frac{1}{|(\tau ^{2})'(a_{1})||(\tau ^2)'(a_{-1})|}\\&=5^{-(2+2)}=5^{-4}. \end{aligned}$$\end{document}

We thus deduce the extremal index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\theta =1-5^{-4}.\end{aligned}$$\end{document}

And the rare event statistics from Theorem 2.2 follows, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _\delta >0$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0}\xi _\delta =1$$\end{document} , such that for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\left| \mu _{\varepsilon ,{\textbf {p}}^*}\Big \{t_\delta \ge \frac{t}{\xi _\delta \mu _{\varepsilon ,{\textbf {p}}^*}(H_\delta ({\textbf {p}}^*))}\Big \}-e^{-\theta t}\right| \le (t\vee 1)e^{-\theta t} C\delta ^2\left| \ln (\delta ^2)\right| .\end{aligned}$$\end{document}

For the compound Poisson process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(t):=\sum _{i=1}^{N(t)}X_i$$\end{document} of Theorem 2.5 we can also conclude that (see the a detailed justification in Sect. 5.2)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{\theta }(s)=1-\sum _{{\textbf{v}},{\textbf{v}}'\in V}\sum _{k\in {\mathbb {K}({\textbf{v}},{\textbf{v}}')}}\beta _{k,\delta }^{(1)}(j,{\textbf{v}},{\textbf{v}}')=\theta \end{aligned}$$\end{document}

and that the law of the integer variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_i$$\end{document} is given by the following characteristic function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{X}(s)=e^{is}$$\end{document} . In other words, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_i$$\end{document} is almost surely constant and equal to 1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(t)\sim \mathcal {P}(\theta t)$$\end{document} , that is a Poisson law of intensity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta t$$\end{document} .

Strategy of the proofs

The proofs are based on spectral techniques of [18, 19, 23], albeit in an infinite dimensional setting. We proceed as follows:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} Show \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon , {\textbf {p}}^*}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathcal {L}}}_{\varepsilon ,\delta , {\textbf {p}}^*}$$\end{document} satisfy a uniform Lasota-Yorke inequality, and recalling (11), show that the difference of the two, acting on measures from the strong space and evaluated in the weak norm, is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(\delta )$$\end{document} . Then show for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} small enough \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon ,{\textbf {p}}^*}$$\end{document} has a spectral gap on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} , and by [19], conclude for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} small enough that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathcal {L}}}_{\varepsilon ,\delta , {\textbf {p}}^*}$$\end{document} also has a spectral gap, with leading eigenvalue \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{\varepsilon ,\delta }\in (0,1)$$\end{document} .

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} Using the notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{\varepsilon ,\delta }$$\end{document} as above, we now differentiate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{\varepsilon ,\delta }$$\end{document} following the abstract results of [23] and obtain a formula for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} , which is the main technical part of this step. To obtain the law for the first collision time, with sharp error term, we follow [18].

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} Finally, as suggested by [1], to obtain a limit law for counting the number of visits to a target set, one can still use the spectral framework of [23]. For this purpose, we study the ‘twisted’ transfer operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathcal {L}}_{\delta ,s}(\cdot ):=\mathcal {L_{\varepsilon ,\delta }}(e^{is1_{H_\delta ({\textbf{p}}^*)}}\cdot )$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L_{\varepsilon ,\delta }}$$\end{document} is the transfer operator of the fully coupled map lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,\delta }$$\end{document} . This will provide the expression of the characteristic function of (13) and its limit as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \rightarrow 0$$\end{document} .

Proof of Theorem 2.1

Recall the definition of the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\mathcal {L}}_{\varepsilon , \delta , {\textbf {p}}^*}$$\end{document} , which was introduced in (10)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int \varphi d\hat{\mathcal {L}}_{\varepsilon , \delta , {\textbf {p}}^*}(\mu )=\int \varphi \circ T_{\varepsilon ,{\textbf {p}}^*} 1_{X_{0,\delta }({\textbf{p}}^*)} d\mu . \end{aligned}$$\end{document}

Note that by its definition, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\mathcal {L}}_{\varepsilon , \delta , {\textbf {p}}^*}$$\end{document} satisfies a uniform Lasota–Yorke inequality (see Proposition 6.1). But since the unit ball of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert $$\end{document} is not compact in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\cdot |$$\end{document} the Lasota-Yorke inequality does not immediately imply that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\mathcal {L}}_{\varepsilon , \delta , {\textbf {p}}^*}$$\end{document} is quasi-compact. However, using perturbation arguments, below we show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\mathcal {L}}_{\varepsilon , \delta , {\textbf {p}}^*}$$\end{document} has in fact a spectral gap when acting on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} .

Spectral gap for the rare event transfer operator

Lemma 3.1

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\sup _{\Vert \mu \Vert \le 1}|(\mathcal {L}_{\varepsilon , {\textbf {p}}^*}-\hat{\mathcal {L}}_{\varepsilon ,\delta , {\textbf {p}}^*})\mu |=\mathcal {O}({\delta }).\end{aligned}$$\end{document}

Proof

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in \mathcal {D}_1$$\end{document} , which depends on a finite number of coordinates that are included in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _0\subset I^{\mathbb {Z}^d}$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int \varphi d\left[ (\mathcal {L}_{\varepsilon ,\delta }-{\hat{\mathcal {L}}}_{\varepsilon ,\delta ,{\textbf{p}}^*})\mu \right] \nonumber \\&\quad =\sum _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf{p}}^*}}\int \varphi \circ T_{\varepsilon ,{\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} 1_{A_{\delta ,{\textbf{v}}}}\circ \pi _{{\textbf{p}}^*} d\mu \nonumber \\&\quad \le \Vert \varphi \Vert _\infty \sum _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf{p}}^*}}\int _{I^\Lambda } 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} 1_{A_{\delta ,{\textbf{v}}}}\circ \pi _{{\textbf{p}}^*} hdm_{\Lambda }\nonumber \\&\quad \le \Vert \varphi \Vert _\infty \sum _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf{p}}^*}}\int _{I^2} 1_{A_{\delta ,-{\textbf{v}}}}(x_{{\textbf{q}}}) 1_{A_{\delta ,{\textbf{v}}}} (y_{{\textbf{p}}^*}) h_{{\textbf{v}}}(x_{{\textbf{q}}},{y_{{\textbf{p}}^*}})dx_{\textbf{q}}dy_{{\textbf{p}}^*}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda =\Lambda _0\cup \mathcal {N}_{{\textbf{p}}^*}$$\end{document} , h is the density of the marginal of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{{\textbf{v}}}(x_{{\textbf{q}}},{y_{{\textbf{p}}^*}})= \int hdm_{\Lambda {\setminus } \mathcal {N}_{{\textbf{p}}^*}\cup \{{\textbf{p}}^*\}}$$\end{document} . Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|h_{{\textbf{v}}}|_{BV}\le |h|_{BV}\le \Vert \mu \Vert <\infty $$\end{document} . Thus, by applying first the Cauchy-Schwarz inequality then the Sobolev inequality, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\int \varphi d\left[ (\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}-{\hat{\mathcal {L}}}_{\varepsilon ,\delta ,{\textbf{p}}^*})\mu \right] \le 2d\Vert \varphi \Vert _{\infty }\Vert \mu \Vert \delta .\end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Theorem 2.1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon ,\delta }$$\end{document} denotes the transfer operator of the fully coupled systems; i.e., for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in \mathcal {D}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} a Borel complex measure

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\int \varphi d\mathcal {L}_{\varepsilon ,\delta }\mu =\int \varphi \circ T_{\varepsilon ,\delta }d\mu .\end{aligned}$$\end{document}

By the results of [21, 22]7 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} sufficiently small, there exists8 a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} such that for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (1)=0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \mathcal {L}_{\varepsilon ,\delta }^n\mu \Vert \le C\gamma ^n\Vert \mu \Vert . \end{aligned}$$\end{document}

Using the above inequality, the Lasota–Yorke inequality for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}$$\end{document} from Proposition 6.1, and the fact that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}|(\mathcal {L}_{\varepsilon , {\textbf {p}}^*}^m- \mathcal {L}_{\varepsilon ,\delta }^m)\mu |\le C_m \delta \Vert \mu \Vert \end{aligned}$$\end{document}

we can also conclude, that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} small enough, there exists a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} such that for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (1)=0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \mathcal {L}_{\varepsilon , {\textbf {p}}^*}^n\mu \Vert \le C\gamma ^n\Vert \mu \Vert . \end{aligned}$$\end{document}

Then by a weak compactness argument (see [20, 21]) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,{\textbf{p}}^*}$$\end{document} admits an invariant probability measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*}\in \mathcal {B}$$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document} , let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\Pi _{1}\mu = \mu (1)\cdot \mu _{\varepsilon ,{\textbf{p}}^*}\quad \text { and }\quad Q=\mathcal {L}_{\varepsilon , {\textbf {p}}^*}-\Pi _1.\end{aligned}$$\end{document}

Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _1^2=\Pi _1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _1Q=Q\Pi _1=0$$\end{document} . Moreover, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}Q\mu =\mathcal {L}_{\varepsilon , {\textbf {p}}^*}\mu - \mu (1)\cdot \mu _{\varepsilon ,{\textbf{p}}^*}=\mathcal {L}_{\varepsilon , {\textbf {p}}^*}(\mu -\mu (1)\cdot \mu _{\varepsilon ,{\textbf{p}}^*}).\end{aligned}$$\end{document}

Therefore, by (16),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\Vert Q^n\mu \Vert \le C\gamma ^n\Vert \mu \Vert \end{aligned}$$\end{document}

and consequently \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {spec}(\mathcal {L}_{\varepsilon , {\textbf {p}}^*})\cap \{|z|>\gamma \}=\{1\}$$\end{document} ; i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon , {\textbf {p}}^*}$$\end{document} admits a spectral gap on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} . Note that the spectral data of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon , {\textbf {p}}^*}$$\end{document} do not depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} . Consequently, by Proposition 6.1 and Lemma 3.1, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} sufficiently small, the Keller-Liverani Stability result9 [19] implies that the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathcal {L}}}_{\varepsilon ,\delta , {\textbf {p}}^*}$$\end{document} admits a spectral gap on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} . We denote the corresponding simple dominant eigenvalue by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{\varepsilon , \delta }$$\end{document} . We denote the corresponding eigenmeasure by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mu }}_{\varepsilon ,\delta , p^*}$$\end{document} . It is obvious that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{\varepsilon , \delta }\in (0,1)$$\end{document} , since for any positive measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}|{\hat{\mathcal {L}}}_{\varepsilon ,\delta , {\textbf {p}}^*}^n\mu | =|\mathcal {L}_{\varepsilon ,{\textbf {p}}^*}^n1_{X_{n,\delta }({\textbf {p}}^*)}\mu |\le |1_{X_{n,\delta }({\textbf {p}}^*)}\mu |<|\mu |.\end{aligned}$$\end{document}

Moreover, we can write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _{\varepsilon , \delta }^{-n}{\hat{\mathcal {L}}}_{\varepsilon ,\delta , {\textbf {p}}^*}^n=\Pi _{\lambda _{\varepsilon , \delta }} + \hat{Q}_{\varepsilon ,\delta , {\textbf {p}}^*}^n, \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \hat{Q}_{\varepsilon ,\delta , {\textbf {p}}^*}^n\Vert <1$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _{\lambda _{\varepsilon , \delta }}$$\end{document} is a rank one operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Pi _{\lambda _{\varepsilon , \delta }}=\Psi (.){\hat{\mu }}_{\varepsilon ,\delta , p^*}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi (.)$$\end{document} is a linear form on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \in \mathcal {B}$$\end{document} be a positive measure and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in \mathcal {D}$$\end{document} be a positive function. By (17), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \int \varphi d\lambda _{\varepsilon , \delta }^{-n}{\hat{\mathcal {L}}}_{\varepsilon ,\delta , {\textbf {p}}^*}^n\nu -\int \varphi d\Pi _{\lambda _{\varepsilon , \delta }}\nu \right|&\le |\varphi |_{\infty }\left\| \lambda _{\varepsilon , \delta }^{-n}{\hat{\mathcal {L}}}_{\varepsilon ,\delta , {\textbf {p}}^*}^n\nu -\Pi _{\lambda _{\varepsilon , \delta }}\nu \right\| \\&\underset{n\rightarrow \infty }{\longrightarrow }\ 0. \end{aligned}$$\end{document}

Thus, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int \varphi d\Pi _{\lambda _{\varepsilon , \delta }}\nu $$\end{document} is positive. In particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int d\Pi _{\lambda _{\varepsilon , \delta }} m_{\mathbb {Z}^d}$$\end{document} is positive. Therefore, by definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathcal {L}}}_{\varepsilon ,\delta , {\textbf {p}}^*}$$\end{document} and (17)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} m_{\mathbb {Z}^d}(X_{n,\delta }({\textbf {p}}^*))&=\int d{\hat{\mathcal {L}}}_{\varepsilon ,\delta , {\textbf {p}}^*}^nm_{\mathbb {Z}^d}\\&=\lambda ^n_{\varepsilon ,\delta }\left[ \int d\Pi _{\lambda _{\varepsilon , \delta }} m_{\mathbb {Z}^d}+\int d Q_{\varepsilon ,\delta , {\textbf {p}}^*}^n m_{\mathbb {Z}^d} \right] , \end{aligned}$$\end{document}

Consequently, using the above result and the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Q_{\varepsilon ,\delta , {\textbf {p}}^*}|\le \Vert Q_{\varepsilon ,\delta , {\textbf {p}}^*}\Vert <1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists C>0$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \frac{1}{n}\ln \left( m_{\mathbb {Z}^d}(X_{n,\delta }({\textbf {p}}^*))\right) -\ln \lambda _{\varepsilon ,\delta }\right|&=\frac{1}{n}\ln \left| \int d\Pi _{\lambda _{\varepsilon , \delta }} m_{\mathbb {Z}^d}+\int d Q_{\varepsilon ,\delta , {\textbf {p}}^*}^n m_{\mathbb {Z}^d}\right| \\&\le \frac{C}{n}. \end{aligned}$$\end{document}

Thus, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{n \rightarrow \infty }\frac{1}{n}\ln m_{\mathbb {Z}^d}(X_{n,\delta }({\textbf {p}}^*))=\ln \lambda _{\varepsilon ,\delta }$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Theorem 2.2

We first introduce the following notation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \eta _{\delta }:=\sup _{\Vert \mu \Vert \le 1}\left| \left( \mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(\mu )-{\hat{\mathcal {L}}}_{\varepsilon ,\delta ,{\textbf{p}}^*}(\mu )\right) (1)\right| , \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*}\in \mathcal {B}$$\end{document} be the unique eigenmeasure corresponding to the eigenvalue 1 for the transfer operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}$$\end{document} , given a finite set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \subset \mathbb {Z}^d$$\end{document} , we will denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,\Lambda }$$\end{document} the density of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Lambda }*\mu _{\varepsilon ,{\textbf {p}}^*}$$\end{document} . We now set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _\delta :=\left( \mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(\mu _{\varepsilon ,{\textbf{p}}^*})-{\hat{\mathcal {L}}}_{\varepsilon ,\delta ,{\textbf{p}}^*}(\mu _{\varepsilon ,{\textbf{p}}^*})\right) (1). \end{aligned}$$\end{document}

The proof of Theorem 2.2 relies on a subtle application of [23]. To check the different assumptions of [23], we first prove the following Lemma which provides a convenient approximation of the quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _\delta $$\end{document} in terms of the Lebesgue measure. This will play later a key ingredient in the proof of Lemma 4.2 and Lemma 4.4.

Lemma 4.1

The following holds:

1. The mass of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\delta ({\textbf {p}}^*)$$\end{document} under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*}$$\end{document} scales as follows:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf {p}}^*))&\underset{\delta \rightarrow 0}{\sim }\sum _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}{\rho _{\tau }(a_{\textbf{v}})m(A_{\delta ,{\textbf{v}}}})m(A_{\delta ,-{\textbf{v}}})\frac{\rho _{\varepsilon ,{\textbf{q}}}(a_{{\textbf{v}}}^+)+\rho _{\varepsilon ,{\textbf{q}}}(a_{{\textbf{v}}}^-)}{2}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\tau }=d\mu _\tau /dm$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _\tau $$\end{document} is the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} -absolutely continuous invariant measure and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,{\textbf{q}}}=d(\pi _{{\textbf{q}}}*\mu _{\varepsilon ,p^*})/dm$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,q}(a_{-{\textbf{v}}}^+):=\lim _{x\rightarrow a_{-{\textbf{v}}}^+}\rho _{\varepsilon ,{\textbf{q}}}(x)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^-)=\lim _{x\rightarrow a_{-{\textbf{v}}}^{-}}\rho _{\varepsilon ,{\textbf{q}}}(x)$$\end{document} . In particular,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\lim _{\delta \rightarrow 0}\frac{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf {p}}^*))}{m_{\mathbb {Z}^d}(H_\delta ({\textbf {p}}^*))}=\frac{1}{|\mathcal {N}_{{\textbf {p}}^*}|} \sum _{(q,{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\rho _{\tau }(a_{\textbf{v}})\frac{\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^+)+\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^-)}{2}.\end{aligned}$$\end{document}

2. In addition for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} small enough,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\frac{1}{|\mathcal {N}_{{\textbf {p}}^*}|} \sum _{(q,{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\rho _{\tau }(a_{\textbf{v}})\frac{\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^+)+\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^-)}{2}>0.\end{aligned}$$\end{document}

Proof

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*}\in \mathcal {B}$$\end{document} , for any finite set10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \subset \mathbb {Z}^d$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _\Lambda *\mu _{\varepsilon ,{\textbf{p}}^*} \ll \pi _\Lambda *m_{\mathbb {Z}^d}$$\end{document} . Thus

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta )&=\sum _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\mu _{\varepsilon ,{\textbf{p}}^*}(A_{\delta ,{\textbf{v}}}({\textbf{q}})\cap A_{\delta ,-{\textbf{v}}}({\textbf {p}}^*))\end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&=\sum _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\int _{A_{\delta ,{\textbf{v}}}}\rho _{\tau }dm\int _{A_{\delta ,-{\textbf{v}}}}d\pi _{{\textbf{q}}}*\mu _{\varepsilon ,{\textbf{p}}^*}\end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&=\sum _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\mu _{\tau }(A_{\delta ,{\textbf{v}}})\int 1_{A_{\delta ,-{\textbf{v}}}}\rho _{\varepsilon ,{\textbf{q}}}dm\end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&=\sum _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\frac{\mu _{\tau }(A_{\delta ,{\textbf{v}}})}{m(A_{\delta ,{\textbf{v}}})}m(A_{\delta ,{\textbf{v}}})m(A_{\delta ,-{\textbf{v}}})\frac{1}{m(A_{\delta ,-{\textbf{v}}})}\int _{A_{\delta ,-{\textbf{v}}}}\rho _{\varepsilon ,{\textbf{q}}}dm \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\underset{\delta \rightarrow 0}{\sim }\sum _{(q,{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\rho _{\tau }(a_{\textbf{v}})m(A_{\delta ,{\textbf{v}}})m(A_{\delta ,-{\textbf{v}}})\frac{\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^+)+\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^-)}{2}. \end{aligned}$$\end{document}

where in (24) we used the fact that the density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _\tau $$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} and the fact that since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*}\in \mathcal {B}$$\end{document} , the limits \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,q}(a_{-{\textbf{v}}}^+):=\lim _{x\rightarrow a_{-{\textbf{v}}}^+}\rho _{\varepsilon ,{\textbf{q}}}(x)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^-)=\lim _{x\rightarrow a_{-{\textbf{v}}}^{-}}\rho _{\varepsilon ,{\textbf{q}}}(x)$$\end{document} are well defined. Note that the passage from (20) to (21) is due to the decoupling of the measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*}$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*$$\end{document} (no collision with the site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} ) which can be seen as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Lambda }*\mu _{\varepsilon ,{\textbf{p}}^*} =\pi _{\Lambda \backslash \{{\textbf {p}}^*\}}*\mu _{\varepsilon ,{\textbf{p}}^*}\otimes \mu _\tau $$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _\tau $$\end{document} being the invariant measure associated to the site transformation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} .

To prove the last statement of Lemma 4.1, i.e

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{|\mathcal {N}_{{\textbf {p}}^*}|} \sum _{(q,{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\rho _{\tau }(a_{\textbf{v}})\frac{\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^+)+\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^-)}{2}>0 \end{aligned}$$\end{document}

we first prove the following two statements

• (i)There is a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \varepsilon >0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\rho _{\varepsilon ,{\textbf{q}}}|_{BV}\le C$$\end{document} .
• (ii)Moreover, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\varepsilon \rightarrow 0}|\rho _\tau -\rho _{\varepsilon ,{\textbf{q}}}|_{L^1(m)}=0$$\end{document} . To prove statement (i), notice that by Proposition 6.1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists \, C>0$$\end{document} , independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} , such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert <C$$\end{document} . Hence, by definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,{\textbf{q}}}$$\end{document} we also have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\rho _{\varepsilon ,{\textbf{q}}}|_{BV}\le \Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert <C$$\end{document} .

We now prove the statement (ii). Define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}=T_0\circ \Phi _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}$$\end{document} the map decoupled in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{q}}$$\end{document} where for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in \mathbb {Z}^d$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\Phi _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}(x))_r={\left\{ \begin{array}{ll} (\Phi _{\varepsilon ,{\textbf{p}}^*}(x))_r \quad \text {if } r-{\textbf{q}}\notin V\cup \{0\}\\ (\Phi _{\varepsilon ,{\textbf{p}}^*}(x))_r \quad \text {if } {\textbf{v}}=r-{\textbf{q}}\text { and } x_{r}\notin A_{\varepsilon ,-{\textbf{v}}}\\ x_r \quad \quad \text {otherwise.} \end{array}\right. } \end{aligned}$$\end{document}

We denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}$$\end{document} the associated operator decoupled at sites \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{q}}$$\end{document} . It satisfies the Lasota-Yorke inequality of Proposition 6.1. We recall from (16) we proved a spectral gap for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}$$\end{document} with the following properties: given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \in \mathcal {B}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}_{\varepsilon ,{\textbf{p}}^*}\nu =\nu (1)\mu _{\varepsilon ,{\textbf{p}}^*}+Q_{\varepsilon ,{\textbf{p}}^*}(\nu ). \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert Q^n_{\varepsilon ,{\textbf{p}}^*}\Vert <C\gamma ^n$$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} . The same strategy also holds for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}_{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}\nu =\nu (1)\mu _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}+Q_{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}(\nu ). \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert Q^n_{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}\Vert <C\gamma ^n$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}$$\end{document} the invariant measure for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}$$\end{document} . Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mu _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}})_{{\textbf{q}}}=\rho _\tau $$\end{document} . Indeed, taking any local test function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi =\phi \circ \pi _{\textbf{q}}$$\end{document} at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{q}}$$\end{document} , the invariant property of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}$$\end{document} implies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int \phi d\mu _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}=\int \phi \circ T_{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}} d\mu _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}&=\int \phi \circ \pi _q\circ T_{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}(\mu _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}})_{{\textbf{q}}} dm\nonumber \\&=\int \phi \circ \tau (\mu _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}})_{{\textbf{q}}} dm. \end{aligned}$$\end{document}

Thus, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mu _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}})_{{\textbf{q}}}$$\end{document} is the unique invariant density for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} absolutely continue with respect to Lebesgue measure. Hence, it coincides with the invariant density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _\tau $$\end{document} . Moreover,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\Vert \mu \Vert \le 1}\left| \left( (\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}-\mathcal {L}_{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}})\mu \right) (1)\right| \le \eta _\varepsilon . \end{aligned}$$\end{document}

Therefore, by the Keller-Liverani [19] stability result

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} | \mu _{\varepsilon ,{\textbf{p}}^*}-\mu _{\varepsilon ,{\textbf{p}}^*,{\textbf{q}}}| =0. \end{aligned}$$\end{document}

and by the linearity of conditioning over the state of all sites except \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{q}}$$\end{document} , we deduce that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}|\rho _\tau -\rho _{\varepsilon ,{\textbf{q}}}|_{L^1(m)}=0, \end{aligned}$$\end{document}

which concludes statement ii) above. We now use (i) and (ii) to prove the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,{\textbf{q}}}(a_{{\textbf{v}}})>0$$\end{document} . To do so, we suppose that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \varepsilon >0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,{\textbf{q}}}$$\end{document} is continuous at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{{\textbf{v}}}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,q}(a_{{\textbf{v}}})=0$$\end{document} and then reach a contradiction.

Recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\inf _{x\in [0,1]}\rho _\tau (x)=\kappa >0$$\end{document} . We suppose that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \varepsilon >0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,{\textbf{q}}}$$\end{document} is continuous at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{{\textbf{v}}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,{\textbf{q}}}(a_{{\textbf{v}}})=0$$\end{document} . Then statement i) implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \beta ,\varepsilon >0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists \gamma _{\beta }>0$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,{\textbf{q}}}(x)\le \gamma _{\beta }$$\end{document} for all x s.t \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|x-a_{{\textbf{v}}}|\le \beta $$\end{document} . Because of statement ii), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\rho _\tau -\rho _{\varepsilon ,{\textbf{q}}}|_{L^1}=\xi _{\varepsilon }=o(1)$$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \xi _{\varepsilon }\ge |\int _{-\beta }^\beta \rho _\tau (x)dm-\int _{-\beta }^\beta \rho _{\varepsilon ,{\textbf{q}}}dm|\ge 2\beta (\kappa -\gamma _\beta ). \end{aligned}$$\end{document}

Consequently, fixing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} small enough so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa -\gamma _\beta >0$$\end{document} we obtain a contradiction by letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} . Thus for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} small enough,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{|\mathcal {N}_{{\textbf {p}}^*}|} \sum _{(q,{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\rho _{\tau }(a_{\textbf{v}})\frac{\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^+)+\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^-)}{2}>0. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.2

The following holds

1. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _\delta \rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \rightarrow 0$$\end{document} ;
2. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists \, C>0$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _\delta \cdot \Vert (\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}-{\hat{\mathcal {L}}}_{\varepsilon ,\delta ,{\textbf{p}}^*})\mu _{\varepsilon ,{\textbf{p}}^*}\Vert \le C |\Delta _\delta |$$\end{document} .

Proof

The first statement is a direct consequence of Lemma 3.1, notice that we actually proved

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \eta _\delta =\sup _{\mu \in \mathcal {B}} \frac{|\mu (H_\delta )({\textbf{p}}^*)|}{\Vert \mu \Vert }\le 4d(m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*)))^{1/2}. \end{aligned}$$\end{document}

To prove the second statement, we first upper bound for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert (\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}-{\hat{\mathcal {L}}}_{\varepsilon ,\delta ,{\textbf{p}}^*})\mu _{\varepsilon ,{\textbf{p}}^*}\Vert $$\end{document} .

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\phi }\in \mathcal {D}_1$$\end{document} a local coordinate map and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _0\subset \mathbb {Z}^d$$\end{document} finite and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :I^{\Lambda _0}\mapsto \mathbb {R}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\phi }=\varphi \circ \pi _{\Lambda _0}$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda :=\Lambda _0\cup \{{\textbf{p}}+{\textbf{v}}, {\textbf{p}}\in \Lambda _0, {\textbf{v}}\in V\}$$\end{document} . Notice that, there is a map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{T}_{\varepsilon ,{\textbf{p}}^*}:I^{\Lambda }\mapsto I^{\Lambda }$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Lambda _0}\circ \tilde{T}_{\epsilon ,{\textbf{p}}^*}\circ \pi _{\Lambda }=\pi _{\Lambda _0}\circ T_{\varepsilon ,{\textbf{p}}^*}$$\end{document} . Thus for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in \mathcal {D}$$\end{document} depending on local coordinates in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _0$$\end{document} .

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int \partial _r \varphi d\left[ (\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}-{\hat{\mathcal {L}}}_{\varepsilon ,\delta ,{\textbf{p}}^*})\mu _{\varepsilon ,{\textbf{p}}^*}\right] \\&\quad =\sum _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf{p}}^*}}\int \partial _r \varphi \circ \pi _{\Lambda _0}\circ T_{\varepsilon , {\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} 1_{A_{\delta ,{\textbf{v}}}}\circ \pi _{{\textbf{p}}^*} d\mu _{\varepsilon ,{\textbf{p}}^*}\\&\quad =\sum _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf{p}}^*}}\int \partial _r \varphi \circ \pi _{\Lambda _0}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} 1_{A_{\delta ,{\textbf{v}}}}\circ \pi _{{\textbf{p}}^*} \rho _{\varepsilon ,{\textbf{p}}^*}\rho _{\varepsilon ,\Lambda \backslash \{{\textbf{p}}^*\}}dm_{\Lambda }\\&\quad =\sum _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf{p}}^*}}\int \int \partial _r \varphi \circ \pi _{\Lambda _0}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} \rho _{\varepsilon ,\Lambda \backslash \{{\textbf{p}}^*\}}dm_{\Lambda \backslash \{{\textbf{p}}^*\}} 1_{A_{\delta ,{\textbf{v}}}}\rho _{\varepsilon ,{\textbf{p}}^*}dm\\ \end{aligned}$$\end{document}

Recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,\Lambda _0}$$\end{document} is the density of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Lambda _0}*\mu _{\varepsilon ,{\textbf{p}}^*}$$\end{document} . We now distinguish different cases according to whether \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r={\textbf{p}}^*$$\end{document} or not. Suppose first that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\ne {\textbf{p}}^*$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int \partial _r \varphi \circ \pi _{\Lambda _0}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} \rho _{\varepsilon ,\Lambda \backslash \{{\textbf{p}}^*\}}dm_{\Lambda \backslash \{{\textbf{p}}^*\}} 1_{A_{\delta ,{\textbf{v}}}}\circ \pi _{{\textbf{p}}^*}\rho _{\varepsilon ,{\textbf{p}}^*}dm\nonumber \\&\quad =\int 1_{A_{\delta ,{\textbf{v}}}}(\mu _{\varepsilon ,{\textbf{p}}^*})_{{\textbf{p}}^*}\int \partial _r \varphi \circ \pi _{\Lambda _0}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} \rho _{\varepsilon ,\Lambda \backslash \{{\textbf{p}}^*\}}dm_{\Lambda \backslash \{{\textbf{p}}^*\}} dm. \end{aligned}$$\end{document}

Then one can introduce for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}'\in I^{\Lambda _0}$$\end{document} the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{x_{{\textbf{p}}^*}} ({\textbf {x}}'):=\varphi (\tau x_{{\textbf{p}}^*},{\textbf {x}}'_{\ne {\textbf{p}}^*})$$\end{document} where we used the representation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}'=( {\textbf {x}}'_{{\textbf{p}}^*},{\textbf {x}}'_{\ne {\textbf{p}}^*})\in I\times I^{\Lambda _0 \backslash \{{\textbf{p}}^*\}}\simeq I^{\Lambda _0}$$\end{document} . Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in \mathcal {D}_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \varphi \Vert _\infty \le 1$$\end{document} , furthermore for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}_1,{\textbf {x}}_2\in I^\Lambda _0$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Lambda \backslash \{{\textbf{p}}^*\}}({\textbf {x}}_1)=\pi _{\Lambda \backslash \{{\textbf{p}}^*\}}({\textbf {x}}_2)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _r\varphi _{{\textbf {x}}_{{\textbf{p}}^*}}\circ \pi _{\Lambda _0}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*}({\textbf {x}}_1)&=\partial _r\varphi _{{\textbf {x}}_{{\textbf{p}}^*}}\circ \pi _{\Lambda _0}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*}({\textbf {x}}_2)\\&=\int \partial _r\varphi _{{\textbf {x}}_{{\textbf{p}}^*}}\circ \pi _{\Lambda _0}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*}(y,\pi _{\Lambda \backslash \{{\textbf{p}}^*\}}({\textbf {x}}_2)) \rho _{\varepsilon ,{\textbf{p}}^*}(y)dm(y). \end{aligned}$$\end{document}

Plugging this relation in Eq. (31),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left| \int \partial _r \varphi \circ \pi _{\Lambda _0}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} \rho _{\varepsilon ,\Lambda \backslash \{{\textbf{p}}^*\}}dm_{\Lambda \backslash \{{\textbf{p}}^*\}} \right| \\&\quad =\left| \int \partial _r \varphi _{{\textbf {x}}_{{\textbf{p}}^*}}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} \rho _{\varepsilon ,\Lambda \backslash \{{\textbf{p}}^*\}}dm_{\Lambda \backslash \{{\textbf{p}}^*\}} \right| \\&\quad =\left| \int \partial _r \varphi _{{\textbf {x}}_{{\textbf{p}}^*}}\circ \pi _{\Lambda _0}\circ T_{\varepsilon , {\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}}d\mu _{\varepsilon ,{\textbf{p}}^*} \right| \\&\quad \le \Vert \mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}}\mu _{\varepsilon ,{\textbf{p}}^*})\Vert \\&\quad \le C \Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert , \end{aligned}$$\end{document}

the last equation being a consequence of Proposition 6.1. Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,{\textbf{p}}^*}$$\end{document} is one dimensional bounded variation marginals, we deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \rho _{\varepsilon ,{\textbf{p}}^*}\Vert _{\infty }\le \Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert $$\end{document} . We then deduce that,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left| \int \partial _r \varphi \circ \pi _{\Lambda _0}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} \rho _{\varepsilon ,\Lambda \backslash \{{\textbf{p}}^*\}}dm_{\Lambda \backslash \{{\textbf{p}}^*\}} 1_{A_{\delta ,{\textbf{v}}}}\circ \pi _{{\textbf{p}}^*}\rho _{\varepsilon ,{\textbf{p}}^*}dm\right| \nonumber \\&\quad \le \Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert \left| \int 1_{A_{\delta ,{\textbf{v}}}}\circ \pi _{{\textbf{p}}^*}\rho _{\varepsilon ,{\textbf{p}}^*}dm\right| \nonumber \\&\quad \le \Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert \left| \int 1_{A_{\delta ,{\textbf{v}}}} \Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert dm\right| \nonumber \\&\quad \le \Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert ^2\left| \int 1_{A_{\delta ,{\textbf{v}}}}dm\right| \nonumber \\&\quad \le \Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert ^2\delta . \end{aligned}$$\end{document}

Now if one suppose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=p^*$$\end{document} , for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}\in I^{\Lambda }$$\end{document} , we can fix the local map defined for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}'_{{\textbf{p}}^*}\in I$$\end{document} as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\varphi _{{\textbf {x}}_{\ne {\textbf{p}}^*}}({\textbf {x}}'_{{\textbf{p}}^*}):=\varphi ({\textbf {x}}'_{{\textbf{p}}^*},\pi _{\Lambda _0\backslash \{{\textbf{p}}^*\}}(\tilde{T}_{\varepsilon ,{\textbf{p}}^*}{\textbf {x}}_{\ne {\textbf{p}}^*})).\end{aligned}$$\end{document}

Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\pi _{\Lambda _0}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*} ({\textbf {x}}'))_{{\textbf{p}}^*}=\tau {\textbf {x}}'_{{\textbf{p}}^*}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}\in I^{\Lambda }$$\end{document} then the same reasoning as when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\ne {\textbf{p}}^*$$\end{document} holds,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left| \int \partial _{{\textbf{p}}^*} \varphi \circ \pi _{\Lambda _0}\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} \rho _{\varepsilon ,\Lambda \backslash \{{\textbf{p}}^*\}}dm_{\Lambda \backslash \{{\textbf{p}}^*\}} 1_{A_{\delta ,{\textbf{v}}}}\circ \pi _{{\textbf{p}}^*}\rho _{\varepsilon ,{\textbf{p}}^*}dm\right| \\&\quad =\left| \int \partial _{{\textbf{p}}^*} \varphi _{{\textbf {x}}_{\ne {\textbf{p}}^*}} (\tau x) 1_{A_{\delta ,{\textbf{v}}}}(x)\rho _{\varepsilon ,{\textbf{p}}^*}(x)dm(x) 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} \rho _{\varepsilon ,\Lambda \backslash \{{\textbf{p}}^*\}}dm_{\Lambda \backslash \{{\textbf{p}}^*\}}({\textbf {x}}_{\ne {\textbf{p}}^*})\right| \\&\quad \le \left| \int \Vert \mathcal {L}_\tau (1_{A_{\delta ,{\textbf{v}}}}\mu _\tau )\Vert 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} \rho _{\varepsilon ,\Lambda \backslash \{{\textbf{p}}^*\}}dm_{\Lambda \backslash \{{\textbf{p}}^*\}}\right| \\&\quad \le C\Vert \mu _\tau \Vert \int 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} \rho _{\varepsilon ,\Lambda \backslash \{{\textbf{p}}^*\}}dm_{\Lambda \backslash \{{\textbf{p}}^*\}}\\&\quad \le C\Vert \mu _\tau \Vert \int 1_{A_{\delta ,-{\textbf{v}}}} \rho _{\varepsilon ,{\textbf{q}}}dm_{{\textbf{q}}}\\&\quad \le \Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert ^2\delta , \end{aligned}$$\end{document}

where we also used the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\rho _{\varepsilon ,{\textbf{q}}})$$\end{document} is a one dimensional map and thus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \rho _{\varepsilon ,{\textbf{q}}}\Vert _{\infty }\le \Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert $$\end{document} . Therefore,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\| (\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}-{\hat{\mathcal {L}}}_{\varepsilon ,\delta ,{\textbf{p}}^*})\mu _{\varepsilon ,{\textbf{p}}^*}\right\| \le 2\Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert ^2\delta , \end{aligned}$$\end{document}

and the above inequality together with (30) gives us, by Lemma 4.1, the second statement of the lemma. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Existence of \documentclass[12pt]{minimal}

			\usepackage{amsmath}
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			\usepackage{amssymb} 
			\usepackage{amsbsy}
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			\setlength{\oddsidemargin}{-69pt}
			\begin{document}$$\theta $$\end{document}θ

We now have all the requirements for the abstract framework of [23]: (A1)–(A4) of [23] follow from Proposition 6.1 and Theorem 2.1, while (A5)–(A6) of [23] follow from Lemma 4.2. We introduce the following notation11

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} q_{k,\delta }:=\frac{\mu _{\varepsilon ,{\textbf {p}}^*}\left( H_\delta ({\textbf {p}}^*)\cap \bigcap _{i=1}^k T_{\varepsilon ,{\textbf {p}}^*}^{-i}X_{0,\delta }({\textbf {p}}^*)\cap T_{\varepsilon ,{\textbf {p}}^*}^{-(k+1)}H_\delta ({\textbf {p}}^*)\right) }{\mu _{\varepsilon ,{\textbf {p}}^*}\left( H_\delta ({\textbf {p}}^*)\right) }. \end{aligned}$$\end{document}

and let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} q_k:=\lim _{\delta \rightarrow 0}q_{k,\delta }. \end{aligned}$$\end{document}

If the previous limit exists we then set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta =1-\sum _{k=0}^{\infty }q_k \end{aligned}$$\end{document}

and, following the abstract framework of [23], we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\delta \rightarrow 0}\frac{1-\lambda _{\varepsilon ,\delta }}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_{\delta }({\textbf{p}}^*)}=\theta . \end{aligned}$$\end{document}

In this subsection we prove, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in \mathbb {N}$$\end{document} , that the limit in (34) exists and we find \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} explicitly. Notice that for fixed k the characteristic function

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}1_{H_\delta ({\textbf {p}}^*)\cap \bigcap _{i=1}^k T_{\varepsilon ,{\textbf {p}}^*}^{-i}X_{0,\delta }({\textbf {p}}^*)\cap T_{\varepsilon ,{\textbf{p}}^*}^{-(k+1)}H_\delta ({\textbf {p}}^*)}\end{aligned}$$\end{document}

depends only on the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+1$$\end{document} coordinates around the site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*$$\end{document} . Thus, we can fix the box \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda :={\textbf {p}}^* + \{-(k+1),\dots ,(k+1)\}^d$$\end{document} around the vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\varepsilon ,\Lambda }$$\end{document} be the density of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Lambda }*\mu _{\varepsilon ,{\textbf {p}}^*}$$\end{document} . Using the above and Eq. (33), we can rewrite \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{k,\delta }$$\end{document} as follows:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&q_{k,\delta }=\frac{1}{\mu _{\varepsilon ,{\textbf {p}}^*}\left( H_\delta ({\textbf {p}}^*)\right) }\int _{I^\Lambda }1_{H_\delta ({\textbf {p}}^*)\cap \bigcap _{i=1}^k T_{\varepsilon ,{\textbf {p}}^*}^{-i}X_{0,\delta }({\textbf {p}}^*) \cap T_{\varepsilon ,{\textbf{p}}^*}^{-(k+1)}H_\delta ({\textbf {p}}^*)} \rho _{\varepsilon ,\Lambda }dm_{\Lambda }, \end{aligned}$$\end{document}

We have already dealt with the denominator of the expression for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{k,\delta }$$\end{document} in (37) in Lemma 4.1.

We introduce the following notation, for any subset \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \subset \mathbb {Z}^d$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\Psi _k^{{\textbf{q}}}\in A):=\{{\textbf {x}}\in I^{\mathbb {Z}^d}, \Psi _k^{{\textbf{q}}}({\textbf {x}})\in A\}. \end{aligned}$$\end{document}

Notice that the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\Psi _k^{{\textbf{q}}}\ne {\textbf{q}}')=(\Psi _k^{{\textbf{q}}}\in \mathbb {Z}^d{\setminus }\{{\textbf{q}}'\})$$\end{document} . We now prove that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\textbf{q}},{\textbf{v}}),({\textbf{q}}',{\textbf{v}}') \in \mathcal {N}_{{\textbf{p}}^*}$$\end{document} , the sets

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}A_{\delta ,-{\textbf{v}}}({\textbf{q}})\cap T_{\varepsilon ,{\textbf{p}}^*}^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{q}}')\cap (\Psi _k^{{\textbf{q}}}\ne {\textbf{q}}')\end{aligned}$$\end{document}

have negligible weight.

Lemma 4.3

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\textbf{q}},{\textbf{v}}) \in \mathcal {N}_{{\textbf {p}}^*}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\delta \rightarrow 0}\frac{1}{\mu _{\varepsilon ,{\textbf{p}}^*}(A_{\delta ,{\textbf{v}}}({\textbf{q}}))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{q}})\cap T_{\varepsilon ,{\textbf {p}}^*}^{-k}A_{\delta ,{\textbf{v}}}({\textbf{q}})\cap (\Psi _k^{{\textbf{q}}}\ne {\textbf{q}})\right) =0 \end{aligned}$$\end{document}

In addition for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\textbf{q}}',{\textbf{v}}')\in \mathcal {N}_{{\textbf {p}}^*}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\delta \rightarrow 0}\frac{1}{\mu _{\varepsilon ,{\textbf{p}}^*}(A_{\delta ,{\textbf{v}}}({\textbf{q}}))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{q}})\cap T_{\varepsilon ,{\textbf {p}}^*}^{-k}A_{\delta ,{\textbf{v}}'}({\textbf{q}}')\cap (\Psi _k^{{\textbf{q}}}\ne {\textbf{q}}')\right) =0. \end{aligned}$$\end{document}

Proof

The proof of Eq. (38) relies on the fact that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}\in (\Psi _k^{{\textbf{q}}}\ne {\textbf{q}})\cap T_{\varepsilon ,{\textbf {p}}^*}^{-k}A_{\delta ,{\textbf{v}}}({\textbf{q}})$$\end{document} , then it means that there is some coordinate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in \mathbb {Z}^d$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^k x_r \in A_{\delta ,-{\textbf{v}}}$$\end{document} . The full event is then a subset of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{{\textbf{q}}}^{-1}A_{\delta ,{\textbf{v}}}\cap \pi _{r}^{-1}(\tau ^{-k}A_{\delta ,-{\textbf{v}}})$$\end{document} .

Therefore, it remains to show that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu _{\varepsilon ,{\textbf{p}}^*}(\pi _{{\textbf{q}}}^{-1}A_{\delta ,{\textbf{v}}}\cap \pi _{r}^{-1}(\tau ^{-k}A_{\delta ,-{\textbf{v}}}))&=o(\mu _{\varepsilon ,{\textbf{p}}^*}(A_{\delta ,{\textbf{v}}}({\textbf{q}}))). \end{aligned}$$\end{document}

Since the density of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\{{\textbf{q}},r\}}*\mu _{\varepsilon ,{\textbf{p}}^*}$$\end{document} is not necessarily bounded, the measure above does not break easily into a product of measures, thus we prove the above control through what follows. We denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho (x,y):=\frac{d\mu _{\varepsilon ,{\textbf{p}}^*}}{dm_{\{{\textbf{p}}^*,{\textbf{q}}\}}}$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_\delta (y):= \int \rho (x,y) \frac{1_{A_{\delta ,{\textbf{v}}}}}{m(A_{\delta ,{\textbf{v}}})}dx$$\end{document} . Then one can choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi : I^{\{{\textbf{p}}^*,{\textbf{q}}\}}\mapsto \mathbb {R}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _x\phi (x,y):=\frac{1_{A_{\delta ,{\textbf{v}}}}}{m(A_{\delta ,{\textbf{v}}})}(x)$$\end{document} . Notice then that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int |\int \rho (x,y) \frac{1_{A_{\delta ,{\textbf{v}}}}}{m(A_{\delta ,{\textbf{v}}})}dx|dy\le \Vert \rho \Vert $$\end{document} .

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(0,\Vert \rho \Vert )$$\end{document} is relatively compact in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( L^\infty (X)\right) '=\{\mu \in \mathcal {B},\mu \ll Leb\}$$\end{document} for the weak \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^*$$\end{document} topology. Therefore, the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_\delta $$\end{document} admits a subsequence that converges to an element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_0\in \left( L^\infty (X)\right) '$$\end{document} .

Thus, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _0>0$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int G_\delta 1_{B_{\delta _0}}dy \underset{\delta \rightarrow 0}{\rightarrow }\int 1_{B_{\delta _0}}G_0(dy)$$\end{document} . And since for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \le \delta _0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int G_\delta 1_{B_{\delta _0}}dy\ge \int G_\delta 1_{B_{\delta }}dy$$\end{document} we deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop {{\overline{{\textrm{lim}}}}}\int G_\delta 1_{B_{\delta }}dy\le \int 1_{B_{\delta _0}}dG_0$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _0>0$$\end{document} . Consequently, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop {{\overline{{\textrm{lim}}}}}\int G_\delta 1_{B_{\delta }}dy=0$$\end{document} since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_0(dy)$$\end{document} is a positive measure absolutely continuous with respect to the Lebesgue measure. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

For each12 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{{\textbf{v}}},a_{-{\textbf{v}}})\in S^{rec}$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}({\textbf{v}},{\textbf{v}}')$$\end{document} be the set of integers k such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau ^{k+1}a_{{\textbf{v}}},\tau ^{k+1}a_{-{\textbf{v}}})=(a_{{\textbf{v}}'},a_{-{\textbf{v}}'})$$\end{document} . To each such k we associate the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_k$$\end{document} composed of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_1\le \dots \le j_r\le k$$\end{document} , the recurrent time into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}$$\end{document} up to time k (i.e, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau ^{j_i}(a_{{\textbf{v}}}),\tau ^{j_i}(a_{-{\textbf{v}}}))=(a_{{\textbf {w}}_{j_i}},a_{-{\textbf {w}}_{j_i}})$$\end{document} ).

Lemma 4.4

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in \mathbb {N}$$\end{document} , the limit in (34) exists:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} q_k:=\lim _{\delta \rightarrow 0}q_{k,\delta }. \end{aligned}$$\end{document}

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}= \emptyset $$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =1$$\end{document} ; otherwise \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in (0,1)$$\end{document} and is given by the following formula

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta =1-\sum _{{\textbf{v}},{\textbf{v}}'\in V}\sum _{k\in {\mathbb {K}({\textbf{v}},{\textbf{v}}')}}q_{k}({\textbf{v}},{\textbf{v}}'), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{k}({\textbf{v}},{\textbf{v}}')$$\end{document} is non zero whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau ^{k+1}(a_{{\textbf{v}}}),\tau ^{k+1}(a_{-{\textbf{v}}}))=(a_{{\textbf{v}}'},a_{-{\textbf{v}}'})$$\end{document} and is defined as follows:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&q_{k}({\textbf{v}},{\textbf{v}}')=\frac{1}{\sum _{(q,{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}\rho _{\tau }(a_{\textbf{v}})\frac{\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^+)+\rho _{\varepsilon ,{\textbf{q}}}(a_{-{\textbf{v}}}^-)}{2}}\times \\&\quad \frac{\rho _{\tau }(a_{{\textbf{v}}})}{|(\tau ^{k+1})'(a_{{\textbf{v}}})|}\left( \frac{1}{2|(\tau ^{k+1})'(a_{-{\textbf{v}}})|}\tilde{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^+)+\frac{1}{2|(\tau ^{k+1})'(a_{-{\textbf{v}}})|}\tilde{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^-)\right) , \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{\rho }_{\varepsilon ,\Lambda ,k}=\mathbb {E}(1_{ (\Psi _k^{{\textbf{q}}}={\textbf{p}}^*+{\textbf{v}}') \cap \bigcap _{i=1}^r \,^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})}\rho _{\varepsilon ,\Lambda }(.)|I^{\{{\textbf{q}}\}}). \end{aligned}$$\end{document}

Proof

We first consider the case when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}=\emptyset $$\end{document} . Then by the expression (33), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0}q_{k,\delta }=0$$\end{document} . Therefore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_k=0$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =1$$\end{document} in this case. We now consider the case when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}\not =\emptyset $$\end{document} and we prove that there is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in \mathbb {N}^*$$\end{document} for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop {{\overline{{\textrm{lim}}}}}_{\delta \rightarrow 0}q_{k,\delta }>0$$\end{document} .

Recall that we already computed the expression of the denominator of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{k,\delta }$$\end{document} in Lemma 4.1 whose expression depends on the term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*))$$\end{document} . To focus on the numerator we now introduce the quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\check{q}}_{k,\delta }$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{k,\delta }=\frac{m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*))}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))}\check{q}_{k,\delta }$$\end{document} and prove the existence of its limit when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \rightarrow 0$$\end{document} . Using the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\delta ({\textbf{p}}^*)=\bigcup _{({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf {p}}^*}}A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap A_{\delta ,-{\textbf{v}}}({\textbf{q}})$$\end{document} , we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\check{q}}_{k,\delta }&:=\frac{1}{m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*))}\nonumber \\&\quad \sum _{{\textbf{v}}\in V}\mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap A_{\delta ,-{\textbf{v}}}({\textbf{q}}) \cap T_{\varepsilon ,{\textbf{p}}^*}^{-(k+1)}H_{\delta }({\textbf{p}}^*) \cap \bigcap _{i=1}^k T_{\varepsilon ,{\textbf {p}}^*}^{-i}X_{0,\delta }({\textbf {p}}^*)\right) \nonumber \\&\quad =\sum _{{\textbf{v}},{\textbf{v}}'\in V^{+}}\hat{q}_{k,\delta }({\textbf{v}},{\textbf{v}}'), \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{q}_{k,\delta }({\textbf{v}},{\textbf{v}}')= \frac{1}{m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( G_{{\textbf{v}},{\textbf{v}}'}\right) , \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{{\textbf{v}},{\textbf{v}}'}$$\end{document} is a short notation for

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap A_{\delta ,-{\textbf{v}}}({\textbf{q}}) \cap T_{\varepsilon ,{\textbf{p}}^*}^{-(k+1)}\left( A_{\delta ,{\textbf{v}}'}({\textbf{p}}^*)\cap A_{\delta ,-{\textbf{v}}'}({\textbf{q}}')\right) \cap \bigcap _{i=1}^k T_{\varepsilon ,{\textbf {p}}^*}^{-i}X_{0,\delta }({\textbf {p}}^*). \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {q}}'$$\end{document} stands for the element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {p}}^*+{\textbf{v}}'$$\end{document} .

According to Lemma 4.3, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0}\frac{1}{m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( G_{{\textbf{v}},{\textbf{v}}'}\right) $$\end{document} can be reduced to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{\delta \rightarrow 0}\frac{1}{m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( G_{{\textbf{v}},{\textbf{v}}'}\right) \nonumber \\&\quad =\lim _{\delta \rightarrow 0}\frac{1}{m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap \right. \nonumber \\&\quad \left. T_0^{-(k+1)}A_{\delta ,{\textbf{v}}'}({\textbf{p}}^*) \cap A_{\delta ,-{\textbf{v}}}({\textbf{q}})\cap T_0^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{q}})\cap (\Psi _k^{{\textbf{q}}}={\textbf{q}})\cap \right. \nonumber \\&\quad \left. \bigcap _{i=1}^r\, ^c\left( A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap T_0^{-j_i}A_{\delta ,{\textbf {w}}_{j_i}}({\textbf{p}}^*)\cap A_{\delta ,-{\textbf{v}}}({\textbf{q}})\cap T_0^{-j_i}A_{\delta ,-{\textbf {w}}_{j_i}}({\textbf{q}})\cap (\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})\right) \right) . \end{aligned}$$\end{document}

Indeed, Lemma 4.3 implies that in the expression of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{{\textbf{v}},{\textbf{v}}'}$$\end{document} the respective sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap T_{\varepsilon ,{\textbf{p}}^*}^{-(k+1)}A_{\delta ,{\textbf{v}}'}({\textbf{p}}^*)$$\end{document} can be replaced by13 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\delta ,-{\textbf{v}}}({\textbf{q}})\cap T_0^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{q}}')\cap (\Psi _k^{{\textbf{q}}}={\textbf{q}}')$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,{\textbf{p}}^*}^{-j}(H_\delta ({\textbf{p}}^*))$$\end{document} by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\bigcap _{i=1}^r\, ^c\left( A_{\delta ,-{\textbf{v}}}({\textbf{q}})\cap T_0^{-j_i}A_{\delta ,{\textbf {w}}_{j_i}}({\textbf{q}}')\cap (\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})\right) .\end{aligned}$$\end{document}

In particular Lemma 4.3 implies that the integers k such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta }\tilde{q}_{k\delta }$$\end{document} is non zero are all contained in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}({\textbf{v}},{\textbf{v}}')$$\end{document} , thus the expression of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} can be simplified into

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta =1-\sum _{{\textbf{v}},{\textbf{v}}'\in V}\sum _{\mathbb {K}({\textbf{v}},{\textbf{v}}')}\lim _{\delta \rightarrow 0} \frac{m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*))}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))} \hat{q}_{k,\delta }({\textbf{v}},{\textbf{v}}'). \end{aligned}$$\end{document}

Note that since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is a uniformly expanding map, the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_0^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{q}})$$\end{document} is included in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_0^{-j_i}A_{\delta ,{\textbf {w}}_{j_i}}({\textbf{q}})$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le i \le r$$\end{document} . Thus

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{\delta \rightarrow 0}\frac{1}{m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( G_{{\textbf{v}},{\textbf{v}}'}\right) \nonumber \\&\quad =\lim _{\delta \rightarrow 0}\frac{1}{m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*))}\nonumber \\&\quad \mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap T_0^{-(k+1)}A_{\delta ,{\textbf{v}}'}({\textbf{p}}^*) \cap A_{\delta ,-{\textbf{v}}}({\textbf{q}})\cap T_0^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{q}})\right. \nonumber \\&\quad \left. \cap (\Psi _k^{{\textbf{q}}}={\textbf{q}}') \cap \bigcap _{i=1}^r\, ^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})\right) . \end{aligned}$$\end{document}

Since the measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*}$$\end{document} is decoupled at site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} , for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} small enough we can split the numerator of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{q}_{k,\delta }({\textbf{v}},{\textbf{v}}')$$\end{document} into two factors.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{\delta \rightarrow 0}\hat{q}_{k,\delta }({\textbf{v}},{\textbf{v}}')=\lim _{\delta \rightarrow 0}\frac{1}{m_{\mathbb {Z}^d}(H_\delta ({\textbf{p}}^*))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap T_0^{-(k+1)}A_{\delta ,{\textbf{v}}'}({\textbf{p}}^*)\right) \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\quad \mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,-{\textbf{v}}}({\textbf{q}})\cap T_0^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{q}})\cap (\Psi _k^{{\textbf{q}}}={\textbf{q}}') \cap \bigcap _{i=1}^r\, ^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})\right) \end{aligned}$$\end{document}

Now we will treat the two quantities (45) and (46) separately starting with the quantity (45):

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap T_0^{-(k+1)}A_{\delta ,{\textbf{v}}'}({\textbf{p}}^*)\right)&=\int _I 1_{A_{\delta ,{\textbf{v}}}\cap \tau ^{-(k+1)}A_{\delta ,{\textbf{v}}'}}\rho _{\tau }dm\nonumber \\&\sim \frac{\rho _{\tau }(a_{{\textbf{v}}})}{|(\tau ^{k+1})'(a_{{\textbf{v}}})|} m(A_{\delta ,{\textbf{v}}}). \end{aligned}$$\end{document}

where we recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\tau }(.)$$\end{document} is the invariant density of the site dynamics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(I,\tau )$$\end{document} . Now we treat the factor (46). Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{\varepsilon ,{\textbf{p}}^*}$$\end{document} has bounded variation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Lambda }*\mu _{\varepsilon ,{\textbf{p}}^*}$$\end{document} is absolutely continuous with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\Lambda $$\end{document} . Thus we only need to look at the following quantity:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,-{\textbf{v}}}({\textbf{q}})\cap T_0^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{q}})\cap (\Psi _k^{{\textbf{q}}}={\textbf{q}}') \cap \bigcap _{i=1}^r\, ^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})\right) \nonumber \\&\quad =\int 1_{A_{\delta ,-{\textbf{v}}}({\textbf{q}})\cap T_0^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{q}})}1_{(\Psi _k^{{\textbf{q}}}={\textbf{q}}') \cap \bigcap _{i=1}^r\, ^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})}\rho _{\varepsilon ,\Lambda }(.)dm \end{aligned}$$\end{document}

Notice that the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1_{(\Psi _k^{{\textbf{q}}}={\textbf{q}}') \cap \bigcap _{i=1}^r\, ^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})}\rho _{\varepsilon ,\Lambda }(.)$$\end{document} is of bounded variation. Therefore,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{\rho }_{\varepsilon ,\Lambda ,k}=\mathbb {E}_{m_{\textbf{q}}}(1_{ (\Psi _k^{{\textbf{q}}}={\textbf{q}}') \cap \bigcap _{i=1}^r\, ^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})}\rho _{\varepsilon ,\Lambda }(.)|I^{\{q\}}) \end{aligned}$$\end{document}

is a one dimensional bounded variation map and thus is cad-lag. Thus,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,-{\textbf{v}}}({\textbf{q}})\cap T_0^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{q}})\cap (\Psi _k^{{\textbf{q}}}={\textbf{q}}') \cap \bigcap _{i=1}^r\, ^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})\right) \nonumber \\&\quad =\int 1_{A_{\delta ,-{\textbf{v}}}\cap \tau ^{-(k+1)}A_{\delta ,-{\textbf{v}}'}} \hat{\rho }_{\varepsilon ,\Lambda ,k}(.)dm\nonumber \\&\quad \sim \hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^+)m(A_{\delta ,-{\textbf{v}}}^+\cap \tau ^{-(k+1)}A_{\delta ,-{\textbf{v}}'})+\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^-)m(A_{\delta ,-{\textbf{v}}}^-\cap \tau ^{-(k+1)}A_{\delta ,-{\textbf{v}}'}). \end{aligned}$$\end{document}

Where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\delta ,-{\textbf{v}}}^+:=A_{\delta ,-{\textbf{v}}}\cap \{y,y\ge a_{-{\textbf{v}}}\}$$\end{document} (resp \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\delta ,-{\textbf{v}}}^-:=A_{\delta ,-{\textbf{v}}}\cap \{y,y\le a_{-{\textbf{v}}}\}$$\end{document} ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\rho }_{\varepsilon ,\Lambda ,k,s}(a_{-{\textbf{v}}}^\pm )=\lim _{x\rightarrow a_{-{\textbf{v}}}^{\pm }} \rho _{\varepsilon ,\Lambda ,k,s}(x)$$\end{document} .

Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{k+1}$$\end{document} is piecewise \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} and piecewise onto. Therefore, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(A_{\delta ,{\textbf{v}}}^-\cap \tau ^{-(k+1)}A_{\delta ,{\textbf{v}}'}){=}m(A_{\delta ,{\textbf{v}}}^-\cap \tau ^{-(k+1)}A_{\delta ,{\textbf{v}}'}^+)$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(A_{\delta ,{\textbf{v}}}^-\cap \tau ^{-(k+1)}A_{\delta ,{\textbf{v}}'}){=}m(A_{\delta ,{\textbf{v}}}^-\cap \tau ^{-(k+1)}A_{\delta ,{\textbf{v}}'}^-)$$\end{document} .

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{k+1}$$\end{document} is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} map at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{{\textbf{v}}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{-{\textbf{v}}}$$\end{document} then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} m(A_{\delta ,{\textbf{v}}}^-\cap \tau ^{-(k+1)}A_{\delta ,{\textbf{v}}'})&=m(A_{\delta ,{\textbf{v}}}^+\cap \tau ^{-(k+1)}A_{\delta ,{\textbf{v}}'})\\&=\frac{1}{2|(\tau ^{k+1})'(a_{{\textbf{v}}})|} m(A_{\delta ,{\textbf{v}}'}). \end{aligned}$$\end{document}

The above is justified as follows: first for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} small enough, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{-(k+1)}A_{\delta ,{\textbf{v}}'}^{\pm } \subset A_{\delta ,{\textbf{v}}}^{\pm } $$\end{document} and there is a unique inverse branch for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{k+1}$$\end{document} between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\delta ,{\textbf{v}}}^{\pm }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{-(k+1)}A_{\delta ,{\textbf{v}}'}^\pm $$\end{document} that we denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _*$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_m$$\end{document} the associated interval in the partition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(I_j)_{j}$$\end{document} .

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} m(A_{\delta ,{\textbf{v}}}^\pm \cap \tau ^{-(k+1)}A_{\delta ,{\textbf{v}}'}^\pm )&=\int _{I_j} 1_{A_{\delta ,{\textbf{v}}'}^\pm }\circ \tau ^{k+1}dm\\&=\int _I 1_{A_{\delta ,{\textbf{v}}'}^\pm }\frac{1}{|(\tau ^{k+1})'\circ \tau _*|}dm\\&\sim \frac{1}{|(\tau ^{k+1})'(a_{\textbf{v}})|}m(A_{\delta ,{\textbf{v}}'}^\pm )\sim \frac{1}{2|(\tau ^{k+1})'(a_{\textbf{v}})|}m(A_{\delta ,{\textbf{v}}'}). \end{aligned}$$\end{document}

By plugging the above into Eq. (49) we get the estimate of the factor (45):

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\mu _{\varepsilon ,{\textbf {p}}^*}(A_{\delta ,-{\textbf{v}}}({\textbf{q}})\cap T_0^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{q}}')\cap (\Psi _k^{{\textbf{q}}}={\textbf{q}}') \cap \bigcap _{i=1}^r\, ^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf {w}}_{j_i}))\\&\quad \sim \frac{1}{2|(\tau ^{k+1})'(a_{\textbf{v}})|} \left( \hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^+)+\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^-)\right) m(A_{\delta ,{\textbf{v}}'}). \end{aligned}$$\end{document}

We then deduce from the expression of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{q}_{k,\delta }$$\end{document} in (45) and the respective behaviour of the term (45), (46) found in (47) that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{q}_{k}({\textbf{v}},{\textbf{v}}'):=\lim _{\delta \rightarrow 0} \hat{q}_{k,\delta }({\textbf{v}},{\textbf{v}}')$$\end{document} is well defined and explicitly given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\hat{q}_{k}({\textbf{v}},{\textbf{v}}')=\frac{1}{|\mathcal {N}_{{\textbf{p}}^*}|}\nonumber \\&\quad \frac{\rho _{\tau }(a_{{\textbf{v}}})}{|(\tau ^{k+1})'(a_{{\textbf{v}}})|}\left( \frac{1}{2|(\tau ^{k+1})'(a_{-{\textbf{v}}})|}\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^+)+\frac{1}{2|(\tau ^{k+1})'(a_{-{\textbf{v}}})|}\hat{\rho }_{\varepsilon ,\Lambda ,k}(a_{-{\textbf{v}}}^-)\right) , \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{\rho }_{\varepsilon ,\Lambda ,k}=\mathbb {E}(1_{ (\Psi _k^{{\textbf{q}}}={\textbf{p}}^*+{\textbf{v}}') \cap \bigcap _{i=1}^r \,^c(\Psi _{j_i}^{{\textbf{q}}}={\textbf{p}}^*+{\textbf {w}}_{j_i})}\rho _{\varepsilon ,\Lambda }(.)|I^{\{q\}}). \end{aligned}$$\end{document}

Using the above Lemma 4.1 and (43), the quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} is then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta =1-\sum _{{\textbf{v}},{\textbf{v}}'\in V}\sum _{k\in \mathbb {K}({\textbf{v}},{\textbf{v}}')}q_{k}({\textbf{v}},{\textbf{v}}'). \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Theorem 2.2

Using (36) and Lemma 4.4 we obtain the first item of the lemma:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _{\varepsilon ,\delta }=1-\mu _{\varepsilon ,{\textbf {p}}^*}(H_\delta ({\textbf {p}}^*))\cdot \theta (1+o(1)). \end{aligned}$$\end{document}

To prove the second item of the theorem, recall that the spectral data of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon , {\textbf {p}}^*}$$\end{document} do not depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\text {spec}(\mathcal {L}_{\varepsilon , {\textbf {p}}^*})\cap \{|z|>\gamma \}=\{1\},\end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} is as in (16). Choose, r small enough, so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{B(1,r)}\cap \overline{B(0,\gamma )}=\emptyset $$\end{document} . Then, by the Keller-Liverani stability result [19], for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} small enough, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{(1-r)-\gamma }{2}$$\end{document} is a lower bound on the spectral gaps of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon , {\textbf {p}}^*}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathcal {L}}}_{\varepsilon , \delta , {\textbf {p}}^*}$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \in [0,\delta ^*]$$\end{document} for small enough \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta ^*$$\end{document} . Therefore,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\kappa _N:=\sum _{k=N}^\infty \sup _{\delta \in [0,\delta ^*]}\Vert \hat{Q}_{\varepsilon ,\delta , {\textbf {p}}^*}\Vert =\mathcal {O}\left( \left( \frac{1+r+\gamma }{2}\right) ^N\right) .\end{aligned}$$\end{document}

Recall

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \eta _{\delta }=\sup _{\Vert \mu \Vert \le 1}\left| \left( (\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}-{\hat{\mathcal {L}}}_{\varepsilon ,\delta ,{\textbf{p}}^*})\mu \right) (1)\right| , \end{aligned}$$\end{document}

which was defined earlier in (18). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=\mathcal {O}\left( \log \eta _{\delta }\right) $$\end{document} . Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\eta _{\delta }+\mathcal {O}(\kappa _N)=\mathcal {O}\left( \eta _\delta \log \eta _{\delta }\right) $$\end{document} . Finally, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _{\delta }=\theta _{N,\delta }+\mathcal {O}(\kappa _N)$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{N,\delta }=1-\sum _{k=0}^{N-1} \lambda _{\varepsilon ,\delta }^{-k}q_{k,\delta }$$\end{document} . Then the rest of the proof follows exactly as in [18, Proposition 2]. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Theorem 2.5

The proof is based on checking linear response of the following perturbed transfer operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{\mathcal {L}}_{\delta ,s}:=\mathcal {L}_{\varepsilon ,\delta }(e^{is1_{H_\delta ({\textbf{p}}^*)}}\cdot ), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in \mathbb {R}\setminus \{0\}$$\end{document} . To be precise about the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathcal {L}}_{\delta ,s}(\cdot )$$\end{document} we mean that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in \mathcal {D}_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} a Borel complex measure,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int \varphi d\tilde{\mathcal {L}}_{\delta ,s}(\mu )=\int \varphi \circ T_{\varepsilon ,\delta } (e^{is1_{H_\delta ({\textbf{p}}^*)}}) d\mu . \end{aligned}$$\end{document}

Note that when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=0$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\tilde{\mathcal {L}}_{\delta ,0}:=\mathcal {L}_{\varepsilon ,\delta };\end{aligned}$$\end{document}

moreover, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in \mathbb {R}$$\end{document} the iterates are given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\tilde{\mathcal {L}}^n_{\delta ,s}(\cdot )=\mathcal {L}^n_{\varepsilon ,\delta }(e^{is\sum _{k=1}^n1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon .\delta }^{k-1}}\cdot ).\end{aligned}$$\end{document}

On the other hand, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\tilde{\mathcal {L}}_{0,s}=\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}.\end{aligned}$$\end{document}

So we first show that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} small enough \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathcal {L}}_{\delta ,s}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}$$\end{document} satisfy assumptions (A1)–(A6) of [22] in order to apply Theorem 2.1 of [1]. Note that by definition

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}|\tilde{\mathcal {L}}_{\delta ,s}\mu |\le |\mu |;\end{aligned}$$\end{document}

Moreover, by Proposition 6.1 using in this case that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\delta =e^{is1_{H_\delta ({\textbf{p}}^*)}}$$\end{document} , we obtain for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} small enough, there exist a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (0,1)$$\end{document} and a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} , independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} and s, such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {N}$$\end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \tilde{\mathcal {L}}_{\delta ,s}^n\mu \Vert \le C\sigma ^n\Vert \mu \Vert +C|\mu |. \end{aligned}$$\end{document}

Note also that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathcal {L}}_{\delta ,s}$$\end{document} can be decomposed as follows:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{\mathcal {L}}_{\delta ,s}(\cdot )&=(e^{is}-1)\mathcal {L}_{\varepsilon ,\delta }(1_{H_\delta ({\textbf{p}}^*)}\cdot )+\mathcal {L}_{\varepsilon ,\delta }(\cdot )\nonumber \\&=e^{is}\mathcal {L}_{\varepsilon ,\delta }(1_{H_\delta ({\textbf{p}}^*)}\cdot )+\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(1_{X_{0,\delta }({\textbf{p}}^*)}\cdot ). \end{aligned}$$\end{document}

We need to prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathcal {L}}_{\delta ,s}(\cdot )$$\end{document} admits a spectral gap, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} small enough. For this purpose we prove the following lemma.

Lemma 5.1

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\mu \in \mathcal {B}, \Vert \mu \Vert \le 1}|\tilde{\mathcal {L}}_{\delta ,s}(\mu )-\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(\mu ) |=\mathcal {O}({\delta }) \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \tilde{\mathcal {L}}_{\delta ,s}(\mu _{\varepsilon ,{\textbf{p}}^*})-\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(\mu _{\varepsilon ,{\textbf{p}}^*}) \Vert =\mathcal {O}({\delta }). \end{aligned}$$\end{document}

Proof

To prove (55) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in \mathcal {D}_1$$\end{document} . By (54), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int \varphi d\left[ \tilde{\mathcal {L}}_{\delta ,s}(\mu )-\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(\mu )\right]&=e^{is}\int \varphi \circ T_{\varepsilon ,\delta }1_{H_\delta ({\textbf{p}}^*)}d\mu -\int \varphi \circ T_{\varepsilon ,{\textbf{p}}^*} 1_{H_\delta ({\textbf{p}}^*)}d\mu . \end{aligned}$$\end{document}

By the same reasoning as in (15), each of the above terms is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(\delta )$$\end{document} above. Thus,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\mu \in \mathcal {B}, \Vert \mu \Vert \le 1}|\tilde{\mathcal {L}}_{\delta ,s}(\mu )-\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(\mu ) |=\mathcal {O}({\delta }). \end{aligned}$$\end{document}

To prove (56) we fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \in \mathcal {D}$$\end{document} a local map depending of coordinate in a finite set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _0\subset \mathbb {Z}^d$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in \mathbb {Z}^d$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left[ \tilde{\mathcal {L}}_{\delta ,s}(\mu _{\varepsilon ,{\textbf{p}}^*})-\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(\mu _{\varepsilon ,{\textbf{p}}^*})\right] (\partial _{r}\phi )&=e^{is}\int \partial _{{\textbf{q}}}\phi \circ T_{\varepsilon ,\delta } 1_{H_\delta ({\textbf{p}}^*)}d\mu _{\varepsilon ,{\textbf{p}}^*}\nonumber \\&\quad \hspace{14.22636pt}-\int \partial _{r}\phi \circ T_{\varepsilon ,{\textbf{p}}^*} 1_{H_\delta ({\textbf{p}}^*)}d\mu _{\varepsilon , {\textbf{p}}^*}. \end{aligned}$$\end{document}

In the above expression, the second integral is bounded in Lemma 3.1. For the other integral, we introduce the following notation. We choose an element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf{p}}^*}$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{{\textbf{q}}}^\tau $$\end{document} be given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\Phi _{{\textbf{q}}}^\tau ({\textbf {x}}))_{{\textbf{p}}}={\left\{ \begin{array}{ll} x_{{\textbf{q}}} \quad \text {if } x_{{\textbf{p}}}\in \tau A_{\delta ,{\textbf{v}}}\text { and } x_{{\textbf{q}}}\in \tau A_{\delta ,-{\textbf{v}}} \text { for some } {\textbf{v}}\in V \text { and } {\textbf{p}}={\textbf{p}}^*\\ x_{{\textbf{p}}^*} \quad \text {if } x_{\textbf{p}}\in \tau A_{\delta ,{\textbf{v}}}\text { and } x_{{\textbf{q}}}\in \tau A_{\delta ,-{\textbf{v}}} \text { for some } {\textbf{v}}\in V \text { and }{\textbf{p}}={\textbf{q}}\\ x_{\textbf{p}}\quad \quad \text {otherwise.} \end{array}\right. } \end{aligned}$$\end{document}

Then one can split the first integral in (57) into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\mathcal {N}_{{\textbf{p}}^*}|$$\end{document} integrals of the following form:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int \partial _{r}\phi \circ T_{\varepsilon ,\delta } 1_{A_{\delta ,{\textbf{v}}}}\circ \pi _{{\textbf{p}}^*}1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}}d\mu _{\varepsilon ,{\textbf{p}}^*}\\&\quad =\int \partial _{r}\phi \circ \Phi _q^\tau \circ T_{\varepsilon ,{\textbf{p}}^*} 1_{A_{\delta ,{\textbf{v}}}}\circ \pi _{{\textbf{p}}^*}1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}}d\mu _{\varepsilon ,{\textbf{p}}^*} \end{aligned}$$\end{document}

for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\textbf{q}},{\textbf{v}})\in \mathcal {N}_{{\textbf{p}}^*}$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}\in I^{\mathbb {Z}^d}$$\end{document} we choose to represent such element as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}:=({\textbf {x}}_1,{\textbf {x}}_2,{\textbf {x}}_3)\in I^{\{{\textbf{p}}^*\}}\times I^{\{{\textbf{q}}\}}\times I^{\mathbb {Z}^d\backslash \{{\textbf{p}}^*,{\textbf{q}}\}}$$\end{document} notice then, that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda :=\Lambda _0\cup \{{\textbf{p}}+{\textbf{v}}, {\textbf{p}}\in \Lambda _0, {\textbf{v}}\in V\}$$\end{document} (similarly as in the proof of Lemma 4.2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\phi } \circ \pi _{\Lambda } ({\textbf {x}}_{\textbf{p}},{\textbf {x}}_{\textbf{q}},{\textbf {x}}_{\ne }):= \phi ({\textbf {x}}_{\textbf{q}},{\textbf {x}}_{\textbf{p}},{\textbf {x}}_{\ne })$$\end{document} satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{r}\phi \circ \Phi _{{\textbf{q}}}^\tau \circ T_{\varepsilon ,{\textbf{p}}^*}({\textbf {x}})=\partial _{r}\tilde{\phi }\circ \pi _{\Lambda _0}\circ T_{\varepsilon ,{\textbf{p}}^*}({\textbf {x}})$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}\in I^{\mathbb {Z}^d}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\phi } \in \mathcal {D}_1$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \phi \Vert _\infty =\Vert \tilde{\phi }\Vert _\infty $$\end{document} . Thus, one can follow the same reasoning as in the second point of Lemma 4.2 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\phi }$$\end{document} instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} . We then reach the same conclusion as Eq. (32):

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left| \int \partial _r \tilde{\phi }\circ \pi _{\Lambda }\circ \tilde{T}_{\varepsilon , {\textbf{p}}^*} 1_{A_{\delta ,-{\textbf{v}}}}\circ \pi _{{\textbf{q}}} \rho _{\varepsilon ,\Lambda _0\backslash \{{\textbf{p}}^*\}}dm_{\Lambda _0\backslash \{{\textbf{p}}^*\}} 1_{A_{\delta ,{\textbf{v}}}}\circ \pi _{{\textbf{p}}^*}\rho _{\varepsilon ,{\textbf{p}}^*}dm\right| \\&\quad \le \Vert \mu _{\varepsilon ,{\textbf{p}}^*}\Vert ^2\delta . \end{aligned}$$\end{document}

which brings us to the conclusion of the second statement

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \tilde{\mathcal {L}}_{\delta ,s}(\mu _{\varepsilon ,{\textbf{p}}^*})-\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(\mu _{\varepsilon ,{\textbf{p}}^*}) \Vert =\mathcal {O}({\delta }). \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The uniform Lasota–Yorke inequality (53), together with (55) of Lemma 5.1, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} sufficiently small, the Keller-Liverani stability result [19] implies that the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\delta ,s}$$\end{document} admits a spectral gap on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} . This proves assumptions (A1)–(A4) of [22]. Now let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{\eta }_\delta&=\sup _{\Vert \mu \Vert \le 1}|\left( \mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(\mu )-\tilde{\mathcal {L}}_{s,\delta }(\mu )\right) (1)|;\\ \tilde{\Delta }_\delta&=\left( \mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(\mu _{\varepsilon ,{\textbf{p}}^*})-\tilde{\mathcal {L}}_{s,\delta }(\mu _{\varepsilon ,{\textbf{p}}^*})\right) (1). \end{aligned}$$\end{document}

By Lemma 5.1 and Lemma 4.1 we have

Corollary 5.2

The following holds

1. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\eta }_\delta \rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \rightarrow 0$$\end{document} ;
2. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists \, C>0$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\eta }_\delta \cdot \Vert (\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}-\tilde{\mathcal {L}}_{s,\delta })\mu _{\varepsilon ,{\textbf{p}}^*}\Vert \le C |\tilde{\Delta }_\delta |$$\end{document} .

Thus, by Corollary 5.2 conditions (A5)–(A6) of [22] are satisfied. Moreover, notice that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{\eta }_\delta&=|e^{is}-1|\eta _\delta \\ \tilde{\Delta }_\delta (t)&=(1-e^{is})\Delta _\delta =(1-e^{is})\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*)), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _\delta $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _\delta $$\end{document} are defined in Eqs. (18) and (19) respectively.

Existence of \documentclass[12pt]{minimal}

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			\begin{document}$$\tilde{\theta }$$\end{document}θ~

We are left to check the existence of the following limit when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \rightarrow 0$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\tilde{q}_{k,\delta }(s):= \frac{((\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}-\tilde{\mathcal {L}}_{\delta ,s})\tilde{\mathcal {L}}_{\delta ,s}^k((\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}-\tilde{\mathcal {L}}_{\delta ,s})\mu _{\varepsilon ,{\textbf{p}}^*})(1)}{\tilde{\Delta }_\delta (s)}\nonumber \\&\quad =(1-e^{is})\frac{\int 1_{H_\delta ({\textbf{p}}^*)}d\tilde{\mathcal {L}}_{\delta ,s}^k((\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(1_{H_\delta ({\textbf{p}}^*)}\cdot )-e^{is}\mathcal {L}_{\varepsilon ,\delta }(1_{H_\delta ({\textbf{p}}^*)}\cdot )\mu _{\varepsilon ,{\textbf{p}}^*})}{\tilde{\Delta }_\delta (s)}\nonumber \\&\quad =\frac{\int 1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^k e^{is\sum _{j=0}^{k-1}1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^j}d((\mathcal {L}_{\varepsilon ,{\textbf{p}}^*}(1_{H_\delta ({\textbf{p}}^*)}\cdot )-e^{is}\mathcal {L}_{\varepsilon ,\delta }(1_{H_\delta ({\textbf{p}}^*)}\cdot )\mu _{\varepsilon ,{\textbf{p}}^*})}{\Delta _\delta }\end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\quad =\frac{1}{\Delta _\delta }\sum _{j=0}^ke^{isj}\mu _{\varepsilon ,{\textbf{p}}^*}\left( T_{\varepsilon ,{\textbf{p}}^*}^{-1}\left( T_{\varepsilon ,\delta }^{-k}H_\delta ({\textbf{p}}^*)\cap \left( \sum _{r=0}^{k-1}1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^r=j\right) \right) \cap H_{\delta }({\textbf{p}}^*)\right) \nonumber \\&\qquad -\frac{1}{\Delta _\delta }\sum _{j=0}^ke^{is(j+1)}\mu _{\varepsilon ,{\textbf{p}}^*}\left( T_\varepsilon ^{-k-1}H_\delta ({\textbf{p}}^*)\cap \left( \sum _{r=1}^{k}1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^r=j\right) \cap H_{\delta }({\textbf{p}}^*)\right) \nonumber \\&\quad =\sum _{j=0}^ke^{isj}\left( \beta _{k,\delta }^{(1)}(j)-e^{is}\beta _{k,\delta }^{(2)}(j)\right) , \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta _{k,\delta }^{(1)}(j):=\frac{\mu _{\varepsilon ,{\textbf {p}}^*}\left( H_\delta ({\textbf {p}}^*)\cap T_{\varepsilon ,{\textbf{p}}^*}^{-1}\left( T_{\varepsilon ,\delta }^{-k}H_\delta ({\textbf {p}}^*)\cap \left( \sum _{i=0}^{k-1}1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^i=j\right) \right) \right) }{\mu _{\varepsilon ,{\textbf {p}}^*}\left( H_\delta ({\textbf {p}}^*)\right) }. \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta _{k,\delta }^{(2)}(j):=\frac{\mu _{\varepsilon ,{\textbf {p}}^*}\left( H_\delta ({\textbf {p}}^*)\cap T_{\varepsilon ,\delta }^{-(k+1)}H_\delta ({\textbf {p}}^*)\cap \left( \sum _{i=1}^k1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^i=j\right) \right) }{\mu _{\varepsilon ,{\textbf {p}}^*}\left( H_\delta ({\textbf {p}}^*)\right) }. \end{aligned}$$\end{document}

In what follows, we just treat the existence of a limit for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{k,\delta }^{(2)}(j)$$\end{document} , the proof for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{k,\delta }^{(1)}(j)$$\end{document} follows the same path. We introduce the maps \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Psi }_k^{{\textbf{q}}}$$\end{document} as follows14:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{r l c} \tilde{\Psi }_1^{{\textbf {p}}}({\textbf {x}})& =& {\textbf{p}}+{\textbf{v}}\text { if } x_{\textbf{p}}\in A_{\varepsilon ,{\textbf{v}}}\text { and } x_{{\textbf{p}}+{\textbf{v}}}\in A_{\varepsilon ,-{\textbf{v}}} \text { for some } {\textbf{v}}\in V\\ \tilde{\Psi }_1^{{\textbf {p}}}({\textbf {x}})& =& {\textbf{p}}\text { otherwise}\\ \tilde{\Psi }_k^{{\textbf {p}}}({\textbf {x}})& =& \tilde{\Psi }_{k-1}^{{\textbf {p}}}({\textbf {x}})+\tilde{\Psi }_1^{\tilde{\Psi }_{k-1}^{{\textbf {p}}}({\textbf {x}})}(T_{\varepsilon ,\delta }^{k-1}{\textbf {x}}) \text { if }k\ne 0. \end{array} \right. \end{aligned}$$\end{document}

Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Psi }_k^{{\textbf {p}}}$$\end{document} depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} .

Proving the existence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0} \beta _{k,\delta }^{(i)}(j,{\textbf{v}},{\textbf{v}}')$$\end{document} follows the same steps as what we have done for the existence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0} q_{k,\delta }$$\end{document} in the proof of Lemma 4.4. In other words we split \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{k,\delta }^{(i)}(j)$$\end{document} into the counterpart of (39).

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta _{k,\delta }^{(i)}(j):=\sum _{{\textbf{v}},{\textbf{v}}'\in V^{+}} \beta _{k,\delta }^{(i)}(j,{\textbf{v}},{\textbf{v}}'), \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta _{k,\delta }^{(i)}(j,{\textbf{v}},{\textbf{v}}')= \frac{1}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( \tilde{G}_{{\textbf{v}},{\textbf{v}}'}\right) , \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{G}_{{\textbf{v}},{\textbf{v}}'}$$\end{document} being the short notation for

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap A_{\delta ,-{\textbf{v}}}({\textbf{q}})&\cap T_{\varepsilon ,\delta }^{-(k+1)}\left( A_{\delta ,{\textbf{v}}'}({\textbf{p}}^*)\cap A_{\delta ,-{\textbf{v}}'}({\textbf{q}}')\right) \nonumber \\&\cap \left( \sum _{i=1}^k1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^i=j\right) . \end{aligned}$$\end{document}

To prove the existence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0} \beta _{k,\delta }(j,{\textbf{v}},{\textbf{v}}')$$\end{document} we will actually prove that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} small enough \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}_{\mu _{\varepsilon ,{\textbf{p}}^*}}\left( 1_{\tilde{G}_{{\textbf{v}},{\textbf{v}}'}}|I^{{\textbf{p}}^*,{\textbf{p}}^*+{\textbf{v}}}\right) $$\end{document} can be decomposed into

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}_{\mu _{\varepsilon ,{\textbf{p}}^*}}\left( 1_{\tilde{G}_{{\textbf{v}},{\textbf{v}}'}}|I^{{\textbf{p}}^*,{\textbf{p}}^*+{\textbf{v}}}\right)&=1_{A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)}1_{\tau ^{-(k+1)}A_{\delta ,{\textbf{v}}'}({\textbf{p}}^*)}\tilde{\rho }_{\varepsilon ,{\textbf{p}}^*}\circ \pi _{{\textbf{p}}^*}\nonumber \\&\quad \hspace{14.22636pt}1_{A_{\delta ,-{\textbf{v}}}({\textbf{p}}^*+{\textbf{v}})}1_{\tau ^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{p}}^*+{\textbf{v}})}\tilde{\rho }_{\varepsilon ,{\textbf{p}}^*+{\textbf{v}}}\circ \pi _{{\textbf{p}}^*+{\textbf{v}}}\nonumber \\&\qquad +o(\delta ^2), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{{\textbf{p}}^*}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\rho }_{\varepsilon ,{\textbf{p}}^*+{\textbf{v}}}$$\end{document} are functions of bounded variation and they are independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} . Once this done one can consider the limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0} \beta _{k,\delta }(j,{\textbf{v}},{\textbf{v}}')$$\end{document} and obtain a formula similar to (50).

Now notice that from the definition of the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_k$$\end{document} associated to a couple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{{\textbf{v}}},a_{-{\textbf{v}}})\in S^{rec}$$\end{document} , for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap A_{\delta ,-{\textbf{v}}}({\textbf{q}})$$\end{document} the sum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i=1}^k1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^i(x)$$\end{document} may be reduced to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i=1}^r1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^{j_i}(x)$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} small enough. Actually one can notice that Lemma 4.3 extends to that setting with maps \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Psi }_{j_i}^{{\textbf{q}}}({\textbf {x}})$$\end{document} into the following lemma:

Lemma 5.3

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\textbf{q}},{\textbf{v}}) \in \mathcal {N}_{{\textbf {p}}^*}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>0$$\end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\textbf{q}}',{\textbf{v}}') \in \mathcal {N}_{{\textbf {p}}^*}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\delta \rightarrow 0}\frac{1}{\mu _{\varepsilon ,{\textbf{p}}^*}(A_{\delta ,{\textbf{v}}}({\textbf{q}}))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{q}})\cap T_{\varepsilon ,{\textbf {p}}^*}^{-k}A_{\delta ,{\textbf{v}}'}({\textbf{q}}')\cap (\tilde{\Psi }_k^{{\textbf{q}}}\ne {\textbf{q}}')\right) =0. \end{aligned}$$\end{document}

In addition and for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\le k$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{\delta \rightarrow 0}\frac{1}{\mu _{\varepsilon ,{\textbf{p}}^*}(A_{\delta ,{\textbf{v}}}({\textbf{q}}))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{q}})\cap T_{\varepsilon ,{\textbf {p}}^*}^{-k}A_{\delta ,{\textbf{v}}}({\textbf{q}})\cap (\tilde{\Psi }_j^{{\textbf{q}}}\ne {\textbf{p}}^*)\cap (\tilde{\Psi }_j^{{\textbf{p}}^*}\ne {\textbf{p}}^*)\right) \nonumber \\&\quad =0. \end{aligned}$$\end{document}

The proof of the above lemma follows the exact same reasoning as Lemma 4.3 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Psi }_j^{{\textbf{p}}^*}$$\end{document} instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _j^{{\textbf{q}}}$$\end{document} (and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,\delta }$$\end{document} instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,{\textbf{p}}^*}$$\end{document} ).

So for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} small enough, the values taken by the sum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i=1}^r1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^{j_i}(x)$$\end{document} (still for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap A_{\delta ,-{\textbf{v}}}({\textbf{q}})$$\end{document} ) are entirely determined by the values taken by the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tilde{\Psi }_{j_i}^{{\textbf{p}}}({\textbf {x}}),\tilde{\Psi }_{j_i}^{{\textbf{q}}}({\textbf {x}}))$$\end{document} which index the site impacted at time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_i$$\end{document} by any small variation of the state of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}$$\end{document} in site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{q}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}$$\end{document} at time 0. When

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{i=1}^k1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^i(x)=\sum _{i=1}^k1_{\{\tilde{\Psi }_{j_i}^{{\textbf{p}}^*}({\textbf {x}}),\tilde{\Psi }_{j_i}^{{\textbf{q}}}\}=\{{\textbf{p}}^*,{\textbf{p}}^*+{\textbf {w}}_{j_i}\}}. \end{aligned}$$\end{document}

Therefore, one can divide the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \sum _{i=1}^r1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^{j_i}(x)=m \right\} $$\end{document} into subsets of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_\delta :=\bigcup _{P\in P_m\{j_1,\dots ,j_r\}} \bigcap _{p_i\in P} T_{\varepsilon ,\delta }^{-p_i}H_{\delta }({\textbf{p}}^*)\bigcap _{p_i\notin P}T_{\varepsilon ,\delta }^{-p_i}\left( X_{0,\delta }({\textbf{p}}^*)\right) $$\end{document} and expand it using Lemma 5.3 and the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is a uniformly expanding map, we deduce that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{\delta \rightarrow 0}\frac{1}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))}\mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap T_{\varepsilon ,\delta }^{-(k+1)}A_{\delta ,{\textbf{v}}'}({\textbf{p}}^*)\cap W_\delta \right) \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\quad =\lim _{\delta \rightarrow 0}\frac{1}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))}\nonumber \\&\quad \mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap A_{\delta ,-{\textbf{v}}}({\textbf{p}}^*+{\textbf{v}})\cap T_0^{-(k+1)}A_{\delta ,{\textbf{v}}'}({\textbf{p}}^*)\cap T_0^{-(k+1)}A_{\delta ,-{\textbf{v}}'}({\textbf{p}}^*+{\textbf{v}})\right. \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\quad \left. \hspace{28.45274pt}\bigcup _{P\in P_m\{j_1,\dots ,j_r\}}\bigcap _{p_i\in P} (\tilde{\Psi }_{j_i}^{{\textbf{q}}},\tilde{\Psi }_{j_i}^{{\textbf{p}}^*})\in \{{\textbf{p}}^*,{\textbf{p}}^*+{\textbf {w}}_{j_i}\}\right. \nonumber \\&\quad \left. \hspace{56.9055pt}\cap \bigcap _{p_i\notin P}\left( (\tilde{\Psi }_{j_i}^{{\textbf{q}}},\tilde{\Psi }_{j_i}^{{\textbf{p}}^*})\notin \{{\textbf{p}}^*,{\textbf{p}}^*+{\textbf {w}}_{j_i}\}\right) \right) \nonumber \\&\quad =\lim _{\delta \rightarrow 0}\frac{1}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))}\nonumber \\&\quad \sum _{P\in P_m\{j_1,\dots ,j_r\}}\mu _{\varepsilon ,{\textbf{p}}^*}\left( A_{\delta ,{\textbf{v}}}({\textbf{p}}^*)\cap T_0^{-(k+1)}A_{\delta ,{\textbf{v}}'}({\textbf{p}}^*)\cap D_P\right) \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D_P:=&\bigcap _{p_i\in P} \left( (\tilde{\Psi }_{j_{p_i}-j_{p_{i-1}}-1}^{{\textbf{q}}}\left( \tilde{T}_{p_{i-1}+1}({\textbf {x}})\right) ={\textbf{p}}^*+{\textbf {w}}_{p_i})\right) \\&\cap \bigcap _{p\in \{j_1,\dots ,j_r\}\backslash P}\left( (\tilde{\Psi }_{j_p-j_{p_i}-1}^{{\textbf{q}}}\left( \tilde{T}_{p_i+1}({\textbf {x}})\right) \ne {\textbf{p}}^*+{\textbf {w}}_{p_i})\right) , \end{aligned}$$\end{document}

where the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\in P_m\{j_1,\dots ,j_r\}$$\end{document} at the last line of (69) is ordered so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P=\{p_1,\dots ,p_{|P|}\}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1\le \dots \le p_{|P|}$$\end{document} and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in \{1,\dots ,r\}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_i$$\end{document} is the largest element of P so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>p_i$$\end{document} and we define the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{T}_{p_i+1}$$\end{document} as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\tilde{T}_{p_i+1}:= \Phi \circ T_{\varepsilon ,{\textbf{p}}^*}^{p_i-p_{i-1}}\circ \dots \circ \Phi \circ T_{\varepsilon ,{\textbf{p}}^*}^{p_2-p_{1}}\circ \Phi \circ T_{\varepsilon ,{\textbf{p}}^*}^{p_1}\circ \Phi \circ T_{\varepsilon ,{\textbf{p}}^*}. \end{aligned}$$\end{document}

From Eq. (69) we can deduce that whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{S}^{rec}=\emptyset $$\end{document} , then15

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\delta \rightarrow 0} \beta _{k,\delta }^{(2)}(j)=0. \end{aligned}$$\end{document}

Notice in addition that on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigcap _{p\in P} T_{\varepsilon ,\delta }^{-p}H_{\delta }({\textbf{p}}^*)\bigcap _{p\notin P,p\le k}T_{\varepsilon ,\delta }^{-p}\left( X_{0,\delta }({\textbf{p}}^*)\right) $$\end{document} , all the interactions with the site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} occur at precise identified time, and from Lemma 5.3, we always have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*\in \{\tilde{\Psi }_{j}^{{\textbf{q}}},\tilde{\Psi }_{j}^{{\textbf{p}}^*}\}$$\end{document} , so we only need to keep trace of one of those two maps at a time, this is what is displayed at the last line of Eq. (69) with the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Psi }_{p_i-p_{i-1}}^{{\textbf{q}}}$$\end{document} . The point here is that on such a set, this map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Psi }_{j_p-j_{p_i}-1}^{{\textbf{q}}}\left( \tilde{T}_{p_i+1}({\textbf {x}})\right) $$\end{document} actually does not depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} : \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Psi }_{j_p-j_{p_i}-1}^{{\textbf{q}}}\left( \tilde{T}_{p_i+1}({\textbf {x}})\right) =\Psi _{j_p-j_{p_i}-1}^{{\textbf{q}}}\left( \tilde{T}_{p_i+1}({\textbf {x}})\right) $$\end{document} . Therefore, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\in \mathcal {P}\{1,\dots ,r\}$$\end{document} , the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\rho }:=\rho _{\varepsilon }\mathbb {E}_{\mu _{\varepsilon ,{\textbf{p}}^*}}\left( 1_{D_P}|I^{\{{\textbf{p}}^*,{\textbf{q}}\}}\right) $$\end{document} is independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} and our last step is to prove that it can be decomposed on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{\{{\textbf{p}},{\textbf{q}}\}}$$\end{document} into a linear combination of product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{{\textbf{p}}^*}(x_{{\textbf{p}}^*})\rho _{{\textbf{q}}}(x_{{\textbf{q}}})$$\end{document} of one dimensional functions of bounded variation. To do this, we first introduce the following notion:

Definition 5.4

We call cylinder map any finite linear combination of maps of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1_{A_1\times \dots \times A_{|\Lambda |}}\circ \pi _{\Lambda }$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \subset \mathbb {Z}^d$$\end{document} is a finite set and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_1,\dots , A_{|\Lambda |}$$\end{document} are a set of sub-intervals of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{{\textbf{q}}}$$\end{document} for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{q}}\in \Lambda $$\end{document} .

Notice that if f is a cylinder map, then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in \mathbb {R}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1_{f=p}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\circ T_0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\circ \Phi $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Psi }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\circ \Phi _\varepsilon $$\end{document} are cylinder map. And for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \subset \mathbb {Z}^d$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}_{\mu _{\varepsilon ,{\textbf{p}}^*}}\left( f| I^{\Lambda }\right) $$\end{document} is a cylinder map on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{\Lambda }$$\end{document} . Therefore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}_{\mu _{\varepsilon ,{\textbf{p}}^*}}\left( 1_{D_P}|I^{\{{\textbf{p}}^*,{\textbf{q}}\}}\right) :=\sum _k 1_{R^P_k}({\textbf {x}}_{{\textbf{p}}^*})1_{B^P_k}({\textbf {x}}_{{\textbf{q}}})$$\end{document} is a cylinder map and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\mathbb {E}_{\mu _{\varepsilon ,{\textbf{p}}^*}}\left( 1_{\tilde{G}_{{\textbf{v}},{\textbf{v}}'}}|I^{{\textbf{p}}^*,{\textbf{p}}^*+{\textbf{v}}}\right) \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\quad =\sum _{P\in \mathcal {P}\{1,\dots ,r\} }\sum _{k} \int 1_{A_{\delta ,{\textbf{v}}}}(x)1_{\tau ^{-(k+1)}A_{\delta ,{\textbf{v}}}}(x)\rho _\tau (x)1_{R^P_k}(x)dx \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\quad \int 1_{A_{\delta ,{\textbf{v}}}}(y)1_{\tau ^{-(k+1)}A_{\delta ,{\textbf{v}}}}(y)\rho _q(y)1_{B^P_k}(y)dy. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _q(\cdot )1_{B^P_k}(\cdot )$$\end{document} corresponds to some piecewise continuous map, through the same reasoning as in (49), the following limit exists

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{\delta \rightarrow 0}\frac{1}{m(A_{\delta ,{\textbf{v}}}\times A_{\delta ,-{\textbf{v}}})}\int 1_{A_{\delta ,{\textbf{v}}}}(x)1_{\tau ^{-(k+1)}A_{\delta ,{\textbf{v}}}}(x)\rho _\tau (x)1_{R^P_k}(x)dx\\&\quad \int 1_{A_{\delta ,{\textbf{v}}}}(y)1_{\tau ^{-(k+1)}A_{\delta ,{\textbf{v}}}}(y)\rho _q(y)1_{B^P_k}(y)dy. \end{aligned}$$\end{document}

Consequently, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in \{1,2\}$$\end{document} , the following limit exists

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta _k^{(i)}(j,{\textbf{v}},{\textbf{v}}'):=\lim _{\delta \rightarrow 0}\beta _{k,\delta }^{(i)}(j,{\textbf{v}},{\textbf{v}}'). \end{aligned}$$\end{document}

Using the above Lemma 4.1 and (43), we deduce that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\delta \rightarrow 0}\beta _{k,\delta }^{(i)}(j)=\sum _{{\textbf{v}},{\textbf{v}}'\in V^{+}} \beta _k^{(i)}(j,{\textbf{v}},{\textbf{v}}'), \end{aligned}$$\end{document}

and the quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\theta }$$\end{document} is then given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{\theta }(s)=\,&1-\sum _{k\ge 1}\tilde{q}_{k,\delta }(s) \\ =\,&1-\sum _{{\textbf{v}},{\textbf{v}}'\in V}\sum _{k\in \mathbb {K}({\textbf{v}},{\textbf{v}}')}\sum _{j=0}^ke^{isj}\left( \beta _k^{(1)}(j,{\textbf{v}},{\textbf{v}}')-e^{is}\beta _k^{(2)}(j,{\textbf{v}},{\textbf{v}}')\right) .\\ \end{aligned}$$\end{document}

Proof of Theorem 2.5

By the above, we have shown that the dominant eigenvalue, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{s,\delta }$$\end{document} , of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathcal {L}}_{s,\delta }$$\end{document} has the following first order approximation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 1-\lambda _{s,\delta }&=\tilde{\Delta }_\delta \tilde{\theta }(s)(1+o(1))\\&=(1-e^{is})\mu _{\varepsilon ,{\textbf{p}}^*}(H_{\delta }({\textbf{p}}^*))\tilde{\theta }(s)(1+o(1)), \end{aligned}$$\end{document}

that is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _{s,\delta }=e^{-\mu _{\varepsilon ,{\textbf{p}}^*}(H_{\delta }({\textbf{p}}^*))(1-e^{is})\tilde{\theta }(s)(1+o(1))}. \end{aligned}$$\end{document}

Recall from (13) the definition of the process

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}Z_\delta (t)=\sum _{k=1}^{\left\lfloor \frac{t}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))}\right\rfloor }1_{H_\delta ({\textbf{p}}^*)}\circ T_{\varepsilon ,\delta }^k\end{aligned}$$\end{document}

and the fact that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{\mathcal {L}}^{\left\lfloor \frac{t}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))}\right\rfloor }_{\delta ,s}(\cdot )=\mathcal {L}^{\left\lfloor \frac{t}{\mu _{\varepsilon ,{\textbf{p}}^*}(H_\delta ({\textbf{p}}^*))}\right\rfloor }_{\varepsilon ,\delta }(e^{isZ_\delta (t)}\cdot ). \end{aligned}$$\end{document}

Therefore, using the spectral decomposition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mathcal {L}}$$\end{document} , (74) and (75) we obtain for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\lim _{\delta \rightarrow 0}\int e^{isZ_\delta (t)}d\mu =e^{-(1-e^{is})\tilde{\theta }(s)t}.\end{aligned}$$\end{document}

Thus, using the Lévy Continuity Theorem we can argue as in the discussion after Theorem 2.1 of [1] to conclude that the process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_\delta (t)$$\end{document} converges in law to a compound Poisson process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(t)=\sum _{i=1}^{N(t)}X_i$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N(t))_t$$\end{document} is a Poisson process of intensity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta Leb_{|[0,\infty )}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X_i)_{i\in \mathbb {N}}$$\end{document} is an iid sequence whose characteristic function is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _X(s)=\frac{(1-e^{is})\tilde{\theta }(s)}{\theta }+1$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Justification of (14) in the example

Here we justify the formula in (14) in the example of Sect. 2.3. To compute the limit law of the compound Poisson process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(t):=\sum _{i=1}^{N(t)}X_i$$\end{document} from Theorem 2.5, one has to compute the limit of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0}\beta _{k,\delta }^{(i)}(j)$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in \{1,2\}$$\end{document} as presented under Eq. (59). Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{S}^{rec}=\emptyset $$\end{document} , we deduce from Eq. (69) that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0}\beta _{k,\delta }^{(2)}(j,\cdot ,\cdot )=0$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in \mathbb {N}$$\end{document} . Although \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{rec}\ne 0$$\end{document} , one can also notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0}\beta _{k,\delta }^{(1)}(j)=0$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\ge 1$$\end{document} .

To see it, fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 0$$\end{document} and choose any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} small enough so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{m}x$$\end{document} remains close to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{{\textbf{v}}}$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\le k+1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{v}}\in V$$\end{document} assume there is an intermediate collision with the site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} at time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\le k$$\end{document} from the neighboring site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{q}}={\textbf{p}}^*+{\textbf{v}}$$\end{document} . Then at time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r+m\ge r+1$$\end{document} for a realisation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}$$\end{document} of the probability in play in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\delta \rightarrow 0}\beta _{k,\delta }^{(1)}(j)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( T_{\epsilon ,\delta }^{m+r}({\textbf {x}})\right) _{{\textbf{q}}}$$\end{document} remains close to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{m}a_{{\textbf{q}}+{\textbf{v}}}=a_{{\textbf{q}}+{\textbf{v}}}$$\end{document} for any m such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Psi }_k^{{\textbf {p}}^*}({\textbf {x}})={\textbf{q}}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Psi }_k^{{\textbf {p}}^*}({\textbf {x}})$$\end{document} given by the formula (61) and playing a similar role for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{k,\delta }^{(1)}(j)$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _k^{{\textbf{q}}}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{k,\delta }$$\end{document} . Thus there is no collision for such values of m. For the other values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\le k+1-r$$\end{document} , Lemma 5.3 ensures that the relative weight of such collision would be negligible as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \rightarrow 0$$\end{document} . We thus can conclude that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{\theta }(s)=1-\sum _{{\textbf{v}},{\textbf{v}}'\in V}\sum _{k\in {\mathbb {K}({\textbf{v}},{\textbf{v}}')}}\beta _{k,\delta }^{(1)}(j,{\textbf{v}},{\textbf{v}}')=\theta \end{aligned}$$\end{document}

and that the law of the integer variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_i$$\end{document} is given by the following characteristic function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{X}(s)=e^{is}$$\end{document} . In other words, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_i$$\end{document} is almost surely constant and equal to 1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(t)\sim \mathcal {P}(\theta t)$$\end{document} , that is a Poisson law of intensity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta t$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =1-5^{-4}$$\end{document} .

An Auxiliary Uniform Lasota–Yorke Inequality

In this section we prove an auxiliary Lasota–Yorke inequality which can be applied to different transfer operators in Sects. 3, 4 and 5. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in \mathcal {D}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} be a Borel complex measure, recall

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int \varphi d\mathcal {L}_{\varepsilon ,\delta }\mu =\int \varphi \circ T_{\varepsilon ,\delta } d\mu , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\varepsilon ,\delta }$$\end{document} is the fully coupled map lattice defined in (2), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \in (0,\varepsilon ]$$\end{document} being the size of collision zones involved with site \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}^*$$\end{document} .

Proposition 6.1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \subset \mathbb {Z}^d$$\end{document} be a finite subset and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(A_i)_{i=1,\dots ,s} \subset (I^{\Lambda })^s$$\end{document} be a finite family of sets with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_i$$\end{document} being a product of subintervals of I whose Lebesgue measure depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_i\in \mathbb {R}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|a_i|\le 1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\dots ,s$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1\in \{0,1\}$$\end{document} . Define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_\delta =\sum _{i=1}^s a_i1_{A_i}$$\end{document} and assume that whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \ge \delta '$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{h_\delta =a_1\}\subset \{h_{\delta '}=a_1\}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{h_{\delta '}=a_i\}\subset \{h_{\delta }=a_i\}$$\end{document} for all other \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i>1$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\delta : I^{\mathbb {Z}^d}\rightarrow \mathbb {R}$$\end{document} be given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\delta ({{\textbf {x}}}):=h_\delta ({{\textbf {x}}}_{\Lambda })$$\end{document} and consider the following operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}_{\varepsilon ,g_\delta }:=\mathcal {L}_{\varepsilon ,\delta }(g_\delta \cdot ). \end{aligned}$$\end{document}

Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} , independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} , such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {N}$$\end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \mathcal {L}_{\varepsilon ,g_\delta }^n\mu \Vert \le C\sigma ^n\Vert \mu \Vert +C|\mu |. \end{aligned}$$\end{document}

Proof

Notice that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}_{\varepsilon ,g_\delta }^n(\cdot )=\mathcal {L}_{\varepsilon ,\delta }^n\left( \prod _{k=0}^{n-1}g_\delta \circ T^k_{\varepsilon ,\delta }\right) , \end{aligned}$$\end{document}

and that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\delta $$\end{document} can take at most r different values then the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\prod _{k=0}^{n-1}g_\delta \circ T^k_{\varepsilon ,\delta }$$\end{document} can take at most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^r$$\end{document} different values that we denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1,\dots ,a_{m_r}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_r\le n^r$$\end{document} .

We first prove the following statement. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\in \{0,1\}$$\end{document} , there is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\sigma _0<\frac{1}{2}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^*\in \mathbb {N}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{n^*}>0$$\end{document} such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_i \in \text {Image}(g_{\delta }^n)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \mathcal {L}_{\varepsilon , \delta }^{n^*}\left( 1_{B_{\delta ,n^*,a_i}^b}\mu \right) \Vert \le 2\sigma _0\Vert \mu \Vert +C_{n^*}|\mu |, \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{\delta ,n,a_i}^0:=\{g_{\delta }^n=a_i\}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{\delta ,n,a_i}^1:=^c\{g_{\delta }^n=a_i\}$$\end{document} and notice that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}_{\varepsilon ,g_\delta }^n(\cdot )=\sum _{i=1}^{m_r}a_i\mathcal {L}_{\varepsilon ,\delta }^n\left( 1_{B_{\delta ,n,a_i}^0}\cdot \right) . \end{aligned}$$\end{document}

Let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\mathcal {L}_{\delta ,n,a_i,0}':=\mathcal {L}_{\varepsilon , \delta }^n\left( 1_{B_{\delta ,n,a_i}^0}\cdot \right) \text { and }\mathcal {L}_{\delta ,n,a_i,1}':=\mathcal {L}_{\varepsilon , \delta }^n\left( 1_{^c\{g_{\delta }^n=a_i\}}\cdot \right) .\end{aligned}$$\end{document}

We start by proving (77) for the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\delta ,n,a_i,0}'$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\ne 1$$\end{document} . We will eventually prove it for any index i. We introduce

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\Delta _{\varepsilon ,{\textbf{q}},{\textbf{v}}}:=A_{\varepsilon ,{\textbf{v}}}({\textbf{q}})\cap A_{\varepsilon ,-{\textbf{v}}}({\textbf{q}}+{\textbf{v}})\end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\Delta _{\varepsilon ,{\textbf {q}},0}:=\, ^c\left( \bigcup _{{\textbf{v}}\in V}\Delta _{\varepsilon ,{\textbf{q}},{\textbf{v}}}\right) .\end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in \mathcal {D}_1$$\end{document} . If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}\in \Delta _{\varepsilon ,{\textbf {q}},0}$$\end{document} , then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{{\textbf{q}}} \left( \frac{\varphi \circ T_{\varepsilon ,\delta }}{\tau '\circ \pi _{\textbf{q}}}\right) =\partial _{{\textbf{q}}} \left( \varphi \circ T_{\varepsilon ,\delta } \right) \frac{1}{\tau '\circ \pi _{\textbf{q}}}- \left( \varphi \circ T_{\varepsilon ,\delta } \right) \frac{\tau ''\circ \pi _{\textbf{q}}}{(\tau ')^2\circ \pi _{\textbf{q}}}. \end{aligned}$$\end{document}

Note that we use the assumption that the site dynamics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is piecewise onto. On the other hand if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}\in \Delta _{\varepsilon ,{\textbf{q}},{\textbf{v}}}$$\end{document} then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{{\textbf{q}}+{\textbf{v}}} \left( \frac{\varphi \circ T_{\varepsilon ,\delta }}{\tau '\circ \pi _{{\textbf{q}}+{\textbf{v}}}}\right)&=\partial _{{\textbf{q}}+{\textbf{v}}} \left( \varphi \circ T_{\varepsilon ,\delta } \right) \frac{1}{\tau '\circ \pi _{{\textbf{q}}+{\textbf{v}}}}- \left( \varphi \circ T_{\varepsilon ,\delta } \right) \frac{\tau ''\circ \pi _{{\textbf{q}}+{\textbf{v}}}}{(\tau ')^2\circ \pi _{{\textbf{q}}+{\textbf{v}}}}. \end{aligned}$$\end{document}

Moreover,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{{\textbf{q}}}(\varphi \circ T_{\varepsilon ,\delta })=\left\{ \begin{array}{l r c} \partial _{{\textbf{q}}} \varphi \circ T_{\varepsilon ,\delta } \cdot \tau '\circ \pi _{\textbf{q}}\text { if } {\textbf {x}}\notin \Delta _{\varepsilon ,{\textbf{q}},{\textbf{v}}}\\ \partial _{{\textbf{q}}+{\textbf{v}}} \varphi \circ T_{\varepsilon ,\delta } \cdot \tau '\circ \pi _{\textbf{q}}\text { otherwise.} \end{array}\right. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{\varepsilon ,{\textbf{q}},{\textbf{v}}}=\Delta _{\varepsilon ,{\textbf{q}}+{\textbf{v}},-{\textbf{v}}}$$\end{document} , the identity applied on the expansion above can be written as:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{\textbf{q}}\varphi \circ T_{\varepsilon ,\delta }=\left\{ \begin{array}{l r c} \partial _{\textbf{q}}\left( \frac{\varphi \circ T_{\varepsilon ,\delta }}{\tau '\circ \pi _{\textbf{q}}}\right) +\left( \varphi \circ T_{\varepsilon ,\delta } \right) \frac{\tau ''}{\tau '^2}\circ \pi _{\textbf{q}}\text { if } {\textbf {x}}\notin \Delta _{\varepsilon ,{\textbf{q}},{\textbf{v}}}\\ \partial _{{\textbf{q}}+{\textbf{v}}} \left( \frac{\varphi \circ T_{\varepsilon ,\delta }}{\tau '\circ \pi _{{\textbf{q}}+{\textbf{v}}}}\right) +\left( \varphi \circ T_{\varepsilon ,\delta } \right) \frac{\tau ''}{\tau '^2}\circ \pi _{{\textbf{q}}+{\textbf{v}}} \text { otherwise.} \end{array}\right. \end{aligned}$$\end{document}

Thus, we recursively get, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \in \mathbb {N}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{v}}_1,\dots ,{\textbf{v}}_n \in V \cup \{0\}$$\end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}\in \Delta _{\varepsilon ,n,{\textbf{q}},{\textbf{v}}_1,\dots ,{\textbf{v}}_{n-1}}:=\bigcap _{k=0}^{n-1} T_{\varepsilon ,\delta }^{-k}\Delta _{\varepsilon ,{\textbf{q}}+\sum _{i=1}^{k}{\textbf{v}}_i,{\textbf{v}}_{k+1}}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{\textbf{q}}\varphi \circ T_{\varepsilon ,\delta }^n= \begin{array}{l r c} \partial _{{\textbf{q}}+\sum _{i=1}^n{\textbf{v}}_i} \left( \frac{\varphi \circ T_{\varepsilon ,\delta }^n}{(\tau ^{n})'\circ \pi _{{\textbf{q}}+\sum _{i=1}^n{\textbf{v}}_i}}\right) +\left( \varphi \circ T_{\varepsilon ,\delta }^n \right) \frac{(\tau ^{n})''}{((\tau ^{n})')^2}\circ \pi _{{\textbf{q}}+\sum _{i=1}^n{\textbf{v}}_i} \end{array} \end{aligned}$$\end{document}

Notice first that for any two distinct tuples \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\textbf{v}}_1,\dots ,{\textbf{v}}_n)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\textbf{v}}_1',\dots ,{\textbf{v}}_n')$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{\varepsilon ,n,{\textbf{q}},{\textbf{v}}_1,\dots ,{\textbf{v}}_n}\cap \Delta _{\varepsilon ,n,{\textbf{q}},{\textbf{v}}_1',\dots ,{\textbf{v}}_n'}=\emptyset $$\end{document} and that any element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{q}}'={\textbf{q}}+\sum _{i=1}^n{\textbf{v}}_i$$\end{document} lies in a d-dimensional box \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{\textbf{q}}^n$$\end{document} around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{q}}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|G_{\textbf{q}}^n|=2dn+1$$\end{document} . Thus, denoting

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\tilde{\Delta }_{\varepsilon ,n,{\textbf{q}},{\textbf{q}}',{\textbf{v}}}:=\bigcup _{({\textbf{v}}_1,\dots ,{\textbf{v}}_n),{\textbf{q}}+\sum {\textbf{v}}_i={\textbf{q}}'}\Delta _{\varepsilon ,n,{\textbf{q}},{\textbf{v}}_1,\dots ,{\textbf{v}}_n,{\textbf{v}}},\end{aligned}$$\end{document}

we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int \partial _{\textbf{q}}\varphi d{\mathcal {L}_{\delta ,n,a_i,0}'\mu }= \int \partial _{\textbf{q}}\varphi \circ T^n_{\varepsilon ,\delta } 1_{ { B_{\delta ,n,a_i}^0}}d\mu \nonumber \\&\quad = \sum _{{\textbf{q}}' \in G_{\textbf{q}}^n}\sum _{{\textbf{v}}\in V\cup \{0\}}\alpha ^{-n}\int _{\tilde{\Delta }_{\varepsilon ,n,{\textbf{q}},{\textbf{q}}',{\textbf{v}}}\cap {B_{\delta ,n,a_i}^0}} \partial _{{\textbf{q}}'} \left( \alpha ^n \frac{\varphi \circ T_{\varepsilon ,\delta }^n}{(\tau ^{n})'\circ \pi _{{\textbf{q}}'}}\right) d\mu \nonumber \\&\qquad +\int _{\tilde{\Delta }_{\varepsilon ,n,{\textbf{q}},{\textbf{q}}',{\textbf{v}}}\cap { B_{\delta ,n,a_i}^0}}\left( \varphi \circ T_{\varepsilon .\delta }^n \right) \frac{(\tau ^{n})''}{((\tau ^{n})')^2}\circ \pi _{q'}d\mu , \end{aligned}$$\end{document}

where we recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\tau '(x)|\ge \alpha >1$$\end{document} . The second term of (79) can be easily bounded by a constant times the weak norm of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} . We now deal with the first term of (79). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(I_k)_{k\in \mathbb {N}}$$\end{document} be the partition of I into intervals of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} -regularity for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^n$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_k$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{k+1}$$\end{document} are consecutive intervals and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_i:=\bigcup _{k}I_{2k+i}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in \{0,1\}$$\end{document} . We have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\sum _{{\textbf{q}}' \in G_{\textbf{q}}^n}\sum _{{\textbf{v}}\in V\cup \{0\}}\alpha ^{-n}\int _{\tilde{\Delta }_{\varepsilon ,n,{\textbf{q}},{\textbf{q}}',{\textbf{v}}}\cap { B_{\delta ,n,a_i}^0}} \partial _{{\textbf{q}}'} \left( \alpha ^n \frac{\varphi \circ T_{\varepsilon ,\delta }^n}{(\tau ^{n})'\circ \pi _{{\textbf{q}}'}}\right) d\mu \\&\quad =\alpha ^{-n} \sum _{{\textbf{q}}' \in G_{\textbf{q}}^n}\sum _{{\textbf{v}}\in V\cup \{0\}}\sum _{i\in \{0,1\}}\int \textbf{1}_{\tilde{\Delta }_{\varepsilon ,n,{\textbf{q}},{\textbf{q}}',{\textbf{v}}}\cap { B_{\delta ,n,a_i}^0}\cap C_{i}^{\textbf{q}}} \partial _{{\textbf{q}}'} \left( \alpha ^n \frac{\varphi \circ T_{\varepsilon ,\delta }^n}{(\tau ^{n})'\circ \pi _{{\textbf{q}}'}}\right) d\mu , \end{aligned} \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}C_{i}^{\textbf{q}}:=\{{\textbf {x}}\in I^{\mathbb {Z}^d}, x_{\textbf{q}}\in K_i\}.\end{aligned}$$\end{document}

Let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\psi _{{\textbf{q}}',n,i}:=\alpha ^n\frac{\varphi \circ T_\varepsilon ^n}{(\tau ^{n})'\circ \pi _{{\textbf{q}}'}}1_{ \tilde{\Delta }_{\varepsilon ,n,{\textbf{q}},{\textbf{q}}',{\textbf{v}}_n}\cap { B_{\delta ,n,a_i}^0}\cap C_{i}^{{\textbf{q}}'} }.\end{aligned}$$\end{document}

Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _{{\textbf{q}}',n, i}$$\end{document} is continuous on its non-zero set since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is piecewise \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} and piecewise onto. To build a good test function out of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _{{\textbf{q}}',n, i}$$\end{document} , let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\tilde{\psi }_{{\textbf{q}}', n, i}=\psi _{{\textbf{q}}', n, i}-\psi ^*_{{\textbf{q}}',n, i}\end{aligned}$$\end{document}

where, for any fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {x}}_{\ne {\textbf{q}}'}\in I^{\mathbb {Z}^d \backslash \{{\textbf{q}}'\}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi ^*_{{\textbf{q}}',n,i}$$\end{document} is piecewise linear in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{{\textbf{q}}'}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi ^*_{{\textbf{q}}', n, i}=0$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Delta }_{\varepsilon ,n,{\textbf{q}},{\textbf{q}}',{\textbf{v}}_n}\cap { B_{\delta ,n,a_i}^0}\cap C_{i}^{{\textbf{q}}'}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi ^*_{{\textbf{q}}', n, i}(x)=\alpha ^n\frac{\varphi \circ T_\varepsilon ^n}{(\tau ^{n})'\circ \pi _{{\textbf{q}}'}}({\textbf {x}})$$\end{document} if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \partial _{{\textbf{q}}'}\tilde{\Delta }_{\varepsilon ,n,{\textbf{q}},{\textbf{q}}',{\textbf{v}}_n}\cap { B_{\delta ,n,a_i}^0}\cap C_{i}^{{\textbf{q}}'}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{{\textbf{q}}'}\tilde{\Delta }_{\varepsilon ,n,{\textbf{q}},{\textbf{q}}',{\textbf{v}}_n}\cap { B_{\delta ,n,a_i}^0}\cap C_{i}^{{\textbf{q}}'}$$\end{document} is the boundary of the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Delta }_{\varepsilon ,n,{\textbf{q}},{\textbf{q}}',{\textbf{v}}_n}\cap { B_{\delta ,n,a_i}^0}\cap C_{i}^{{\textbf{q}}'}$$\end{document} for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{{\textbf{q}}'}$$\end{document} coordinates. Therefore, by its definition, the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\psi }_{{\textbf{q}}',n,i}\in \mathcal {D}_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\partial _{{\textbf{q}}',n,i}\psi ^*|\le \frac{|\varphi |_\infty }{l_{{\textbf{q}}',n^*,a_1,0,\delta }}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{{\textbf{q}}',n^*,a_1,0,\delta }$$\end{document} is the minimal distance between two connected components of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{{\textbf{q}}',0}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{{\textbf{q}}',1}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{{\textbf{q}}',i}:=\{x_{{\textbf{q}}'},(x_{{\textbf{q}}'},{\textbf {x}}_{\ne {\textbf{q}}'})\in \tilde{\Delta }_{\varepsilon ,n,{\textbf{q}},{\textbf{q}}',{\textbf{v}}_n}\cap { B_{\delta ,n,a_i}^0}\cap C_{i}^{{\textbf{q}}'}\}$$\end{document} . By construction, there is a finite number of connected subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{{\textbf{q}}',i}$$\end{document} , thus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{{\textbf{q}}',n^*,a_1,0,\delta }>0$$\end{document} . Consequently,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int \partial _{{\textbf{q}}'} \psi _{{\textbf{q}}',n,i}d\mu&=\int \partial _{{\textbf{q}}'}\tilde{\psi }_{{\textbf{q}}',n,i}d\mu +\int \partial _{{\textbf{q}}'}\psi ^*_{{\textbf{q}}',n,i}d\mu \nonumber \\&\le \Vert \mu \Vert +\frac{1}{l_{{\textbf{q}}',n^*,a_1,0,\delta }}|\mu |. \end{aligned}$$\end{document}

Notice that when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\delta :=1$$\end{document} and setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1:=1$$\end{document} then the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{\delta ,n,a_i}^0=I^{\mathbb {Z}^d}$$\end{document} and the distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{{\textbf{q}}',n^*,a_1,0,\delta }$$\end{document} does not depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} . Thus we obtain from the equation above the uniform Lasota–Yorke iequality for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{\varepsilon ,\delta }$$\end{document} : There is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {N}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \mathcal {L}_{\varepsilon ,\delta }^n\mu \Vert \le \sigma _0^n\Vert \mu \Vert +C|\mu |. \end{aligned}$$\end{document}

Now, choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^*\in \mathbb {N}$$\end{document} large enough so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\alpha ^{-n^*} (2d+1)|G_{{\textbf{q}}'}^{n^*}|<\sigma _0<\frac{1}{2}$$\end{document} and use (80) in inequality (79) to obtain for any couple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_i,0)$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\ne 1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\int \partial _{\textbf{q}}\varphi d{\mathcal {L}_{\delta ,n^*,a_i,0}'}\mu \\&\quad \le \sigma _0\Vert \mu \Vert +2 (2d+1)|G_{{\textbf{q}}'}^{n^*}|\left( \frac{1}{l_{{\textbf{q}}', n^*,a_i,0,\delta }}+2\left| \frac{(\tau ^{n^*})''}{((\tau ^{n^*})')^2}\right| _\infty \right) |\mu |\\&\quad \le \sigma _0\Vert \mu \Vert + C_{n^*}|\mu |, \end{aligned} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{n^*}>0$$\end{document} can be chosen independently of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} , for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} sufficiently small. Indeed, for such couple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_i,b)$$\end{document} , for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \le \delta '$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{\delta ,n,a_i}^0\subset B_{\delta ',n,a_i}^0$$\end{document} are made of shrinking subintervals, and thus eventually \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{{\textbf{q}}',n^*,a_i,0,\delta }\ge l_{{\textbf{q}}',n^*,a_i,0,\delta '}$$\end{document} . Similarly, since for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \le \delta '$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{\delta ,n,a_1}^1\subset B_{\delta ',n,a_1}^1$$\end{document} , and thus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{{\textbf{q}}',n^*,a_1,1,\delta }\ge l_{{\textbf{q}}',n^*,a_1,1,\delta '}$$\end{document} we also obtain the above Lasota-Yorke inequality for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_1,1)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\int \partial _{\textbf{q}}\varphi d{\mathcal {L}_{\delta ,n^*,a_1,1}'}\mu \\&\quad \le \sigma _0\Vert \mu \Vert +2 (2d+1)|G_{{\textbf{q}}'}^{n^*}|\left( \frac{1}{l_{{\textbf{q}}',n^*,a_1,1,\delta }}+2\left| \frac{(\tau ^{n^*})''}{((\tau ^{n^*})')^2}\right| _\infty \right) |\mu |\\&\quad \le \sigma _0\Vert \mu \Vert + C_{n^*}|\mu |, \end{aligned} \end{aligned}$$\end{document}

We deduce from Eqs. (77) and (81) that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \mathcal {L}_{\delta ,n^*,a_1,0}^{n^*}\mu \Vert&\le \Vert \mathcal {L}_{\varepsilon ,{\textbf {p}}^*}\mu -\mathcal {L}_{\delta ,n^*,a_1,1}\mu \Vert \nonumber \\&\le 2\sigma _0\Vert \mu \Vert + 2C_{n^*}|\mu |. \end{aligned}$$\end{document}

Now we choose m such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2(mn^*)^r\sigma _0^m<1$$\end{document} and sum the m-iterates of inequality (77) over every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_i$$\end{document} to get a Lasota–Yorke inequality uniform in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} . There is some constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K>0$$\end{document} such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \mathcal {L}_{\varepsilon ,g_\delta }^{mn^*}\mu \Vert&\le \sum _{i\le m_{mn^*}}|a_i|\Vert \mathcal {L}_{\delta ,n^*,a_i,0}^{n^*}\mu \Vert \\&\le 2(mn^*)^r\sigma _0^m \Vert \mu \Vert + 2K(mn^*)^rC_{n^*}|\mu |. \end{aligned}$$\end{document}

Thus, iterating the above inequality, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {N}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathcal {B}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\Vert \mathcal {L}_{\varepsilon ,g_\delta }^{n}\mu \Vert \le C \sigma ^n\Vert \mu \Vert + C|\mu |,\end{aligned}$$\end{document}

for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}