# Potential theory and $\mathbb{Z}^{d}$-extensions

Damien THOMINE

###### Abstract

We study hitting probabilities for $\mathbb{Z}^{d}$-extensions of Gibbs-Markov
maps. The goal is to estimate, given a finite $\Sigma\subset\mathbb{Z}^{d}$
and $p$, $q\in\Sigma$, the probability $P_{pq}$ that the process starting from
$p$ returns to $\Sigma$ at site $q$.

Our study generalizes the methods available for random walks. We are able to
give in many settings (square integrable jumps, jumps in the basin of a Lévy
or Cauchy random variable) asymptotics for the transition matrix
$(P_{pq})_{p,q\in\Sigma}$ when the elements of $\Sigma$ are far apart.

We use three main tools: a variant of the balayage identity using a transfer
operator as a Markov transition kernel, a study inspired from fast-slow
systems and the hitting time of small sets in hyperbolic systems to relate
transfer operators and the transition matrices we seek to compute, and finally
Fourier transform and perturbations of transfer operators à la Nagaev-
Guivarc’h to effectively compute these transition matrices in an asymptotic
regime.

###### Contents

  1. Introduction
  2. 1 Setting and main results
    1. 1.1 Hitting probabilities for recurrent random walks
    2. 1.2 Dynamical systems: extension and induction
    3. 1.3 Gibbs-Markov maps
    4. 1.4 Shapes and matrices
    5. 1.5 Strategy and main results
  3. 2 From transition probabilities to potential
    1. 2.1 Cone contraction at time scale $\Theta(1)$
    2. 2.2 Product of matrices
    3. 2.3 Cone contraction at time scale $\Theta(\varepsilon^{-1})$
  4. 3 The ergodic case
    1. 3.1 Structure of extensions with non-mixing first return maps
    2. 3.2 End of the proof of Lemma 1.20
  5. 4 From potential to transition probabilities
    1. 4.1 Bootstrapping
    2. 4.2 Irreducibility of the bi-L-matrix
  6. 5 Fourier transform and perturbations of the transfer operator
    1. 5.1 Twisted Poisson equations
    2. 5.2 Transfer operators and periodicities
    3. 5.3 Fourier transform and twisted transfer operators
  7. 6 Computation of the potential operators
    1. 6.1 Computation of the potential: $d=1$, square integrable jumps
    2. 6.2 Computation of the potential: $d=1$, Lévy jumps
    3. 6.3 Computation of the potential: $d=1$, Cauchy jumps
    4. 6.4 Computation of the potential: $d=2$, square integrable jumps
  8. A Gibbs-Markov maps
    1. A.1 Markov partitions and stopping times
    2. A.2 Stopping times and regularizing effect of $\mathcal{L}_{\varphi}$
    3. A.3 Induced Gibbs-Markov maps

## Introduction

Large classes of uniformly hyperbolic dynamical systems, when endowed with
suitable probability measures, behave similarly to Markov chains. Many
theorems pertaining to sequences of i.i.d. random variables have found
generalizations for sequences of observables of such systems: the law of large
numbers (Birkhoff’s ergodic theorem), the central limit theorem [36, 47, 37,
22], the local central limit theorem and asymptotics of occupation times [11,
43, 1, 48, 16] and large deviation principles [38, 39, 53], to cite only the
most widely known.

This article is part of a project to understand how an important part of the
general theory of Markov processes, namely the probabilistic theory of
potentials, can be adapted to dynamical systems. The specific problem we are
interested in is inspired by the computation of _hitting probabilities for
random walks_. Given a recurrent random walk $(S_{n})_{n\geq 0}$ on
$\mathbb{Z}^{d}$ and a finite set $\Sigma\subset\mathbb{Z}^{d}$, one can
compute the transition probabilities between points of $\Sigma$ – in other
words, the probability, given any $p$, $q\in\Sigma$, that a random walk
starting from point $p$ returns to $\Sigma$ at point $q$.

$A$$B$$C$

Figure 1: Three sites $A$, $B$, $C$ in $\mathbb{Z}^{2}$, and a transition from
$A$ to $C$.

This problem has an interpretation in terms of resistances in electrical
networks, and may be solved using potential theory, more precisely the so-
called _balayage identity_ (see e.g. [45, Corollary 1.11], or Theorem 1.1
below). With this method, for the simple random walk on $\mathbb{Z}^{2}$ and
the subset $\Sigma=\\{A,B,C\\}$ of Figure 1, one finds that the matrix of
transition probabilities is

$P_{\\{A,B,C\\}}=\frac{1}{-\pi^{2}+8\pi-4}\left(\begin{array}[]{ccc}-\frac{1}{2}\pi^{2}+4\pi-4&-\frac{1}{2}\pi^{2}+3\pi&\pi\\\
-\frac{1}{2}\pi^{2}+3\pi&-\pi^{2}+6\pi-4&\frac{1}{2}\pi^{2}-\pi\\\
\pi&\frac{1}{2}\pi^{2}-\pi&-\frac{3}{2}\pi^{2}+8\pi-4\\\ \end{array}\right).$
(0.1)

A natural generalization of random walks is the class of
$\mathbb{Z}^{d}$-extensions of dynamical systems. Starting from a probability-
preserving ergodic dynamical system $(A,\mu,T)$ and a jump function
$F:A\to\mathbb{Z}^{d}$, its $\mathbb{Z}^{d}$-extensions is the system
$([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ defined by:

  * •

State space: $[\mathbb{Z}^{d}]:=A\times\mathbb{Z}^{d}$;

  * •

Measure: $\widetilde{\mu}:=\mu\otimes\sum_{p\in\mathbb{Z}^{d}}\delta_{p}$;

  * •

Transformation: $\widetilde{T}(x,p):=(T(x),p+F(x))\in[\mathbb{Z}^{d}]$ for all
$(x,p)\in[\mathbb{Z}^{d}]$.

This class of systems include $\mathbb{Z}^{d}$-valued random walks,
$\mathbb{Z}^{d}$-extensions of Markov chains [31, 32, 33] (sometimes used as
toy models for more realistic processes), $\mathbb{Z}^{d}$-extensions of
subshifts of finite type [43] as well as the collision map for Lorentz gases
[48, 49]. Using well-chosen sections, the study of the Lorentz gas flow or of
the geodesic flow on Abelian covers of negatively curved compact manifolds may
also be reduced to the study of such extensions [44, 16].

In the setting of $\mathbb{Z}^{d}$-extensions, the study of hitting
probabilities can be transposed as follows. Given a finite subset
$\Sigma\subset\mathbb{Z}^{d}$ and $p$, $q\in\Sigma$, what is the probability
(under $\mu$) that the trajectory $(\widetilde{T}^{n}(x,p))_{n\geq 0}$ returns
to $A\times\Sigma$ at the site $A\times\\{q\\}$?

While an explicit computation of the matrix of transition probabilities, as in
Equation (0.1), was possible for the simple random walk, such an explicit
formula seems to be out of reach for more general dynamical systems. As for
the central limit theorem, the local central limit theorem or the principle of
large deviations, elementary formulas are available in an asymptotic regime.
For hitting probabilities, we propose to study the asymptotics of the
transition probabilities for families $(\Sigma_{t})_{t>0}$ of subsets of
$\mathbb{Z}^{d}$ whose points get farther and farther apart when the parameter
$t$ goes to infinity:

Figure 2: Some elements of a family of subset $(A\times\Sigma_{t})_{t>0}$
growing farther apart while keeping the same shape.

One particularly interesting case, pictured in Figure 2, occurs when those
points keep the same shape, that is

$\Sigma_{t}=\left\\{t\sigma(i)+o(t):\ i\in I\right\\}$

for some finite set $I$ and some injective function
$\sigma:I\to\mathbb{R}^{d}$. This kind of limit theorems is in line with a few
former studies [32, 40].

In this setting, hitting probabilities could be obtained in [40] for families
of subsets of size $2$. However, these results were byproducts of
distributional limit theorems for additive functionals of the process
$(\widetilde{T}^{n})_{n\geq 0}$. As a consequence, the methods involved were
extremely heavy, limited in scope as they could not produce the matrix of
transition probabilities but only its symmetrization, and obscured the
underlying potential theory. A second article [41] laid down the foundation
for the use of probabilistic potential theory for hyperbolic dynamical
systems; in particular, we need to use the transfer operator as a Markov
operator instead of the Koopman operator.

The current contribution has two main parts. First, parametrizing by
$\varepsilon=t^{-1}$ instead of $t$, we define _potential operators_
$(Q_{\varepsilon})_{\varepsilon>0}$ acting on Lipschitz functions on
$A\times\Sigma_{\varepsilon}$. We are able to relate these potential operators
to the matrices of transition probabilities when $\varepsilon$ vanishes:

######  Theorem 0.1.

Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be an ergodic and
recurrent Markov $\mathbb{Z}^{d}$-extension of a Gibbs-Markov map. Let $I$ be
finite. For $\varepsilon>0$, let
$\sigma_{\varepsilon}:I\hookrightarrow\mathbb{Z}^{d}$. Let $P_{\varepsilon}$
be the matrix of transition probabilities on $\Sigma_{\varepsilon}$.

The following properties are equivalent:

  * •

there exists an irreducible (in the sense of Definition 1.8) bi-L-matrix $R$
such that $P_{\varepsilon}=\operatorname{Id}-\varepsilon R+o(\varepsilon)$;

  * •

there exists an irreducible (in the sense of Definition 1.13) matrix $S$ such
that $Q_{\varepsilon}=\varepsilon^{-1}S+o(\varepsilon^{-1})$.

If any of these properties holds, then, in addition, $S=R_{0}^{-1}$.

One of the main features of this theorem is that its hypotheses are quite
general: recurrence is necessary for $P_{\varepsilon}$ to be a stochastic
matrix, while ergodicity ensures (among other things) that $P_{\varepsilon}$
is irreducible. No assumption is made on the shape of the subsets
$(\Sigma_{\varepsilon})_{\varepsilon>0}$ nor on the jumps $F$. Note that, for
random walks, we have
$Q_{\varepsilon}=(\operatorname{Id}-P_{\varepsilon})^{-1}$, so that the
conclusion of Theorem 0.1 then holds even in the non-asymptotic regime.

The proof of Theorem 0.1 relies on the theory of _hitting time of small
targets_ [42, 26, 9, 6]: when $\varepsilon$ vanishes, the transitions between
different sites of $A\times\Sigma_{\varepsilon}$ become rare events111This
point of view explains the use of the parameter $\varepsilon$, which is the
size of the targets, instead of $t$, for most of this article., so the process
induced by $(\widetilde{T}^{n})_{n\geq 0}$ looks like a Markov process on
$\Sigma_{\varepsilon}$ with transitions at times of order $\varepsilon^{-1}$.
We give a probabilistic proof of Theorem 0.1, inspired by previous works on
fast-slow systems [9, 12] and using a coupling argument to control these
transitions. Note however, that a more analytic approach could be possible,
along the line adopted by G. Keller and C. Liverani [29], or by D. Dolgopyat
and P. Wright [17] (the later work being closer to our setting, as the
operators involved have a degeneracy of higher order than those studied in
[29]).

The potential operators $(Q_{\varepsilon})_{\varepsilon>0}$ can be estimated,
in the asymptotic regime $\varepsilon\to 0$, from the perturbations of the
main eigenvalue of the transfer operator associated with $(A,\mu,T)$. When the
jump function $F$ is well-behaved, we get explicit expressions for the
operator $S$ in Theorem 0.1, and from there, a first order expansion of the
transition matrices $(P_{\varepsilon})_{\varepsilon>0}$. We investigate four
cases, when $d=1$ and $F$ is in the basin of attraction of a Lévy stable
distribution (distinguishing the $\mathbb{L}^{2}$, general Lévy and Cauchy
cases), and when $d=2$ and $F$ is square integrable. For instance, in the
later case, we get

######  Proposition 0.2.

Let $([\mathbb{Z}^{2}],\widetilde{\mu},\widetilde{T})$ be an ergodic and
recurrent Markov $\mathbb{Z}^{2}$-extension of a Gibbs-Markov map $(A,\mu,T)$
with jump function $F:A\to\mathbb{Z}^{2}$. Assume furthermore that
$\left\|F\right\|\in\mathbb{L}^{2}(A,\mu)$. Let $\operatorname{Cov}>0$ be that
asymptotic covariance matrix of $F$ defined by Equation (6.14).

Let $I$ be non-empty and finite and $\sigma:I\hookrightarrow\mathbb{R}^{2}$ be
injective. For $t>0$, let $\sigma_{t}:I\hookrightarrow\mathbb{Z}^{2}$ be such
that $\sigma_{t}(i)=t\sigma(i)+o(t)$ for all $i\in I$. Let $P_{t}$ be the
transition matrix of the Markov chain induced on $\Sigma_{t}:=\sigma_{t}(I)$.
Then

$P_{t}=_{t\to+\infty}\operatorname{Id}-\frac{\pi\sqrt{\det(\operatorname{Cov})}}{\ln(t)}R+o(\ln(t)^{-1}),$

where $R$ is the irreducible, symmetric bi-L-matrix defined by Equation
(6.12):

$\left\\{\begin{array}[]{lcll}R_{ij}&=&-\frac{1}{|I|}&\quad\text{whenever
}i\neq j\in I,\\\ R_{ii}&=&\frac{|I|-1}{|I|}&\quad\forall i\in
I.\end{array}\right.$

This text also includes a few results which may be of independent interest,
such as a structural result on the periodic components of a
$\mathbb{Z}^{d}$-extension of a Gibbs-Markov map, which has the curious
particularity of being different for $\mathbb{Z}$ and
$\mathbb{Z}^{2}$-extensions (see Proposition 3.3).

The structure of this article is as follows. Section 1 is an introduction. We
present a general method to compute transition matrices for random walks, ours
setting ($\mathbb{Z}^{d}$-extensions, induced systems, Gibbs-Markov maps and
transfer operators), and give our strategy to get ours results. Sections 2, 3
and 4 give a proof of Theorem 0.1; this decomposition in three steps shall be
explained in Sub-subsection 1.5.3. Afterwards, we shall obtain Proposition 0.2
and similar propositions in Sections 5 and 6; the former will give a general
formula for the potential operator $Q_{\varepsilon}$ using Fourier transform,
while the later will apply this formula to various settings. Finally, an
appendix compiles some technical results on Gibbs-Markov maps.

## Acknowledgements

The authors thanks Françoise Pène for the useful discussions while working on
this project and her comments on the current version of this article.

##  1 Setting and main results

###  1.1 Hitting probabilities for recurrent random walks

####  1.1.1 General theory

Consider a sequence of independent and identically distributed
$\mathbb{Z}^{d}$-valued random variables $(X_{n})_{n\geq 0}$, and set
$S_{n}:=\sum_{k=0}^{n-1}X_{k}$. The sequence of random variables
$(S_{n})_{n\geq 0}$ is a random walk in $\mathbb{Z}^{d}$ starting from $0$.
Let $P$ be its transition kernel:

$P_{pq}=\mathbb{P}(X_{0}=q-p).$

This transition kernel acts on functions by
$P(f)(p)=\sum_{q\in\mathbb{Z}^{d}}P_{pq}f(q)$.

In what follows, we shall assume that $(S_{n})_{n\geq 0}$ is recurrent and
ergodic, which implies in particular that $d\in\\{1,2\\}$. Given
$\Sigma\subset\mathbb{Z}^{d}$ and $p\in\mathbb{Z}^{d}\setminus\Sigma$, we
define the hitting time of $\Sigma$ as the random variable

$\overline{\varphi}_{\Sigma}(p):=\inf\\{n\geq 0:\ p+S_{n}\in\Sigma\\},$

and the _hitting point_ as $p+S_{\overline{\varphi}_{\Sigma}(p)}\in\Sigma$.
Then, by recurrence, the hitting point is a $\Sigma$-valued random variable.
Its distribution can be computed in the following way: given any bounded
function $f:\Sigma\to\mathbb{R}$, extend it to a bounded function
$g:\mathbb{Z}^{d}\to\mathbb{R}$ such that

$\left\\{\begin{array}[]{lll}g(r)&=f(r)&\quad\forall r\in\Sigma,\\\
(\operatorname{Id}-P)(g)(r)&=0&\quad\forall
r\in\mathbb{Z}^{d}\setminus\Sigma.\end{array}\right.$ (1.1)

Then, by a martingale argument,

$g(p)=\mathbb{E}(f(p+S_{\overline{\varphi}_{\Sigma}(p)}));$

specializing e.g. to $f=\mathbf{1}_{q}$ for $q\in\Sigma$ gives in principle
the distribution of the hitting point, as long as one is able to solve
Equation (1.1).

In this article, we are interested in a slightly different problem: $\Sigma$
is finite, and we want to understand transition probabilities ; that is, the
starting point $p$ belongs to $\Sigma$, and we consider the _hitting time_

$\varphi_{\Sigma}(p):=\inf\\{n\geq 1:\ p+S_{n}\in\Sigma\\}.$

The problem of computing transition probabilities can be reframed in the
following way. Taking a trajectory $(S_{n})_{n\geq 0}$ of the random walk and
keeping only the times at which the random walk belongs to $\Sigma$ yields, by
the strong Markov property, a $\Sigma$-valued _induced Markov chain_.

$A$$B$$C$$A$$B$$C$

Figure 3: A transition on $\\{A,B,C\\}$ induced by a path in $\mathbb{Z}^{2}$.

This induced Markov chain has transition matrix

$P_{\Sigma,pq}=\mathbb{P}(p+S_{\varphi_{\Sigma}(p)}=q),$

and we want to compute the operator $P_{\Sigma}$. This can be done thanks to a
variation of Equation (1.1) adapted to starting points belonging to $\Sigma$,
the so-called _balayage identity_. The version we quote is close to [45,
Corollary 1.11], and can be found in [41]:

######  Theorem 1.1 (Balayage identity).

Let $\Omega$ be a Polish space, $P$ a Markov kernel on $\Omega$, and $\mu$ a
measure which is stationary, recurrent and ergodic for $P$.

Let $\Sigma\subset\Omega$ be measurable, with $\mu(\Sigma)>0$. Let
$g\in\mathbb{L}^{\infty}(\Omega,\mu)$ and
$f\in\mathbb{L}^{\infty}(\Sigma,\mu_{|\Sigma})$. Let
$f\mathbf{1}_{\Sigma}:\mathbb{Z}^{d}\to\mathbb{C}$ be the function which
coincides with $f$ on $\Sigma$ and is null everywhere else.

If

$(\operatorname{Id}-P)g=f\mathbf{1}_{\Sigma},$

then:

$(\operatorname{Id}-P_{\Sigma})g_{|\Sigma}=f.$

Theorem 1.1 relates the solution of the Poisson equation for the kernel $P$ on
$\mathbb{Z}^{d}$ to the solutions of the Poisson equation for the kernel
$P_{\Sigma}$ on $\Sigma$. In the framework of electrodynamics, $f$ is an
intensity, while $g$ is an associated potential.

A first remark is that $(S_{n})_{n\geq 0}$, seen as a Markov chain on
$\mathbb{Z}^{d}$, preserves the counting measure, is ergodic and recurrent.
Moreover, for general recurrent Markov chains with stationary measure $\mu$,
the restricted measure $\mu_{|\Sigma}$ is stationary for the Markov chain
induced on $\Sigma$. Consequently, in the context of random walks,
$P_{\Sigma}$ preserves the uniform measure $\mu_{\Sigma}$ on $\Sigma$.

When acting on functions, $P_{\Sigma}$ preserves constant functions. Since
$(S_{n})_{n\geq 0}$ is ergodic, so is $P_{\Sigma}$, so
$\operatorname{Ker}(\operatorname{Id}-P_{\Sigma})=\operatorname{Vect}(\mathbf{1})$
has dimension $1$. Finally, since $P_{\Sigma}$ preserves $\mu_{\Sigma}$, the
eigenprojection onto $\operatorname{Vect}(\mathbf{1})$ is given by
$f\mapsto\mu_{\Sigma}(f)\mathbf{1}$, whence we have a $P_{\Sigma}$-invariant
splitting
$\mathbb{C}^{\Sigma}=\operatorname{Vect}(\mathbf{1})\oplus\mathbb{C}_{0}^{\Sigma}$,
with $\mathbb{C}_{0}^{\Sigma}:=\operatorname{Ker}(\mu_{\Sigma})$. Hence, all
we need is to understand the action of $P_{\Sigma}$ on
$\mathbb{C}_{0}^{\Sigma}$.

Now, take any $f\in\mathbb{C}_{0}^{\Sigma}$. Extend it to
$f\mathbf{1}_{\Sigma}:\mathbb{Z}^{d}\to\mathbb{C}$. Find a function
$g:\mathbb{Z}^{d}\to\mathbb{C}$ such that
$(\operatorname{Id}-P)g=f\mathbf{1}_{\Sigma}$ and $\mu_{|\Sigma}(g)=0$ ; such
a function always exists under our hypotheses. Then
$g_{|\Sigma}\in\mathbb{C}_{0}^{\Sigma}$ and
$(\operatorname{Id}-P_{\Sigma})g_{|\Sigma}=f$ by the balayage identity. Doing
this operation for all $f$ in a basis of $\mathbb{C}_{0}^{\Sigma}$ allows us,
in theory, to compute the operator $(\operatorname{Id}-P_{\Sigma})^{-1}$ on
$\mathbb{C}_{0}^{\Sigma}$. Then, invert $(\operatorname{Id}-P_{\Sigma})^{-1}$,
and from there get the action of $P_{\Sigma}$ on $\mathbb{C}_{0}^{\Sigma}$.

All is left is to effectively compute $g$ given $f$. At this point the
algebraic structure of the random walk plays a role, as we can use the Fourier
transform. The main idea is that the operator $\widehat{P}$, which corresponds
to the action of $P$ in the frequency domain, is merely the multiplication
operator by the characteristic function
$\Psi(\xi):=\mathbb{E}(e^{i\langle\xi,X\rangle})$ of $X$. Up to an additive
constant,

$\hat{g}(\xi)=\sum_{n=0}^{+\infty}\widehat{P}^{n}(\xi)\hat{f}(\xi)=\sum_{n=0}^{+\infty}\Psi(\xi)^{n}\hat{f}(\xi)=\frac{\hat{f}(\xi)}{1-\Psi(\xi)},$

whence, writing $\mathbb{T}^{d}=\mathbb{R}^{d}_{/\mathbb{Z}^{d}}$,

$g(p)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}\frac{\hat{f}(\xi)}{1-\Psi(\xi)}e^{i\langle\xi,p\rangle}\;{\rm
d}\xi,$ (1.2)

still up to an additive constant. If the kernel $P$ is nice enough, as is the
case for the simple random walk on $\mathbb{Z}^{d}$, such integrals can even
be explicitly computed.

To sum up, this strategy to compute the Markov kernel $P_{\Sigma}$ has three
main ingredients:

  * •

the balayage identity (Theorem 1.1) which relates the Markov kernels $P$ and
$P_{\Sigma}$ through the solutions to their respective Poisson equations;

  * •

the ability to solve the Poisson equation on $\mathbb{Z}^{d}$, using for
instance the Fourier transform;

  * •

the ability to invert $(\operatorname{Id}-P_{\Sigma})^{-1}$.

In the case of random walks, $\mathbb{C}_{0}^{\Sigma}$ is finite dimensional,
so the last step may seem comparatively insignificant. However, in the
remainder of this article, we shall work with operators acting on infinite-
dimensional Banach spaces, for which getting explicit information on their
inverse is much tougher.

####  1.1.2 An example with the simple random walk

Now, let us delve deeper into the method described in Sub-subsection 1.1.1 by
applying it to the example of Figure 1. We take for $(S_{n})_{n\geq 0}$ the
simple random walk on $\mathbb{Z}^{2}$, and $\Sigma=\\{A,B,C\\}$ with
$A=(0;0)$, $B=(-1;0)$ and $C=(1;1)$.

The simple random walk has a transition kernel
$P_{p}=\frac{1}{4}\sum_{q\leftrightarrow p}\delta_{q}$, where
$q\leftrightarrow p$ if and only if $p$ an $q$ are neighbours. Its action on
functions $f:\mathbb{Z}^{2}\to\mathbb{C}$ is thus

$P(f)(p)=\frac{1}{4}\sum_{q\leftrightarrow p}f(q),$

and the simple random walk is indeed ergodic, recurrent, and preserves the
counting measure on $\mathbb{Z}^{2}$.

Let $P_{\Sigma}$ be the transition matrix for the Markov chain induced on
$\Sigma$. Then $P_{\Sigma}$ preserves the uniform measure on $\Sigma$ denoted
by $\mu_{\Sigma}$. Thus we have a $P_{\Sigma}$-invariant splitting
$\mathbb{C}^{\Sigma}=\operatorname{Ker}(\mu_{\Sigma})\oplus\operatorname{Vect}(\mathbf{1})$.

Now, we need to understand the action of $P_{\Sigma}$ on
$\operatorname{Ker}(\mu_{\Sigma})=\mathbb{C}_{0}^{\Sigma}$. It is enough to
understand its action on a basis, say, $e_{1}:=\mathbf{1}_{A}-\mathbf{1}_{B}$
and $e_{2}:=\mathbf{1}_{A}-\mathbf{1}_{C}$. Note however that Equation (1.2)
only let us compute $(\operatorname{Id}-P_{\Sigma})^{-1}e_{j}$ up to an
additive constant. To sidestep this issue, we compute the quantities

$\langle
e_{i},(\operatorname{Id}-P_{\Sigma})^{-1}e_{j}\rangle_{\ell^{2}(\Sigma)}:=\sum_{p\in\Sigma}e_{i}(p)\cdot\left[(\operatorname{Id}-P_{\Sigma})^{-1}e_{j}\right](p)$

for $i$, $j\in\\{1,2\\}$, which is enough to reconstruct the action of
$(\operatorname{Id}-P_{\Sigma})^{-1}$ on $\mathbb{C}_{0}^{\Sigma}$. Since all
the $e_{i}$ are in $\mathbb{C}_{0}^{\Sigma}$, the indetermination of
$(\operatorname{Id}-P_{\Sigma})^{-1}$ up to constants does not matters. We
illustrate this computation with $i=j=1$. By Theorem 1.1,

$\displaystyle\langle
e_{1},(\operatorname{Id}-P_{\Sigma})^{-1}e_{1}\rangle_{\ell^{2}(\Sigma)}$
$\displaystyle=\sum_{p\in\Sigma}e_{1}(p)\cdot\left[(\operatorname{Id}-P_{\Sigma})^{-1}e_{1}\right](p)$
$\displaystyle=\sum_{p\in\mathbb{Z}^{2}}(e_{1}\mathbf{1}_{\Sigma})(p)\cdot\left[(\operatorname{Id}-P)^{-1}(e_{1}\mathbf{1}_{\Sigma})\right](p)$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}\widehat{(\mathbf{1}_{A}-\mathbf{1}_{B})}(-\xi)\cdot\frac{\widehat{(\mathbf{1}_{A}-\mathbf{1}_{B})}(\xi)}{1-\Psi(\xi)}\;{\rm
d}\xi$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}(1-e^{-i\xi_{1}})\cdot\frac{(1-e^{i\xi_{1}})}{1-\frac{e^{i\xi_{1}}+e^{-i\xi_{1}}+e^{i\xi_{2}}+e^{-i\xi_{2}}}{4}}\;{\rm
d}\xi$
$\displaystyle=\frac{1}{\pi^{2}}\int_{\mathbb{T}^{2}}\frac{1-\cos(\xi_{1})}{2-\cos(\xi_{1})-\cos(\xi_{2})}\;{\rm
d}\xi$ $\displaystyle=2.$

The other three computations are similar. We recover from there the action of
$(\operatorname{Id}-P_{\Sigma})^{-1}$ on $\mathbb{C}_{0}^{\Sigma}$, then the
action on $P_{\Sigma}$ on $\mathbb{C}_{0}^{\Sigma}$ after inversion. Finally,
adding the rank one map $\mathbf{1}\otimes\mu_{\Sigma}$ yields the matrix of
Equation (0.1).

###  1.2 Dynamical systems: extension and induction

####  1.2.1 $\mathbb{Z}^{d}$-extensions of dynamical systems

A common generalization of random walks on $\mathbb{Z}^{d}$, in the context of
ergodic theory, is that of _$\mathbb{Z}^{d}$ -extension_. Let $(A,\mu,T)$ be a
probability-preserving dynamical system, and $F:A\to\mathbb{Z}^{d}$ a
measurable function (which we call the jump function). Let
$([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be the corresponding
$\mathbb{Z}^{d}$-extension as defined at the beginning of this text.

Let us write $S_{n}F(x):=\sum_{k=0}^{n-1}F(T^{k}(x))$ the Birkhoff sums of
$F$. Then

$\widetilde{T}^{n}(x,p)=(T^{n}(x),p+S_{n}F(x)).$

This setting includes random walks: take $A=(\mathbb{Z}^{d})^{\mathbb{N}}$,
then $T((x_{n})_{n\geq 0})=(x_{n+1})_{n\geq 0}$ and
$\mu=\mu_{0}^{\otimes\mathbb{N}}$ a product measure on $A$. Choose
$F((x_{n})_{n\geq 0})=x_{0}$; then, under
$\mu\otimes\nu\in\mathcal{P}([\mathbb{Z}^{d}])$, the second coordinate is a
random walk whose starting point has distribution $\nu$ and whose jumps have
distribution $\mu_{0}$. If one replaces the full shift by a subshift, or take
a Markov measure instead of the product measure, we get a random walk with
internal states, which is a special case of hidden Markov models [31, 32, 33].
If $(A,\mu,T)$ is the collision map of a Sinai billiard, then
$([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ is the collision map of a
Lorentz gas, a popular diffusion model.

####  1.2.2 Inductions of dynamical systems

Let $(B,\nu,S)$ be a measure-preserving dynamical system. For any measurable
$C\subset B$ with positive measure, we define the first hitting time of $C$
as:

$\varphi_{C}(x):=\inf\\{n\geq 1:\widetilde{T}(x)\in C\ \\}.$

The system $(B,\nu,S)$ is said to be ergodic if it has no non-trivial strictly
invariant subset, and recurrent if $\varphi_{C}(x)<+\infty$ for almost every
$x\in C$, for any measurable $C\subset[\mathbb{Z}^{d}]$.

If $(B,\nu,S)$ is recurrent and $C\subset B$ has positive measure, the induced
map

$T_{C}:\left\\{\begin{array}[]{rcl}C&\to&C\\\
x&\mapsto&\widetilde{T}^{\varphi_{C}(x)}(x)\end{array}\right.$

is well-defined almost everywhere, recurrent, and preserves the restricted
measure $\nu_{C}:=\nu_{|C}$ [41, Corollary 1.1]. If in addition $(B,\nu,S)$ is
ergodic, then so is $(C,\nu_{C},T_{C})$. In what follows, we shall often make
this assumption:

######  Hypothesis 1.2 (Recurrence).

The $\mathbb{Z}^{d}$-extension
$([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ is ergodic and recurrent.

Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be a recurrent
$\mathbb{Z}^{d}$-extension. For any $\Sigma\subset\mathbb{Z}^{d}$, we shall
write $[\Sigma]:=A\times\Sigma\subset[\mathbb{Z}^{d}]$. If
$\Sigma=\\{\sigma\\}$ is a singleton, we shall dispense with the brackets, and
write $[p]=A\times\\{p\\}$. In this article, we will work with three types of
systems:

  * •

the initial probability-preserving dynamical system $(A,\mu,T)$;

  * •

a $\mathbb{Z}^{d}$-extension
$([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ with jump function $F$;

  * •

induced maps $([\Sigma],\mu_{[\Sigma]},T_{[\Sigma]})$ for
$\Sigma\subset\mathbb{Z}^{d}$.

We try as much as possible to keep our notations coherent: objects with a
tilde refer to the $\mathbb{Z}^{d}$-extension, and objects with a set as an
index refer to induced maps.

In this context, we define transition probabilities as follows:

######  Definition 1.3 (Transition probabilities).

Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be a recurrent
$\mathbb{Z}^{d}$-extension of a dynamical system $(A,\mu,T)$. Let
$\Sigma\subset\mathbb{Z}^{d}$ be finite.

The matrix of transition probabilities $P_{\Sigma}$ is defined by:

$P_{\Sigma,pq}:=\mu\left(x\in A:T_{[\Sigma]}(x,p)\in[q]\right).$

As long as the extension is recurrent, the matrix $P_{\Sigma}$ is bi-
stochastic: the sum of each of its rows and of each of its columns is $1$.

######  Lemma 1.4.

Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be a recurrent
$\mathbb{Z}^{d}$-extension of a probability preserving dynamical system. Let
$\Sigma\subset\mathbb{Z}^{d}$ be non-empty. Then $P_{\Sigma}$ is bi-
stochastic.

###### Proof.

The fact that $P_{\Sigma}$ is stochastic is a translation of the fact that
$\widetilde{T}$ is recurrent. For any $p\in\Sigma$,

$\displaystyle\sum_{q\in\Sigma}P_{\Sigma,pq}$
$\displaystyle=\sum_{q\in\Sigma}\mu\left(x\in
A:T_{[\Sigma]}(x,p)\in[q]\right)$ $\displaystyle=\mu\left(x\in
A:T_{[\Sigma]}(x,p)\in[\Sigma]\right)$
$\displaystyle=\widetilde{\mu}\left([p]\cap\\{\varphi_{[\Sigma]}<+\infty\\}\right)$
$\displaystyle=1.$

The bi-stochasticity of $P_{\Sigma}$ comes from the fact that $T_{[\Sigma]}$
preserves $\mu_{[\Sigma]}$. For any $q\in\Sigma$,

$\displaystyle\sum_{p\in\Sigma}P_{\Sigma,pq}$
$\displaystyle=\sum_{p\in\Sigma}\mu\left(x\in
A:T_{[\Sigma]}(x,p)\in[q]\right)$
$\displaystyle=\mu_{[\Sigma]}\left(T_{[\Sigma]}^{-1}([q])\right)$
$\displaystyle=\mu_{[\Sigma]}\left([q]\right)$ $\displaystyle=1.\qed$

###  1.3 Gibbs-Markov maps

If no assumption is made on $(A,\mu,T)$ nor on $F$, the behaviour of the
trajectories of the system $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$
can be extremely diverse: for instance, the rate of diffusion in Ehrenfest’s
wind-tree model depends on the geometry of the scatterers [14, 19]. However,
if $(A,\mu,T)$ is in some sense chaotic and $F$ regular, the trajectories of
the system $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ will be very
similar to random walks on $\mathbb{Z}^{d}$.

In this article, we choose to work with Gibbs-Markov maps. This family of
transformations is Markovian, which brings the benefits of avatars of the
strong Markov property, for instance Corollary A.7. These transformations are
still flexible enough to give rise to a wide variety of examples and, using
Young’s construction [54], they appear in many geometric examples such as the
geodesic flow on negatively curved manifolds, finite horizon Sinai billiards,
and Collet-Eckmann unimodal transformations.

The family of $\mathbb{Z}^{d}$-extensions we get is much more general than
random walks and random walks with internal states as in [32, 33], while still
more restrictive than Sinai billiards or unimodal maps, since we do not work
with Young towers.

####  1.3.1 Definition

In what follows, we shall assume that $(A,\mu,T)$ is a Gibbs-Markov map:

######  Definition 1.5 (Measure-preserving Gibbs-Markov maps).

Let $(A,d^{\prime})$ be a metric bounded Polish space, $\mathcal{A}$ its Borel
$\sigma$-algebra and $\mu$ a probability measure on $(A,\mathcal{A})$. Let
$\alpha$ be a partition of $A$ in subsets of positive measure (up to a null
set). Let $T:\ A\to A$ be a $\mu$-preserving map, Markov with respect to the
partition $\alpha$, and such that $\alpha$ is generating. Such a map is said
to be _Gibbs-Markov_ if it also satisfies:

  * •

Big image property: $\inf_{a\in\alpha}\mu(Ta)>0$;

  * •

Expansivity: there exists $\Lambda>1$ such that $d^{\prime}(Tx,Ty)\geq\Lambda
d(x,y)$ for all $a\in\alpha$ and $(x,y)\in a\times a$;

  * •

Bounded distortion: there exists $C>0$ such that, for all $a\in\alpha$, for
almost every $(x,y)\in a\times a$:

$\left|\frac{\;{\rm d}\mu}{\;{\rm d}\mu\circ T}(x)-\frac{\;{\rm d}\mu}{\;{\rm
d}\mu\circ T}(y)\right|\leq Cd^{\prime}(Tx,Ty)\frac{\;{\rm d}\mu}{\;{\rm
d}\mu\circ T}(x).$ (1.3)

Given a Gibbs-Markov map $(A,\alpha,d^{\prime},\mu,T)$, a function
$F:A\to\mathbb{Z}^{d}$ is said to be _Markov_ if it is constant on elements of
the partition $\alpha$. In this case, we also say that the corresponding
$\mathbb{Z}^{d}$-extension $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$
is Markov.

By encoding, Gibbs-Markov maps can model sequences of independent and
identically distributed $\mathbb{Z}^{d}$-valued random variables, as well as
finite state Markov chains. As such, Markov extensions of Gibbs-Markov maps
include all random walks, with or without internal states, on
$\mathbb{Z}^{d}$. They are however more general, including for instance the
Gauss maps

$T:\left\\{\begin{array}[]{rcl}(0,1]&\to&[0,1]\\\ x&\mapsto&1/x-\lfloor
1/x\rfloor\end{array}\right.,$

and appear in various encoding schemes used to study geodesic flows in
negative sectional curvature as well as Sinai billiards.

A comprehensive presentation of the properties of Gibbs-Markov maps we use in
this article can be found in Appendix A, in particular their properties
related to the extension and induction operations described in Subsection 1.2.

####  1.3.2 Transfer operators

One of our main tools is the transfer operator $\mathcal{L}$, defined as the
dual of the composition (or Koopman) operator $\mathcal{K}:g\mapsto g\circ T$:

$\int_{A}\mathcal{L}(f)\cdot g\;{\rm d}\mu:=\int_{A}f\cdot g\circ T\;{\rm
d}\mu,$ (1.4)

for all $p\in[1,\infty]$, all $f\in\mathbb{L}^{p}(A,\mu;\mathbb{C})$ and all
$g\in\mathbb{L}^{q}(A,\mu;\mathbb{C})$, with $p^{-1}+q^{-1}=1$. Since the
Koopman operator is an isometry on $\mathbb{L}^{q}$, the transfer operator is
a weak contraction on $\mathbb{L}^{p}$.

As specified in Sub-subsection 1.2.2, the operator $\mathcal{L}$ acts on
$\mathbb{L}^{p}(A,\mu;\mathbb{C})$, the operator $\widetilde{\mathcal{L}}$ on
$\mathbb{L}^{p}([\mathbb{Z}^{d}],\widetilde{\mu};\mathbb{C})$ and the operator
$\mathcal{L}_{[\Sigma]}$ on
$\mathbb{L}^{p}([\Sigma],\mu_{[\Sigma]};\mathbb{C})$.

One of the main feature of Gibbs-Markov maps is that the transfer operator
acts quasi-compactly on a Banach space of Lipschitz functions. We present here
the most important properties of the action; again, details and references can
be found in Appendix A.

######  Definition 1.6 (Lipschitz functions).

Let $\alpha^{*}:=\bigvee\\{T(a):\ a\in\alpha\\}$ be the image partition222In
many applications, $\alpha^{*}$ is trivial – consider for instance the Gauss
map, where any interval $((n+1)^{-1},n^{-1}]$ is sent onto $[0,1)$. of
$\alpha$ by $T$. The Lipschitz seminorm of a function $f:A\to\mathbb{C}$ is

$|f|_{\mathcal{B}}:=\sup_{a\in\alpha^{*}}|f|_{\operatorname{Lip}(a,d^{\prime})},$

where $|f|_{\operatorname{Lip}(a,d^{\prime})}$ is the Lipschitz constant of
$f$ on $a$. The Lipschitz norm is then given by

$\left\|f\right\|_{\mathcal{B}}:=\left\|f\right\|_{\infty}+|f|_{\mathcal{B}},$

and the space of Lipschitz functions $\mathcal{B}$ by $\\{f:A\to\mathbb{C},\
\left\|f\right\|_{\mathcal{B}}<+\infty\\}$.

Finally, we write $\mathcal{B}_{0}$ for the kernel of $\mu$ in $\mathcal{B}$.

A typical result (see [21, Proposition 1.1.17 and Corollary 1.1.21]) is then:

######  Theorem 1.7.

Let $(A,\alpha,d^{\prime},\mu,T)$ be a Gibbs-Markov map with expansivity
constant $\Lambda>1$. The transfer operator $\mathcal{L}$ acts on
$\mathcal{B}$, with essential spectral radius
$\rho_{\operatorname{ess}}(\mathcal{L}\curvearrowright\mathcal{B})\leq\Lambda^{-1}$.

If $(A,\mu,T)$ is also mixing, then the spectral radius of
$\mathcal{L}\curvearrowright\mathcal{B}_{0}$ is strictly less than $1$: there
exist constants $C>0$ and $\rho\in(0,1)$ such that
$\left\|\mathcal{L}^{n}\right\|_{\mathcal{B}_{0}\to\mathcal{B}_{0}}\leq
C\rho^{n}$.

Another important feature of Gibbs-Markov maps is that they behave very well
under induction. Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be a
recurrent and ergodic Markov $\mathbb{Z}^{d}$-extension of a Gibbs-Markov map,
and $\Sigma\subset\mathbb{Z}^{d}$ be finite. Following the strategy of e.g.
[51, Lemma 1.6], one may endow the system
$([\Sigma],\mu_{[\Sigma]},T_{[\Sigma]})$ with a Gibbs-Markov structure, with
expansivity and distortion constants independent from $\Sigma$. From there one
could get a family of Banach spaces depending on $\Sigma$. However, we shall
not use directly this construction, but choose a closely related strategy: we
make all the transfer operators $\mathcal{L}_{[\Sigma]}$ act on the same
Banach space, with relevant bounds (such as the essential spectral radius)
uniform in $\Sigma$. This let us compare more easily all the operators
$\mathcal{L}_{[\Sigma]}$.

We associate to $([\Sigma],\mu_{[\Sigma]},T_{[\Sigma]})$ a Banach space of
Lipschitz functions

$\mathcal{B}_{[\Sigma]}:=\left\\{f:[\Sigma]\to\mathbb{C},\
f(\cdot,i)\in\mathcal{B}\ \forall i\in\Sigma\right\\},$ (1.5)

endowed with the norm $\left\|f\right\|_{\mathcal{B}_{[\Sigma]}}=\max_{i\in
I}\left\|f(\cdot,i)\right\|_{\mathcal{B}}$. We denote by
$\mathcal{B}_{[\Sigma],0}$ the kernel of $\mu_{[\Sigma]}$.

Not that the spaces $\mathcal{B}_{[\Sigma]}$ depend only on $\Sigma$ through
its cardinality: if two subsets $\Sigma_{1}$, $\Sigma_{2}$ have the same
cardinality, then $\mathcal{B}_{[\Sigma_{1}]}$ and
$\mathcal{B}_{[\Sigma_{2}]}$ are naturally isomorphic. By Proposition A.8, the
operator $\mathcal{L}_{[\Sigma]}$ acts quasi-compactly on
$\mathcal{B}_{[\Sigma]}$, with essential spectral radius at most
$\Lambda^{-1}$. In particular, by ergodicity, $1$ is a simple eigenvalue of
$\mathcal{L}_{[\Sigma]}$ corresponding to constant functions, so that
$(\operatorname{Id}-\mathcal{L}_{[\Sigma]})\curvearrowright\mathcal{B}_{[\Sigma],0}$
is invertible.

###  1.4 Shapes and matrices

####  1.4.1 Limit shapes

As we argued in the introduction, it seems extremely hard to state anything
non-trivial about the matrix of transition probabilities $P_{[\Sigma]}$ for
any fixed $\Sigma$. Instead, we shall be interested in subsets $\Sigma$ whose
sites are very far apart. We formalize this by fixing a finite set $I$,
considering a family of injections
$\sigma_{\varepsilon}:I\hookrightarrow\mathbb{Z}^{d}$ whose values get apart
when $\varepsilon$ vanishes, and looking at the matrix of transition
probabilities of $[\Sigma_{\varepsilon}]:=[\sigma_{\varepsilon}(I)]$ .

A simple way of generating such functions $\sigma_{\varepsilon}$ is to fix
some injection $\sigma:I\hookrightarrow\mathbb{R}^{d}$, and to define
$\sigma_{\varepsilon}(i)$ as one of the points of $\mathbb{Z}^{d}$ closest to
$\varepsilon^{-1}\sigma(i)$ – or, more generally, any point at distance
$o(\varepsilon^{-1})$ of $\varepsilon^{-1}\sigma(i)$. As $\varepsilon$
vanishes, the subsets $\Sigma_{\varepsilon}=\sigma_{\varepsilon}(I)$ become
growing realizations of a fixed shape (see Figure 2).

For any $\varepsilon>0$, there is a canonical bijection
$\sigma_{\varepsilon*}:(x,i)\mapsto(x,\sigma_{\varepsilon}(i))$ between
$[I]:=A\times I$ and $[\Sigma_{\varepsilon}]=A\times\Sigma_{\varepsilon}$. We
use this bijection to transport objects from $[\Sigma_{\varepsilon}]$ to
$[I]$:

  * •

$\mu_{I}:=|I|^{-1}\sum_{i\in I}\mu\otimes\delta_{i}$ is conjugated with
$\mu_{[\Sigma_{\varepsilon}]}$ and does not depend on $\varepsilon$;

  * •

$T_{\varepsilon}:[I]\to[I]$ is conjugated with $T_{[\Sigma_{\varepsilon}]}$;

  * •

$\mathcal{L}_{\varepsilon}\curvearrowright\mathbb{L}^{p}(|I],\mu_{I})$ is
conjugated with
$\mathcal{L}_{[\Sigma_{\varepsilon}]}\curvearrowright\mathbb{L}^{p}(|\Sigma],\mu_{[\Sigma]})$;

  * •

$\mathcal{B}_{I}$ is conjugated with $\mathcal{B}_{[\Sigma_{\varepsilon}]}$,
and by definition does not depend on $\varepsilon$;

  * •

$P_{\varepsilon}$ is conjugated with $P_{[\Sigma_{\varepsilon}]}$, etc.

The general rule is that objects defined on $[I]$ have $I$ as a subscript if
they do not depend on $\varepsilon$, and $\varepsilon$ as a subscript if they
do. This point of view allows us to keep some objects fixed (the space, the
measure, the Banach space $\mathcal{B}_{I}$, and later on functions) while
$\varepsilon$, and thus $\Sigma_{\varepsilon}$, changes.

####  1.4.2 Transfer operator and transition probabilities

The matrices $(P_{\varepsilon})_{\varepsilon>0}$ can be recovered from the
Koopman operators $(\mathcal{K}_{\varepsilon})_{\varepsilon>0}$. Given
$\varepsilon>0$ and $i$, $j\in I$,

$\displaystyle P_{\varepsilon,ij}$ $\displaystyle=\mu\left([i]\cap
T_{\varepsilon}^{-1}([j])\right)$
$\displaystyle=|I|\int_{[I]}\mathbf{1}_{[i]}\cdot\mathcal{K}_{\varepsilon}(\mathbf{1}_{[j]})\;{\rm
d}\mu_{I}.$ (1.6)

Using transfer operators, this relationship can be rewritten

$P_{\varepsilon,ij}=|I|\int_{[I]}\mathcal{L}_{\varepsilon}(\mathbf{1}_{[i]})\cdot\mathbf{1}_{[j]}\;{\rm
d}\mu_{I},$ (1.7)

and made more systematic by introducing an averaging operator. Let:

$\Pi_{*}:\left\\{\begin{array}[]{rcl}\mathbb{L}^{1}([I],\mu_{I})&\to&\mathbb{C}^{I}\\\
f&\mapsto&\left(\int_{A}f(x,i)\;{\rm d}\mu(x)\right)_{i\in
I}\end{array}\right.,$

which to a function $f$ associate its averages over each $[i]$, and define a
right inverse $\Pi^{*}$ of $\Pi_{*}$:

$\Pi^{*}:\left\\{\begin{array}[]{rcl}\mathbb{C}^{I}&\to&\mathbb{L}^{1}([I],\mu_{I})\\\
f&\mapsto&\sum_{i\in I}f_{i}\mathbf{1}_{[i]}\end{array}\right..$

The operators $\Pi_{*}$ and $\Pi^{*}$ satisfy the relation
$\Pi_{*}\Pi^{*}=\operatorname{Id}_{\mathbb{C}^{I}}$, while $\Pi^{*}\Pi_{*}$ is
a rank $|I|$ averaging operator on $\mathbb{L}^{1}([I],\mu_{I})$:

$(\Pi^{*}\Pi_{*}f)(x,i)=\int_{A}f(y,i)\;{\rm d}\mu(y).$

Equations (1.6) and (1.7) can then be recast as

$P_{\varepsilon}=\Pi_{*}\mathcal{K}_{\varepsilon}\Pi^{*}\text{ and
}{\vphantom{P}}^{\mathit{t}}{P}_{\varepsilon}=\Pi_{*}\mathcal{L}_{\varepsilon}\Pi^{*}.$
(1.8)

####  1.4.3 Asymptotics of the transition matrices and irreductibility

By [40, Lemma 3.17], if the minimal distance between distinct sites in
$\Sigma_{\varepsilon}$ increases to $+\infty$ when $\varepsilon$ vanishes,
then the transitions from one site to another becomes increasingly unlikely,
so that $\lim_{\varepsilon\to 0}P_{\varepsilon}=\operatorname{Id}$. In this
article, we are interested in the next order asymptotics of the family
$(P_{\varepsilon})_{\varepsilon>0}$ when $\varepsilon$ vanishes. By Lemma 1.4,
the matrices $P_{\varepsilon}$ are bi-stochastic, so that the differences
$(\operatorname{Id}-P_{\varepsilon})$ are what we shall call bi-L-matrices. In
other words, up to a sign, bi-L-matrices make up the tangent cone at
$\operatorname{Id}$ to the space of bi-stochastic matrices.

######  Definition 1.8 (L-matrix).

A _L-matrix_ is a square matrix such that333The usual definition of L-matrix
does not require that the sum of each line be zero.:

  * •

$R_{ii}\geq 0$ for all $i\in I$;

  * •

$R_{ij}\leq 0$ for $i\neq j\in I$;

  * •

$\sum_{j\in I}R_{ij}=0$ for all $i\in I$.

We shall call a L-matrix a _bi-L-matrix_ if, in addition, $\sum_{i\in
I}R_{ij}=0$ for all $j\in I$.

A L-matrix is _irreducible_ if, for all $i$, $j\in I$, there exists a finite
sequence $i=i_{0}$, $i_{1}$, $\ldots$, $i_{n}=j$ such that
$R_{i_{k}i_{k+1}}<0$ for all $0\leq k<n$.

Using Perron-Frobenius theory, eigenvalues of a L-matrix always have non-
negative real part. For irreducible L-matrices, we can be more accurate.

######  Lemma 1.9.

Let $R$ be a $n\times n$ irreducible L-matrix. Then:

  * •

$0$ is a simple eigenvalue of $R$ with eigenfunction $\mathbf{1}$.

  * •

there exists a unique probability vector $\nu$ such that $\nu R=0$.

  * •

the matrix $R$ acts on $\operatorname{Ker}(\nu)$, and
$\operatorname{Sp}(R\curvearrowright\operatorname{Ker}(\nu))\subset\\{z\in\mathbb{C}:\
\Re(z)>0\\}$.

###### Proof.

Let $R$ be an irreducible L-matrix. Set $t:=\max_{i}R_{ii}$. Then
$\operatorname{Id}-t^{-1}R$ is an irreducible nonnegative matrix. The lemma
follows from Perron-Frobenius theory applied to $\operatorname{Id}-t^{-1}R$. ∎

Let us write $R_{0}$ for the restriction of $R$ to $\operatorname{Ker}(\nu)$.
A L-matrix $R$ is never invertible. However, if $R$ is also irreducible, then
$R_{0}$ is invertible. Given an irreducible L-matrix $R$, we shall write

$\rho_{R}:=\min\\{\Re(\lambda):\
\lambda\in\operatorname{Sp}(R\curvearrowright\operatorname{Ker}(\nu))\\}=\min\\{\Re(\lambda):\
\lambda\in\operatorname{Sp}(R_{0})\\}.$

We shall often assume that the family $(P_{\varepsilon})_{\varepsilon>0}$
admits a first order asymptotics at $\varepsilon=0$.

######  Hypothesis 1.10 (Asymptotics of the transition matrices).

There exists an irreducible bi-L-matrix $R$ such that

$P_{\varepsilon}=\operatorname{Id}-\varepsilon R+o(\varepsilon).$

####  1.4.4 Asymptotics of the potential and irreductibility

One issue with Hypothesis 1.10 is that we do not know a priori the matrices
$(P_{\varepsilon})_{\varepsilon>0}$; indeed, our goal is to compute them! As
described in Subsection 1.1, what can be computed are potentials. We shall now
define matrices related to these potential, as well as a related
irreducibility condition.

Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be a Markov
$\mathbb{Z}^{d}$-extension satisfying Hypothesis 1.2. By the discussion at the
very end of Sub-subsection 1.3.2, for any function $f\in\mathcal{B}_{I,0}$ and
any $\varepsilon>0$, there exists a unique function $g\in\mathcal{B}_{I,0}$,
its associated potential, such that
$(\operatorname{Id}-\mathcal{L}_{\varepsilon})g=f$.

Let $\langle\cdot,\cdot\rangle_{\ell^{2}(I)}$ be the canonical scalar – and
not Hermitian – product on $\mathbb{C}^{I}$. In order to get a finite-
dimensional version of $(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}$,
we define an operator acting on $\mathbb{C}^{I}_{0}=\left\\{f:I\to\mathbb{C},\
\sum_{i\in I}f_{i}=0\right\\}=\operatorname{Ker}(\mu_{I})$ by:

$\langle
f,Q_{\varepsilon}(g)\rangle_{\ell^{2}(I)}:=\langle\Pi_{*}(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}\Pi^{*}(f),g\rangle_{\ell^{2}(I)}$

where the inverse $(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}$ is
taken in $\mathcal{B}_{I,0}$. This operator is well-defined since
$\Pi^{*}(\mathbb{C}^{I}_{0})\subset\mathcal{B}_{I,0}$.

The operator $Q_{\varepsilon}$ is characterized by the integrals:

$\langle
f,Q_{\varepsilon}(g)\rangle_{\ell^{2}(I)}=|I|\int_{[I]}(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}\left(\Pi^{*}(f)\right)\cdot\Pi^{*}(g)\;{\rm
d}\mu_{I}=|I|\langle(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}\Pi^{*}(f),\Pi^{*}(g)\rangle_{\mathbb{L}^{2}([I],\mu_{I})}.$
(1.9)

######  Remark 1.11.

Let us justify our choice of definition for $Q_{\varepsilon}$. We want to
adapt the construction we have for Markov maps to dynamical systems, using the
Koopman operator $\mathcal{K}_{\varepsilon}$ instead of the Markov operator.
Since the Koopman operator does not acts well on function spaces, we introduce
its dual, defined by Equation (1.4). The operator
$\Pi_{*}(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}\Pi^{*}$ is thus a
discretization of the potential operator associated with the dual of
$\mathcal{K}_{\varepsilon}$. We make it act of the left to cancel those
dualities, so that the resulting operator $Q_{\varepsilon}$ acts as a
discretization of the potential operator associated with
$\mathcal{K}_{\varepsilon}$.

As we formulated asymptotics and irreducibility conditions for the family
$(P_{\varepsilon})_{\varepsilon>0}$, let us do so for the family
$(Q_{\varepsilon})_{\varepsilon>0}$.

######  Hypothesis 1.12 (Asymptotics of the potential matrices).

There exists a matrix $S$ acting on $\mathbb{C}_{0}^{I}$ such that

$Q_{\varepsilon}=\varepsilon^{-1}S+o(\varepsilon^{-1}).$

######  Definition 1.13 (Irreducible potentials).

A matrix $S$ acting on $\mathbb{C}_{0}^{I}$ is said to be irreducible if, for
all $i\neq j\in I$,

$\langle(\mathbf{1}_{i}-\mathbf{1}_{j}),S(\mathbf{1}_{i}-\mathbf{1}_{j})\rangle_{\ell^{2}(I)}>0.$

The notion of irreducibility of Definition 1.13 is very easy to check in the
computations of Section 6, and quite convenient in the proof of Theorem 0.1.
In the later theorem, we shall conclude that $S=R_{0}^{-1}$ for an irreducible
bi-L-matrix $R$; it is thus expected that this notion of irreducibility is
related to the notion of irreducibility of bi-L-matrices of Sub-subsection
1.4.3.

######  Lemma 1.14.

Let $R$ be an irreducible bi-L-matrix in the sense of Definition 1.8. Then
$R_{0}^{-1}$ is irreducible in the sense of Definition 1.13.

###### Proof.

Our proof is probabilistic in nature, using potential theory for finite state
Markov chains. Let $I$ be the index set of the matrix $R$. Fix $\varepsilon>0$
such that $P:=\operatorname{Id}-\varepsilon R$ is a stochastic matrix. Since
$R$ is irreducible, so is $P$. Since $R$ is a bi-L-matrix, $P$ preserves the
counting measure on $I$.

Let $i\neq j\in I$. Let $P_{\\{i,j\\}}$ be the transition matrix for the
induced Markov chain on $\\{i,j\\}$. Then, by [41, Corollary 1.1], and using
the fact that all solutions of the Poisson equation on $\\{i,j\\}$ differ by a
constant since $P$ (and thus $P_{\\{i,j\\}}$) is irreducible,

$\sum_{k\in\\{i,j\\}}(\mathbf{1}_{i}-\mathbf{1}_{j})(k)\cdot(\operatorname{Id}-P)^{-1}(\mathbf{1}_{i}-\mathbf{1}_{j})(k)=\sum_{k\in\\{i,j\\}}(\mathbf{1}_{i}-\mathbf{1}_{j})(k)\cdot(\operatorname{Id}-P_{\\{i,j\\}})^{-1}(\mathbf{1}_{i}-\mathbf{1}_{j})(k).$
(1.10)

On the one hand, $(\operatorname{Id}-P)^{-1}=\varepsilon^{-1}R_{0}^{-1}$. On
the other hand, since $P_{\\{i,j\\}}$ is stochastic, irreducible and preserves
the counting measure, there exist $t>0$ such that

$P_{\\{i,j\\}}=\left(\begin{array}[]{cc}1-t&t\\\ t&1-t\end{array}\right).$

In particular, $P_{\\{i,j\\}}f=(1-2t)f$ for all
$f\in\mathbb{C}_{0}^{\\{i,j\\}}$. Equation (1.10) becomes:

$\langle(\mathbf{1}_{i}-\mathbf{1}_{j}),R_{0}^{-1}(\mathbf{1}_{i}-\mathbf{1}_{j})\rangle_{\ell^{2}(I)}=\varepsilon\langle(\mathbf{1}_{i}-\mathbf{1}_{j}),(2t)^{-1}(\mathbf{1}_{i}-\mathbf{1}_{j})\rangle_{\ell^{2}(I)}=\frac{\varepsilon}{t}>0.\qed$

###  1.5 Strategy and main results

Now that all the necessary objects have been introduced, let us present the
strategy we follow and the main results that come along. We recall that, as
sketched at the end of Sub-subsection 1.1.1, the strategy to compute the
Markov kernel $P_{\Sigma}$ on $[\Sigma]$ from a Markov kernel $\widetilde{P}$
on $[\mathbb{Z}^{d}]$ has three main steps:

  * •

the balayage identity, which relates $(\operatorname{Id}-\widetilde{P})^{-1}$
and $(\operatorname{Id}-P_{\Sigma})^{-1}$;

  * •

the ability to solve the Poisson equation on $[\mathbb{Z}^{d}]$, using the
Fourier transform;

  * •

the ability to invert $(\operatorname{Id}-P_{\Sigma})^{-1}$, or at least
extract some information from it.

####  1.5.1 Koopman operator and Poisson equation

First, one must choose the right Markov kernel. A first choice is to use

$\widetilde{P}_{x}:=\delta_{\widetilde{T}(x)},$

whose action on functions is the Koopman operator $\widetilde{\mathcal{K}}$.

However, we then want to apply Theorem 1.1, which requires us to find
solutions $g\in\mathbb{L}^{\infty}([\mathbb{Z}^{d}],\widetilde{\mu})$ to the
equation

$f\mathbf{1}_{[\Sigma]}=(\operatorname{Id}-\widetilde{P})(g)=(\operatorname{Id}-\widetilde{\mathcal{K}})(g)=g-g\circ\widetilde{T}\quad\widetilde{\mu}-\text{almost
everywhere},$ (1.11)

where $f\mathbf{1}_{[\Sigma]}$ is supported on $[\Sigma]$. The existence of
solutions to this equation means that $f\mathbf{1}_{[\Sigma]}$ is a coboundary
for $\widetilde{T}$. Let us assume for now that the system $(A,\mu,T)$ is one
of the simplest examples of Gibbs-Markov map: a subshift of finite type
endowed with a Gibbs measure. Then the regularity of $g$ can be improved and
the coboundary equation holds everywhere using a classical argument [21,
Chapter 1.2].

On the other hand, we want to take $f\in\mathcal{B}_{[\Sigma],0}$, hence
Lipschitz. By Theorem 1.1, we then have $\mu_{[\Sigma]}$-almost everywhere

$f=g_{|[\Sigma]}-g_{|[\Sigma]}\circ T_{[\Sigma]},$

so that

$\mathcal{L}_{[\Sigma]}(f)=(\mathcal{L}_{[\Sigma]}-\operatorname{Id})(g_{|[\Sigma]}).$

Then, since $(\operatorname{Id}-\mathcal{L}_{[\Sigma]})$ in invertible on
$\mathcal{B}_{[\Sigma],0}$, the later equation admits a solution

$g_{0}:=-(\operatorname{Id}-\mathcal{L}_{[\Sigma]})^{-1}\mathcal{L}_{[\Sigma]}(f)\in\mathcal{B}_{[\Sigma]}.$

As the kernel of the action of $(\operatorname{Id}-\mathcal{L}_{[\Sigma]})$ on
$\mathbb{L}^{\infty}([\Sigma],\mu_{[\Sigma]})$ is the space of constant
functions by ergodicity, $g_{|[\Sigma]}-g_{0}$ is constant almost everywhere,
so that $g_{|[\Sigma]}$ admits a Lipschitz version.

The same reasoning goes through with any finite $\Sigma\subset\Sigma^{\prime}$
and yields that $g_{|[\Sigma^{\prime}]}$ admits a Lipschitz version on
$[\Sigma^{\prime}]$, whence $g$ admits a locally Lipschitz version on
$[\mathbb{Z}^{d}]$. In particular, taking for $g$ this continuous version, the
equality

$f\mathbf{1}_{[\Sigma]}=g-g\circ\widetilde{T}$

holds everywhere, and not only $\widetilde{\mu}$-almost everywhere. As a
consequence, for any point $(x,p)\in[\mathbb{Z}^{d}]$ of period $n\geq 1$,

$\sum_{k=0}^{n-1}(f\mathbf{1}_{[\Sigma]})\circ\widetilde{T}^{k}(x,p)=0.$
(1.12)

This gives countably many obstructions to the existence of solutions to
Equation (1.11). In addition, if the sites of $\Sigma$ are far apart and
$p\in\Sigma$, then there exist periodic orbits which intersect $[\Sigma]$ only
in $[p]$. Then, if $f$ is constant on $[p]$, the sum of Equation (1.12) is
zero if and only if $f(\cdot,p)\equiv 0$. As this holds for all $p\in\Sigma$,
we get $f\equiv 0$: the only function $f$ which is constant on all sites and
satisfy the coboundary equation is the null function.

A solution of this quandary is to use not the Koopman operator
$\widetilde{\mathcal{K}}$, but its dual $\widetilde{\mathcal{L}}$.

####  1.5.2 Transfer operator and Poisson equation

The transformation $\widetilde{T}$ is assumed to be ergodic and recurrent,
which implies that the Markov kernel $\widetilde{\mathcal{K}}$ is also ergodic
and recurrent. The operator $\widetilde{\mathcal{L}}$ is dual to
$\widetilde{\mathcal{K}}$ with respect to $\widetilde{\mu}$, and thus is also
ergodic and recurrent by [41]. The relevant version of the balayage identity
is [41, Proposition 0.1]:

######  Proposition 1.15 (Balayage identity for the transfer operator).

Let $(\widetilde{A},\widetilde{\mu},\widetilde{T})$ be measure-preserving,
with $\widetilde{\mu}$ a recurrent $\sigma$-finite measure. Let
$B\subset\widetilde{A}$ be such that $0<\widetilde{\mu}(B)\leq+\infty$. Let
$f$, $g\in\mathbb{L}^{\infty}(\widetilde{A},\widetilde{\mu})$ be such that
$f\equiv 0$ on $B^{c}$ and:

$(\operatorname{Id}-\widetilde{\mathcal{L}})(g)=f.$

Then:

$(\operatorname{Id}-\mathcal{L}_{B})(g_{|B})=f_{|B}.$

Note that, in Section 5, we shall actually use the more general [41, Lemma
1.7], which applies to equations such as
$(\operatorname{Id}-z\widetilde{\mathcal{L}})(g)=f$ with $|z|<1$. This
generalization is only used to deal with convergence issues.

By the discussion at the end of Sub-subsection 1.3.2, for any
$f\in\mathcal{B}_{[\Sigma],0}$, one can find a locally Lipschitz solution $g$
to the equation $(\operatorname{Id}-\widetilde{\mathcal{L}})(g)=f$, which
makes the balayage identity potentially useful. We still have two hurdles to
cross:

  * •

If we can construct (approximations of) solutions to the equation
$(\operatorname{Id}-\mathcal{L}_{\Sigma})(g)=f_{|[\Sigma]}$, how do we use
them to extract data on $\mathcal{L}_{\Sigma}$, and in particular an
approximation of the transition matrix $P_{\Sigma}$?

  * •

How do we construct (approximations of) solutions to the equation
$(\operatorname{Id}-\widetilde{\mathcal{L}})(g\mathbf{1}_{[\Sigma]})=f$?

Both of these problems will be addressed in an asymptotic regime, when the
sites of $\Sigma$ are far away one from the others. We delve deeper into these
questions in the next two sub-subsections.

####  1.5.3 Towards the main theorem

Our solution to the first of these two problem is Theorem 0.1. Let us recall
that it states that, for an ergodic and recurrent Markov
$\mathbb{Z}^{d}$-extension of a Gibbs-Markov map and a family
$\sigma_{\varepsilon}:I\hookrightarrow\mathbb{Z}^{d}$, there is equivalence
between:

  * •

there exists an irreducible (in the sense of Definition 1.8) bi-L-matrix $R$
such that $P_{\varepsilon}=\operatorname{Id}-\varepsilon R+o(\varepsilon)$;

  * •

there exists an irreducible (in the sense of Definition 1.13) matrix $S$ such
that $Q_{\varepsilon}=\varepsilon^{-1}S+o(\varepsilon^{-1})$.

If in addition any of these properties holds, then $S=R_{0}^{-1}$.

######  Remark 1.16 (Theorem 0.1 and averaging operator).

Using the operators $\Pi_{*}$ and $\Pi^{*}$ as well as transfer operators,
Theorem 0.1 states that, under an irreducibility condition, the finite-
dimensional operators
$(\operatorname{Id}-\Pi_{*}\mathcal{L}_{\varepsilon}\Pi^{*})^{-1}$ and
$\Pi_{*}(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}\Pi^{*}$ are
equivalent.

######  Remark 1.17 (Intrinsic version of Theorem 0.1).

The statement of Theorem 0.1 depends both on the parametrization of the
subsets $\Sigma_{\varepsilon}$ and the precise speed of convergence of
$P_{\varepsilon}$ to $\operatorname{Id}$. While easy to state and convenient
for computations, a more intrisic version can be crafted. First, notice that
the irreductibility conditions 1.10 and 1.12 are satisfied on open cones in
the relevant subspaces of $M_{n}(\mathbb{R})$, or in other words, on
projectively open sets. We can then restate Theorem 0.1 as follows.

Consider an ergodic and recurrent Markov $\mathbb{Z}^{d}$-extension of a
Gibbs-Markov map and a family of finite subsets
$(\Sigma_{\varepsilon})_{\varepsilon>0}$ all of the same size and such that

$\lim_{\varepsilon\to 0}\min_{p\neq q\in\Sigma_{\varepsilon}}|p-q|=+\infty.$

Then there is equivalence between:

  * •

$(\operatorname{Id}-P_{\varepsilon})_{\varepsilon>0}$ is projectively
relatively compact for Hypothesis 1.10.

  * •

$(Q_{\varepsilon})_{\varepsilon>0}$ is projectively relatively compact for
Hypothesis 1.12.

If either condition is satisfied, then
$(I-P_{\varepsilon})^{-1}_{|\mathbb{C}_{0}^{I}}\sim_{\varepsilon\to
0}Q_{\varepsilon}$.

The matrix $Q_{\varepsilon}$ can be computed when one evaluates the quantities

$\langle\Pi_{*}(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}\Pi^{*}(f),g\rangle_{\ell^{2}(I)},$

for $f$, $g\in\mathbb{C}_{0}^{I}$. Theorem 0.1 gives the desired asymptotics
for $R$. The initial infinite-dimensional problem of inverting
$(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}$ is reduced to the finite-
dimensional problem of inverting $Q_{\varepsilon}$.

The proof of Theorem 0.1 spans three sections. First, it will be very
convenient to assume that the induced system $(A,\mu,T_{[0]})$ is mixing. This
shall be denoted as a separate hypothesis:

######  Hypothesis 1.18 (Mixing).

The system $(A,\mu,T_{[0]})$ is mixing.

Section 2 is devoted to the proof of the direct implication in Theorem 0.1
under Hypothesis 1.18:

######  Lemma 1.19.

Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be an ergodic and
recurrent Markov $\mathbb{Z}^{d}$-extension of a Gibbs-Markov map (Hypothesis
1.2). Let $I$ be finite and, and let
$\sigma_{\varepsilon}:I\hookrightarrow\mathbb{Z}^{d}$ for all $\varepsilon>0$.

Assume furthermore that $([0],\mu_{[0]},T_{[0]})$ is mixing (Hypothesis 1.18).

If there exists an irreducible bi-L-matrix $R$ such that
$P_{\varepsilon}=\operatorname{Id}-\varepsilon R+o(\varepsilon)$ (Hypothesis
1.10), then $Q_{\varepsilon}=\varepsilon^{-1}R_{0}^{-1}+o(\varepsilon^{-1})$.
In addition, $R_{0}^{-1}$ is irreducible in the sense of Definition 1.13.

Lemma 1.19 is the heart of the proof of Theorem 0.1. Its proof is inspired by
works on the hitting time of small target for hyperbolic systems [42, 26] and
invariant cones for fast-slow systems [12]; we refer the reader to the
introduction of Section 2 for more details.

Once this is done, in Section 3 we shall remove this mixing assumption. If
$([0],\mu_{[0]},T_{[0]})$ is not mixing, then it has a decomposition into
periodic components. This step requires us to understand what effects the
existence of such periodic components have on the whole extension
$([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$; to our knowledge, these
structural results (in particular Propositions 3.2 and 3.3) are new and of
independent interest.

This analysis is independent from Section 2. That said, we think it is clearer
to give the proof of Lemma 1.19 with the additional mixing assumption, and
then point out the differences non-mixing implies, than to write directly the
proof of Lemma 1.19 in a non-mixing context. At the end of Section 3, we shall
have proved:

######  Lemma 1.20.

Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be an ergodic and
recurrent Markov $\mathbb{Z}^{d}$-extension of a Gibbs-Markov map (Hypothesis
1.2). Let $I$ be finite and, and choose
$\sigma_{\varepsilon}:I\hookrightarrow\mathbb{Z}^{d}$ for all $\varepsilon>0$.

If there exists an irreducible bi-L-matrix $R$ such that
$P_{\varepsilon}=\operatorname{Id}-\varepsilon R+o(\varepsilon)$ (Hypothesis
1.10), then $Q_{\varepsilon}=\varepsilon^{-1}R_{0}^{-1}+o(\varepsilon^{-1})$.
In addition, $R_{0}^{-1}$ is irreducible in the sense of Definition 1.13.

The final step is to reverse the implication, so as to get Theorem 0.1. This
is the goal of Section 4. This uses the $|I|=2$ case of the theorem (which is
deduced directly from Lemma 1.20) and a compactness argument.

####  1.5.4 Applications

We are given an ergodic an recurrent Markov $\mathbb{Z}^{d}$-extension of a
Gibbs-Markov map and a family $(\sigma_{\varepsilon})_{\varepsilon>0}$ of
injection from $I$ into $\mathbb{Z}^{d}$. All is left is, given $f$,
$g\in\mathbb{C}_{0}^{\Sigma}$, to estimate the integrals

$\langle
f,Q_{\varepsilon}(g)\rangle_{\ell^{2}(I)}=|I|\langle(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}\Pi^{*}(f),\Pi^{*}(g)\rangle_{\mathbb{L}^{2}([I],\mu_{I})}$

when $\varepsilon$ vanishes. Thanks to the balayage identity, this reduces
formally to the computation of the quantities (identifying $I$ with
$\Sigma_{\varepsilon}$)

$\langle\Pi_{*}(\operatorname{Id}-\widetilde{\mathcal{L}})^{-1}\left(\Pi^{*}(f)\mathbf{1}_{[\Sigma_{\varepsilon}]}\right),\Pi^{*}(g)\mathbf{1}_{[\Sigma_{\varepsilon}]}\rangle_{\mathbb{L}^{2}([\mathbb{Z}^{d}],\widetilde{\mu})}.$

As with random walks, this can be done using the Fourier transform. The main
difference is that this operation make so-called twisted transfer operators
appear.

######  Definition 1.21 (Twisted transfer operators).

For $w\in\mathbb{T}^{d}:=\mathbb{R}^{d}/2\pi\mathbb{Z}^{d}$, define the
twisted transfer operator:

$\mathcal{L}_{w}(\cdot):=\mathcal{L}(e^{i\langle w,F\rangle}\cdot),$ (1.13)

which acts on $\mathbb{L}^{p}(A,\mu)$ for all $p\in[1,\infty]$, as well as on
$\mathcal{B}$.

Our applications are done in two steps. In Section 5, we shall use these
twisted transfer operators to give a general expression for the quantities
$\langle f,Q_{\varepsilon}(g)\rangle_{\ell^{2}(I)}$. Proposition 5.3 states
that, in our setting, for all $\varepsilon>0$,

$\langle f,Q_{\varepsilon}(g)\rangle_{\ell^{2}(I)}=\lim_{\rho\to
1^{-}}\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}\widecheck{\mathcal{F}}_{\sigma_{\varepsilon}}(f)\mathcal{F}_{\sigma_{\varepsilon}}(g)\left(\int_{A}(\operatorname{Id}-\rho\mathcal{L}_{w})^{-1}(\mathbf{1})\;{\rm
d}\mu\right)\;{\rm d}w,$

where $\mathcal{F}_{\sigma_{\varepsilon}}(g)$ is the Fourier transform of
$(g\circ\sigma_{\varepsilon}^{-1})\mathbf{1}_{\Sigma_{\varepsilon}}$.

In the same spirit as the Nagaev-Guivarc’h proof of the central limit theorem
for dynamical systems [36, 37, 22], when the sites are far apart, the main
contribution to these integrals will be given by the main eigenvalue
$\lambda_{w}$ of $\mathcal{L}_{w}$, for $w$ close to $0$. We shall prove that,
for any small enough neighborhood $U$ of $0$,

$\langle f,Q_{\varepsilon}(g)\rangle_{\ell^{2}(I)}=\lim_{\rho\to
1^{-}}\frac{1}{(2\pi)^{d}}
\int_{U}\widecheck{\mathcal{F}}_{\sigma_{\varepsilon}}(f)\mathcal{F}_{\sigma_{\varepsilon}}(g)\frac{1+\delta_{w}}{1-\rho\lambda_{w}}\;{\rm
d}w+O_{U}(1)\left\|f\right\|\cdot\left\|g\right\|,$

with $\delta_{w}$ a small error term.

In Section 6, we apply the later equation to different classes of jump
function $F$ when $\sigma_{t}(i)=t\sigma(i)+o(1)$ for all $i\in I$. For $d=2$
and $\left\|F\right\|\in\mathbb{L}^{2}(A,\mu)$, this yields Proposition 0.2.
For $d=1$ and $F$ in the basin of attraction of a Cauchy random variable444In
addition to some technical constraints, we get a similar asymptotics, stated
in Proposition 6.8. We are also able to give asymptotics for $P_{\varepsilon}$
when $d=1$ and $F$ is square-integrable, in Proposition 6.1. Finally, we give
some partial results when $d=1$ and $F$ is in the basin of attraction of a
Lévy random variable (i.e. when $F$ has regularly varying tails of index in
$(0,1)$). We are able to give an explicit expression for $S$, stated in
Proposition 6.4, but we could not find an explicit expression for its inverse.
We are still able to compute the asymptotics for a fixed extension and fixed
$(\sigma_{t})_{t>0}$, as done in Examples 6.5 and 6.6.

##  2 From transition probabilities to potential

The goal of this section is to prove Lemma 1.19: if $([0],\mu_{[0]},T_{[0]})$
is mixing and $P_{\varepsilon}=\operatorname{Id}-\varepsilon R+o(\varepsilon)$
for some irreducible bi-L-matrix $R$, then
$Q_{\varepsilon}=\varepsilon^{-1}R_{0}^{-1}+o(\varepsilon^{-1})$.

The transition matrix $P_{\varepsilon}$ can be computed from the transfer
operator $\mathcal{L}_{\varepsilon}$ using the relation
${\vphantom{P}}^{\mathit{t}}{P}_{\varepsilon}=\Pi_{*}\mathcal{L}_{\varepsilon}\Pi^{*}$
(Equation (1.8)). From another point of view,
$\Pi^{*}{\vphantom{P}}^{\mathit{t}}{P}_{\varepsilon}\Pi_{*}=(\Pi^{*}\Pi_{*})\mathcal{L}_{\varepsilon}$
could be seen as a finite-rank approximation of $\mathcal{L}_{\varepsilon}$.
Such an approximation is reminiscent of works on metastable states by Keller
and Liverani [29] (for two communicating systems), as well as by Dolgopyat and
Wright [17] (for interval maps).

At first sight, this approximation is not very good: in general,
$\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})\mathcal{L}_{\varepsilon}\right\|_{\mathcal{B}\to\mathcal{B}}$
is of order $1$, because of the non-trivial behaviour of
$\mathcal{L}_{\varepsilon}$ on each site $[i]$. A solution would be to focus
on functions which are constant on each site; however, this subspace is not
stable under $\mathcal{L}_{\varepsilon}$.

On each site, the effect of $\mathcal{L}_{\varepsilon}$ is roughly that of
$\mathcal{L}_{[0]}$, which mixes exponentially quickly. The transitions
between sites introduce an error of order $\varepsilon$. The interaction of
these two dynamical effects imply the existence of cones of functions
$C_{K}(\varepsilon)$ (functions which are $\varepsilon$-close to being
constant on each site) which are invariant under $\mathcal{L}_{\varepsilon}$.
By restricting ourselves to such cones, we shall be able improve the bounds on
$(\operatorname{Id}-\Pi^{*}\Pi_{*})\mathcal{L}_{\varepsilon}$. The goal of
Subsection 2.1 is thus to prove that such cones are stable under the action of
$\mathcal{L}_{\varepsilon}$, and moreover to prove a cone contraction property
which shall be instrumental in the next steps.

######  Remark 2.1 (Fast-slow systems).

Such families of cones of functions have some precedent in ergodic theory. One
feature of the sequence of operators $\mathcal{L}_{\varepsilon}$ is that the
evolution of $(\mathcal{L}_{\varepsilon})_{n\geq 0}$ has two characteristic
time scales. At a time scale of $\Theta(1)$, the mixing property of
$([0],\mu_{[0]},T_{[0]})$ homogeneizes each site $[i]$, but there is little
communication between distinct sites. The transitions between sites occur at a
time scale of $\Theta(\varepsilon^{-1})$. When $\varepsilon$ vanishes, these
two time scales become decoupled, so that the transitions become independent.

The existence of two distinct time scales is also a feature of fast-slow
systems. For instance, consider a family of maps $F_{\varepsilon}$ acting on
$\Omega\times\mathbb{S}_{1}$ such that

$F_{\varepsilon}(x,\theta)=(f_{\varepsilon}(x,\theta),\theta+\varepsilon
g(x,\theta))$

and $x\mapsto f_{\varepsilon}(x,\theta)$ is a family of uniformly hyperbolic
transformations on $\Omega$. Then the first variable, $x$, evolves on a time
scale of $\Theta(1)$, and the second variable, $\theta$, on a time scale of
$\Theta(\varepsilon^{-1})$. When $\varepsilon$ vanishes, these two time scales
become decoupled, so that the evolution of $\theta$ is averaged over the
values of $x$.

In the context of fast-slow systems, such parametric families of cones – or,
similarly, standard families – have been introduced. We refer the interested
reader to [12] for an exposition of these techniques, and [15, 9, 13] for
applications to partially hyperbolic systems and Sinai billiard maps. In
particular, we stress the similarity between [12, Proposition 3.1] and
Corollary 2.6.

Once we have proved the stability of these cones, the error term between
$\mathcal{L}_{\varepsilon}$ and $(\Pi^{*}\Pi_{*})\mathcal{L}_{\varepsilon}$
will be improved. In Subsection 2.2, together with a version of the Baker-
Campbell-Hausdorff formula, this shall let us control the error between their
products on a time scale of $\Theta(\varepsilon^{-1})$. In particular, if $f$
is constant on each site, then, for all positive $t$, the operator
$\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}(f)$ converges to
$\Pi^{*}e^{-t{\vphantom{R}}^{\mathit{t}}{R}}\Pi_{*}f$. This exponential decay
is typical of previous work on the hitting time of small targets [42, 26]. Let
us also point out [30, 6] for settings where rare events correspond to strong
changes in the system, and where the behaviour of the system after those
changes has to be controlled.

Finally, in Subsection 2.3, the exponential contraction at time scale
$\Theta(1)$ together with a coupling argument gets us exponential bounds on
the decay of correlations which are uniform in $\varepsilon^{-1}t$. Note that
this coupling argument is classical for Markov chains [35, Chapter 5] and has
a long tradition in ergodic theory (present for instance in most applications
of standard pairs; see e.g. [10] for applications to Sinai billiards maps).
Once this is done, we are able to relate
$(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}$ with the inverse of
$R^{-1}$ through a Laplace transform, finishing the proof of Lemma 1.19.

###  2.1 Cone contraction at time scale $\Theta(1)$

Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be an ergodic and
recurrent Markov $\mathbb{Z}^{d}$-extension of a Gibbs-Markov map (Hypothesis
1.2). Let $I$ be finite and, and choose
$\sigma_{\varepsilon}:I\hookrightarrow\mathbb{Z}^{d}$ for all $\varepsilon>0$.

As announced earlier, we introduce cones of functions which are
$\varepsilon$-close to being constant on each site, and prove that they are
stable under some iterate of $\mathcal{L}_{\varepsilon}$, as well as a cone
contraction property.

######  Definition 2.2 (Cones of functions).

For all $K\geq 0$ and $\varepsilon>0$, let:

$C_{K}(\varepsilon):=\left\\{f:\ [I]\to\mathbb{R}:\ f\geq 0\text{ and
}\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})f\right\|_{\mathcal{B}_{I}}\leq
K\varepsilon\left\|f\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\right\\}.$

These subsets of $\mathcal{B}_{I}$ are indeed convex cones, as we can quickly
check:

######  Lemma 2.3.

For all $K$, $\varepsilon\geq 0$, the set $C_{K}(\varepsilon)$ is a convex
cone.

###### Proof.

Let $K$, $\varepsilon\geq 0$. Given $f\in C_{K}(\varepsilon)$ and $\lambda\geq
0$,

$\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})(\lambda
f)\right\|_{\mathcal{B}_{I}}=\lambda\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})f\right\|_{\mathcal{B}_{I}}\leq\lambda
K\varepsilon\left\|f\right\|_{\mathbb{L}^{1}([I],\mu_{I})}=K\varepsilon\left\|\lambda
f\right\|_{\mathbb{L}^{1}([I],\mu_{I})},$

so that $\lambda f\in\mathcal{B}_{I}$. In addition, for all $f$, $g\in
C_{K}(\varepsilon)$ and $t\in[0,1]$:

$\displaystyle\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})((1-t)f+tg)\right\|_{\mathcal{B}_{I}}$
$\displaystyle\leq(1-t)\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})f\right\|_{\mathcal{B}_{I}}+t\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})g\right\|_{\mathcal{B}_{I}}$
$\displaystyle\leq
K\varepsilon\left((1-t)\left\|f\right\|_{\mathbb{L}^{1}([I],\mu_{I})}+t\left\|g\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\right)$
$\displaystyle=K\varepsilon\left\|(1-t)f+tg\right\|_{\mathbb{L}^{1}([I],\mu_{I})},$

where we used the fact that
$\left\|\cdot\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$ is linear on nonnegative
functions. Hence $(1-t)f+tg$ also belongs to $C_{K}(\varepsilon)$. ∎

Another fundamental property of these cones is that, since any function in
$C_{K}(\varepsilon)$ is close for the $\mathcal{B}_{I}$ norm to the finite
dimensional space of functions constant on each $[i]$, many norms are
equivalent on $C_{K}(\varepsilon)$. In particular, we can control the
$\mathcal{B}_{I}$ norm (the strongest norm we shall use) with the
$\mathbb{L}^{1}([I],\mu_{I})$ norm (the weakest norm we shall use).

######  Lemma 2.4.

Let $K$, $\varepsilon\geq 0$. For all $f\in C_{K}(\varepsilon)$,

$\left\|f\right\|_{\mathcal{B}_{I}}\leq(|I|+K\varepsilon)\left\|f\right\|_{\mathbb{L}^{1}([I],\mu_{I})}.$
(2.1)

###### Proof.

Let $K$, $\varepsilon\geq 0$. For all $f\in C_{K}(\varepsilon)$,

$\displaystyle\left\|f\right\|_{\mathcal{B}_{I}}$
$\displaystyle\leq\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})f\right\|_{\mathcal{B}_{I}}+\left\|\Pi^{*}\Pi_{*}f\right\|_{\mathcal{B}_{I}}$
$\displaystyle\leq
K\varepsilon\left\|f\right\|_{\mathbb{L}^{1}([I],\mu_{I})}+|I|\left\|\Pi^{*}\Pi_{*}f\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$
$\displaystyle=(|I|+K\varepsilon)\left\|f\right\|_{\mathbb{L}^{1}([I],\mu_{I})},$

where we used the fact that
$\left\|\cdot\right\|_{\mathcal{B}_{I}}=\left\|\cdot\right\|_{\infty}\leq|I|\left\|\cdot\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$
on functions which are constant on each $[i]$. ∎

Given these preliminary results, we shall now prove that, under the
assumptions of Lemma 1.19, the cones $C_{K}(\varepsilon)$ are contracted under
some iterate of $\mathcal{L}_{\varepsilon}$ whenever $K$ is large enough.
Taking $\sigma=1$ in the following proposition also yield their stability.

######  Proposition 2.5.

Assume Hypotheses 1.2, 1.18 and 1.10.

For all $\sigma>0$, there exist $K_{\sigma}$, $n_{\sigma}\geq 0$ such that,
for all $K\geq K_{\sigma}$, for all $\varepsilon>0$ and $n\geq n_{\sigma}$,

$\mathcal{L}_{\varepsilon}^{n}(C_{K}(\varepsilon))\subset C_{\sigma
K}(\varepsilon).$

###### Proof.

Let $K\geq 0$ and $\varepsilon$, $n$ be positive. Define:

$B_{n}:=\\{(x,i)\in[I]:\ T_{\varepsilon}^{k}(x,i)\in[i]\ \forall\ 0\leq k\leq
n\\}=\bigsqcup_{i\in I}\bigcap_{k=0}^{n}T_{\varepsilon}^{-k}([i]),$

the set of points which stay in the same site until time $n$. By Hypothesis
1.10, there exists a constant $C$ such that, for all $i\in I$ and all
$\varepsilon>0$,

$\mu(x\in A:\ T_{\varepsilon}(x,i)\notin[i])=\sum_{j\neq
i}(\delta_{ij}-\varepsilon R_{ij}+o(\varepsilon))=-\varepsilon\sum_{j\neq
i}R_{ij}+o(\varepsilon)\leq C\varepsilon.$

Hence,

$\displaystyle\mu_{I}\left(\left(\bigcap_{k=0}^{n}T_{\varepsilon}^{-k}([i])\right)^{c}\right)$
$\displaystyle=\sum_{k=1}^{n}\mu_{I}\left(x:\
T_{\varepsilon}^{0}(x,i),\ldots,T_{\varepsilon}^{k-1}(x,i)\in[i],\
T_{\varepsilon}^{k}(x,i)\notin[i]\right)$
$\displaystyle\leq\sum_{k=1}^{n}\mu_{I}\left(x:T_{\varepsilon}^{k-1}(x,i)\in[i],\
T_{\varepsilon}^{k}(x,i)\notin[i]\right)$
$\displaystyle=n\mu_{I}\left(x:T_{\varepsilon}(x,i)\notin[i]\right)$
$\displaystyle\leq C|I|^{-1}n\varepsilon.$

Summing over $i\in I$ yields $\mu_{I}(B_{n}^{c})\leq Cn\varepsilon$.

Let $i\in I$ and $f\in C_{K}(\varepsilon)$. As the transfer operator preserves
nonnegative functions, $\mathcal{L}_{\varepsilon}^{n}(f)\geq 0$. Let us focus
on the regularity of $\mathcal{L}_{\varepsilon}^{n}(f)$.

First, by definition of $B_{n}$, given any point $(x,i)\in B_{n}$, the
trajectories of the points $(x,\sigma_{\varepsilon}(i))$,
$T_{[\Sigma_{\varepsilon}]}(x,\sigma_{\varepsilon}(i))$, $\ldots$,
$T_{[\Sigma_{\varepsilon}]}^{n-1}(x,\sigma_{\varepsilon}(i))$ under
$\widetilde{T}$ all return to $[\sigma_{\varepsilon}(i)]$ before hitting any
$[\sigma_{\varepsilon}(j)]$ with $j\neq i$. Hence, by recursion, for all
$0\leq k\leq n$ and all $x\in B_{n}$,

$T_{[\Sigma_{\varepsilon}]}^{k}(x,\sigma_{\varepsilon}(i))=(T_{[0]}^{k}(x),\sigma_{\varepsilon}(i)),$

so that:

$\mathcal{L}_{\varepsilon}^{n}(f\mathbf{1}_{B_{n}})(\cdot,i)=\mathcal{L}_{[0]}^{n}((f\mathbf{1}_{B_{n}})(\cdot,i)).$
(2.2)

From Equation (2.2) we get:

$\displaystyle\mathcal{L}_{\varepsilon}^{n}(f)(\cdot,i)$
$\displaystyle=\mathcal{L}_{\varepsilon}^{n}(f\mathbf{1}_{B_{n}})(\cdot,i)+\mathcal{L}_{\varepsilon}^{n}(f\mathbf{1}_{B_{n}^{c}})(\cdot,i)$
$\displaystyle=\mathcal{L}_{[0]}^{n}((f\mathbf{1}_{B_{n}})(\cdot,i))+\mathcal{L}_{\varepsilon}^{n}(f\mathbf{1}_{B_{n}^{c}})(\cdot,i)$
$\displaystyle=\mathcal{L}_{[0]}^{n}(f(\cdot,i))-\mathcal{L}_{[0]}^{n}((f\mathbf{1}_{B_{n}^{c}})(\cdot,i))+\mathcal{L}_{\varepsilon}^{n}(f\mathbf{1}_{B_{n}^{c}})(\cdot,i).$

We shall now use Corollaries A.7 and A.9 to control each term. Stopping times
are defined in terms of the canonical filtration on $(A,\mu,T)$, while
$\mathcal{L}_{[0]}$ and $\mathcal{L}_{\varepsilon}$ are defined via the
extension $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$. We thus need to
express some characteristics of the trajectories
$(T_{\varepsilon}^{n}(x,i))_{n\geq 0}$ in terms of cylinders for $(A,\mu,T)$,
as is done in the proof of Proposition A.8.

Let $\varphi_{[0]}(x):=\inf\\{n\geq 1:\ \sum_{k=0}^{n-1}S_{k}F(x)=0\\}$. Then
$\varphi_{[0]}$ is a stopping time in the sense of Definition A.2. Since
$([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ is measure-preserving and
recurrent, the map $T_{\varphi_{[0]}}=T_{[0]}$ preserves $\mu$ and is ergodic.
Finally, by Hypothesis 1.18, the map $T_{[0]}$ is also mixing for $\mu$. Hence
we can apply Corollary A.9, which asserts that there exist constants $C>0$ and
$\rho\in(0,1)$ such that, for all $i\in I$,

$\left\|\mathcal{L}_{[0]}^{n}(f(\cdot,i))-\int_{A}f(\cdot,i)\;{\rm
d}\mu\right\|_{\mathcal{B}}\leq
C\rho^{n}\left\|f(\cdot,i)-\int_{A}f(\cdot,i)\;{\rm
d}\mu\right\|_{\mathcal{B}}\leq
C\rho^{n}\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})f\right\|_{\mathcal{B}_{I}}.$
(2.3)

Let $i\in I$. The set $B_{n}^{c}\cap[I]$ is measurable with respect to the
$\sigma$-algebra $\alpha(\varphi_{[0]}^{(n)})$, since, using the
identification $[i]\simeq A$,

$B_{n}\cap[I]=\bigcup_{k=0}^{n-1}T_{[0]}^{-k}\left(\left\\{\exists
0\leq\ell<\varphi_{[0]}(x),\ \exists j\in I\setminus\\{i\\},\
S_{\ell}F(x)=\sigma_{\varepsilon}(j)-\sigma_{\varepsilon}(i)\right\\}\right).$

As $\varphi_{[0]}^{(n)}$ is a stopping time by Lemma A.3, we can apply
Corollary A.7 to get, for some constant $C$ and all $i\in I$,

$\left\|\mathcal{L}_{[0]}^{n}((f\mathbf{1}_{B_{n}^{c}})(\cdot,i))\right\|_{\mathcal{B}}\leq
C\mu(B_{n}^{c}\cap[i])\left\|f(\cdot,i)\right\|_{\mathcal{B}}.$ (2.4)

In the same way, fix $i$, $j\in I$ and let
$\varphi_{j,n,\varepsilon}(x):=\inf\\{\ell\geq 1:\
\operatorname{Card}\\{S_{\ell}F(x)\in\\{j^{\prime}-j:\ j^{\prime}\in
I\\}=n\\}$. The set $B_{n}^{c}\cap[j]$ is measurable with respect to the
$\sigma$-algebra $\alpha(\varphi_{j,n,\varepsilon})$. By Corollary A.7, for
some constant $C$,

$\displaystyle\left\|\mathcal{L}_{\varepsilon}^{n}(f\mathbf{1}_{B_{n}^{c}\cap[j]})(\cdot,i)\right\|_{\mathcal{B}}$
$\displaystyle=\left\|\mathcal{L}_{\varphi_{j,n,\varepsilon}}(f\mathbf{1}_{B_{n}^{c}\cap[j]\cap\\{S_{\varphi_{j,n,\varepsilon}}F=i-j\\}})\right\|_{\mathcal{B}}$
$\displaystyle\leq
C\mu(B_{n}^{c}\cap[j])\left\|f(\cdot,j)\right\|_{\mathcal{B}}$
$\displaystyle=C|I|\mu_{I}(B_{n}^{c}\cap[j])\left\|f(\cdot,j)\right\|_{\mathcal{B}}.$

Summing over $j\in I$ finally yields

$\left\|\mathcal{L}_{\varepsilon}^{n}(f\mathbf{1}_{B_{n}^{c}})(\cdot,i)\right\|_{\mathcal{B}}\leq
C|I|\mu_{I}(B_{n}^{c})\left\|f\right\|_{\mathcal{B}_{I}}.$ (2.5)

Equations (2.1), (2.3), (2.4) and (2.5) together yield, for some positive
constant $C$ and all $n\geq 1$:

$\displaystyle\left\|\mathcal{L}_{\varepsilon}^{n}(f)(\cdot,i)-\int_{A}f(\cdot,i)\;{\rm
d}\mu\right\|_{\mathcal{B}}$ $\displaystyle\leq
C\rho^{n}\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})f\right\|_{\mathcal{B}_{I}}+C\mu_{I}(B_{n}^{c})\left\|f\right\|_{\mathcal{B}_{I}}$
$\displaystyle\leq
C(K\rho^{n}+n)\varepsilon\left\|f\right\|_{\mathbb{L}^{1}([I],\mu_{I})}.$
(2.6)

As $T_{\varepsilon}$ preserves $\mu_{I}$ and $f$ is nonnegative,
$\left\|\mathcal{L}_{\varepsilon}^{n}(f)\right\|_{\mathbb{L}^{1}([I],\mu_{I})}=\left\|f\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$.
By Equation (2.6),

$\mathcal{L}_{\varepsilon}^{n}(C_{K}(\varepsilon))\subset
C_{C(K\rho^{n}+n)}(\varepsilon).$ (2.7)

Now, we finish the proof of Proposition 2.5. Let $\sigma>0$. Without loss of
generality, we assume that $\sigma\leq 1$. Let $n_{\sigma}\geq 1$ be such that
$C\rho^{n_{\sigma}}\leq\sigma/2$. Let $K_{\sigma}:=4C\sigma^{-1}n_{\sigma}$.
Then, for all $K\geq K_{\sigma}$ and $\varepsilon>0$, for all $n_{\sigma}\leq
n\leq 2n_{\sigma}$, by Equation (2.7),

$C(K\rho^{n}+n)\leq C(K\rho^{n_{\sigma}}+2n_{\sigma})\leq\frac{\sigma
K}{2}+\frac{\sigma K_{\sigma}}{2}\leq\sigma K,$

so that $\mathcal{L}_{\varepsilon}^{n}(C_{K}(\varepsilon))\subset C_{\sigma
K}(\varepsilon)$.

By recursion, let us prove that
$\mathcal{L}_{\varepsilon}^{mn_{\sigma}}(C_{K}(\varepsilon))\subset
C_{K}(\varepsilon)$ for all $m\geq 0$. This is true for $m=0$, and if it holds
for some $m\geq 0$, then

$\mathcal{L}_{\varepsilon}^{(m+1)n_{\sigma}}(C_{K}(\varepsilon))=\mathcal{L}_{\varepsilon}^{n_{\sigma}}\mathcal{L}_{\varepsilon}^{mn_{\sigma}}(C_{K}(\varepsilon))\subset\mathcal{L}_{\varepsilon}^{n_{\sigma}}(C_{K}(\varepsilon))\subset
C_{\sigma K}(\varepsilon)\subset C_{K}(\varepsilon),$

which finishes this recursion.

Finally, for all $n\geq n_{\sigma}$, since $n_{\sigma}\leq n-n_{\sigma}\lfloor
n/n_{\sigma}\rfloor+n_{\sigma}\leq 2n_{\sigma}$,

$\mathcal{L}_{\varepsilon}^{n}(C_{K}(\varepsilon))=\mathcal{L}_{\varepsilon}^{n-n_{\sigma}\lfloor
n/n_{\sigma}\rfloor+n_{\sigma}}\mathcal{L}_{\varepsilon}^{n_{\sigma}(\lfloor
n/n_{\sigma}\rfloor-1)}(C_{K}(\varepsilon))\subset\mathcal{L}_{\varepsilon}^{n-n_{\sigma}\lfloor
n/n_{\sigma}\rfloor+n_{\sigma}}(C_{K}(\varepsilon))\subset C_{\sigma
K}(\varepsilon).\qed$

As a consequence, if $K$ is large enough and up to the loss of a constant
factor, $C_{K}(\varepsilon)$ is stable under $\mathcal{L}_{\varepsilon}^{n}$
for all $n\geq 0$. This corollary shall give us a good control on
$\left\|\mathcal{L}_{\varepsilon}^{n}f\right\|_{\mathcal{B}_{I}}$ provided we
know that $f\in C_{K}(\varepsilon)$.

######  Corollary 2.6.

Assume Hypotheses 1.2, 1.18 and 1.10.

There exists $K_{1}$ and $\lambda_{1}\geq 1$ such that, for all $K\geq K_{1}$,
for all $n\geq 0$ and $\varepsilon>0$,

$\mathcal{L}_{\varepsilon}^{n}(C_{K}(\varepsilon))\subset
C_{\lambda_{1}K}(\varepsilon).$

###### Proof.

Let $n_{1}$, $K_{1}$ be given by Proposition 2.5. By Proposition 2.5, for all
$K\geq K_{1}$, for all $n\geq n_{1}$ and all $\varepsilon>0$,

$\mathcal{L}_{\varepsilon}^{n}(C_{K}(\varepsilon))\subset C_{K}(\varepsilon).$

In addition, by Equation (2.7), for all $n\leq n_{1}$ and $\varepsilon>0$,

$\mathcal{L}_{\varepsilon}^{n}(C_{K}(\varepsilon))\subset
C_{C(K+n_{1})}(\varepsilon).$

We choose $\lambda_{1}:=C(1+n_{1}/K_{1})$. ∎

###  2.2 Product of matrices

Given $f\in C_{K}(\varepsilon)$, Corollary 2.6 asserts that
$\mathcal{L}_{\varepsilon}^{n}f$ is close to being constant on each site for
all $n\geq 0$; the operator $\mathcal{L}_{\varepsilon}$ is then well
approximated by the finite rank operator
$\Pi^{*}{\vphantom{P}}^{\mathit{t}}{P}_{\varepsilon}\Pi_{*}$. However, to
understand what happens at time scale $\varepsilon^{-1}$, we need to control a
composition of a large number (of order $\varepsilon^{-1}$) of such operators.
To keep reasonably sharp error bounds, we use a version of the Baker-Campbell-
Hausdorff formula.

######  Lemma 2.7.

Let $N\geq 1$. There exists a neighbourhood $U$ of $0$ in $M_{N}(\mathbb{C})$
and a constant $C>0$ such that, for all matrices $A$ and $B$ in $U$,

$\left\|\ln(e^{A}e^{B})-A-B\right\|\leq C\left\|[A,B]\right\|.$ (2.8)

The ability to bound $\ln(e^{A}e^{B})-A-B$ using only commutators is natural
if one thinks about the Baker-Campbell-Hausdorff formula:

$e^{A}e^{B}=e^{A+B+\frac{1}{2}[A,B]+\frac{1}{12}[A,[A,B]]+\frac{1}{12}[B,[B,A]]+\ldots}.$

We prove it using an integral version of this formula.

###### Proof.

Whenever $A$ and $B$ are small enough, $\ln(e^{A}e^{B})$ can be written with
an integral Baker-Campbell-Hausdorff formula [23, Theorem 3.3]:

$\ln(e^{A}e^{B})=A+\int_{0}^{1}u\left(e^{\operatorname{ad}_{A}}e^{t\operatorname{ad}_{B}}\right)\;{\rm
d}t\ (B),$

where $u(z)=\frac{z\ln(z)}{z-1}$. Set:

$v(z):=\frac{z\ln(z)-z+1}{(z-1)^{2}}=\frac{u(z)-1}{z-1},\quad
w(z):=\frac{e^{z}-1}{z}.$

The functions $v$ and $w$ are analytic on a neighbourhood of the identity, and
for all small enough $A$ and $B$:

$\displaystyle\ln(e^{A}e^{B})-A-B$
$\displaystyle=\int_{0}^{1}u\left(e^{\operatorname{ad}_{A}}e^{t\operatorname{ad}_{B}}\right)\;{\rm
d}t\ (B)-B$
$\displaystyle=\int_{0}^{1}(u-1)\left(e^{\operatorname{ad}_{A}}e^{t\operatorname{ad}_{B}}\right)\;{\rm
d}t\ (B)$
$\displaystyle=\int_{0}^{1}v\left(e^{\operatorname{ad}_{A}}e^{t\operatorname{ad}_{B}}\right)\
(e^{\operatorname{ad}_{A}}e^{t\operatorname{ad}_{B}}-1)\ (B)\;{\rm d}t$
$\displaystyle=\int_{0}^{1}v\left(e^{\operatorname{ad}_{A}}e^{t\operatorname{ad}_{B}}\right)\
(e^{\operatorname{ad}_{A}}-1)\ (B)\;{\rm d}t$
$\displaystyle=\int_{0}^{1}v\left(e^{\operatorname{ad}_{A}}e^{t\operatorname{ad}_{B}}\right)\;{\rm
d}t\ w(\operatorname{ad}_{A})\ ([A,B]).$

Since both $v\left(e^{\operatorname{ad}_{A}}e^{t\operatorname{ad}_{B}}\right)$
and $w(\operatorname{ad}_{A})$ are uniformly bounded for $A$ and $B$ on a
neighbourhood of the identity, we get the claim. ∎

Using Lemma 2.7 recursively, we can control the product of a large number (of
order $\Theta(\varepsilon^{-1})$) of matrices which are $o(\varepsilon)$-close
from a reference matrix $R$.

######  Lemma 2.8.

Let $N\geq 1$ and $R\in M_{N}(\mathbb{C})$. Let $\omega$ be a nonnegative
function such that $\omega(\varepsilon)=_{\varepsilon\to 0}o(\varepsilon)$.
Let $(R_{\varepsilon,n})_{\varepsilon>0,n\geq 0}$ be a family of matrices in
$M_{N}(\mathbb{C})$ such that

$\sup_{n\geq 0}\left\|R_{\varepsilon,n}\right\|\leq\omega(\varepsilon).$

For all $\varepsilon>0$ and $t\geq 0$, let:

$M_{\varepsilon}(t):=\prod_{n=0}^{\lfloor\varepsilon^{-1}t\rfloor-1}(\operatorname{Id}-\varepsilon
R+R_{\varepsilon,n}),$

where the matrices are multiplied in any order. Then the family of matrix-
valued functions $(M_{\varepsilon})_{\varepsilon>0}$ converges to $t\mapsto
e^{-tR}$ when $\varepsilon$ vanishes, uniformly on all compacts of
$\mathbb{R}_{+}$, and for fixed $\omega$ uniformly in
$(R_{\varepsilon,n})_{\varepsilon>0,n\geq 0}$.

###### Proof.

Let $R$ and $(R_{\varepsilon,n})_{\varepsilon>0,n\geq 0}$ be as in the
hypotheses of the lemma. Define:

$R^{\prime}_{\varepsilon,n}:=\ln(\operatorname{Id}-\varepsilon
R+R_{\varepsilon,n})+\varepsilon R.$

Note that $\omega^{\prime}(\varepsilon):=\sup_{n\geq
0}\left\|R^{\prime}_{\varepsilon,n}\right\|=O(\omega(\varepsilon))$.

Let $U$ be a neighbourhood of $0$ in $M_{N}(\mathbb{C})$ and $C>0$ as given by
Lemma 2.7. We put $K:=2C\max\\{\left\|R\right\|,1\\}$. Let $t_{0}$ be small
enough that $U$ contains $\\{-tR:\ 0\leq t\leq t_{0}\\}$.

We claim that, for all small enough $\varepsilon$, for all
$n\leq\lfloor\varepsilon^{-1}t_{0}\rfloor$, the matrix
$\prod_{k=0}^{n-1}(\operatorname{Id}-\varepsilon R+R_{\varepsilon,k})$ belongs
to $U$, and:

$\left\|\ln\left(\prod_{k=0}^{n-1}(\operatorname{Id}-\varepsilon
R+R_{\varepsilon,k})\right)+n\varepsilon
R\right\|\leq(1+Kn\varepsilon)\omega^{\prime}(\varepsilon)\frac{(1+2K\varepsilon)^{n}-1}{2K\varepsilon}.$
(2.9)

The claim is true for $n=0$, as the left-hand side vanishes. Assume that the
claim holds for some $n\leq\lfloor\varepsilon^{-1}t_{0}\rfloor-1$, and put:

$S_{\varepsilon,n}:=\ln\left(\prod_{k=0}^{n-1}(\operatorname{Id}-\varepsilon
R+R_{\varepsilon,k})\right)+n\varepsilon R.$

Define in the same way $S_{\varepsilon,n+1}$. We assume without loss of
generality that $\varepsilon$ is small enough that
$\omega^{\prime}(\varepsilon)\leq\varepsilon$. By Lemma 2.7,

$\displaystyle\left\|S_{\varepsilon,n+1}\right\|$
$\displaystyle=\left\|\ln\left(\prod_{k=0}^{n}(\operatorname{Id}-\varepsilon
R+R_{\varepsilon,k})\right)+(n+1)\varepsilon R\right\|$
$\displaystyle=\left\|\ln\left(e^{-\varepsilon
R+R^{\prime}_{\varepsilon,n}}e^{-n\varepsilon
R+S_{\varepsilon,n}}\right)+(n+1)\varepsilon R\right\|$
$\displaystyle\leq\left\|-\varepsilon
R+R^{\prime}_{\varepsilon,n}-n\varepsilon R+S_{\varepsilon,n}+(n+1)\varepsilon
R\right\|+C\left\|[-\varepsilon R+R^{\prime}_{\varepsilon,n},-n\varepsilon
R+S_{\varepsilon,n}]\right\|$
$\displaystyle\leq\left\|R^{\prime}_{\varepsilon,n}\right\|+\left\|S_{\varepsilon,n}\right\|+2C\left(\varepsilon\left\|R\right\|\left\|S_{\varepsilon,n}\right\|+\left\|R^{\prime}_{\varepsilon,n}\right\|\left\|S_{\varepsilon,n}\right\|+n\varepsilon\left\|R^{\prime}_{\varepsilon,n}\right\|\left\|R\right\|\right)$
$\displaystyle\leq\omega^{\prime}(\varepsilon)+\left\|S_{\varepsilon,n}\right\|+K\varepsilon\left\|S_{\varepsilon,n}\right\|+Kn\varepsilon\omega^{\prime}(\varepsilon)+K\omega^{\prime}(\varepsilon)\left\|S_{\varepsilon,n}\right\|$
$\displaystyle\leq(1+2K\varepsilon)\left\|S_{\varepsilon,n}\right\|+(1+Kn\varepsilon)\omega^{\prime}(\varepsilon).$

Using the recursion hypothesis, we get:

$\displaystyle\left\|S_{\varepsilon,n+1}\right\|$
$\displaystyle\leq(1+2K\varepsilon)(1+Kn\varepsilon)\omega^{\prime}(\varepsilon)\frac{(1+2K\varepsilon)^{n}-1}{2K\varepsilon}+(1+Kn\varepsilon)\omega^{\prime}(\varepsilon)$
$\displaystyle\leq(1+K(n+1)\varepsilon)\omega^{\prime}(\varepsilon)\frac{(1+2K\varepsilon)^{n+1}-1}{2K\varepsilon}.$

Hence, we get the bound we want on $S_{\varepsilon,n+1}$. In addition,
$n+1\leq\lfloor\varepsilon^{-1}t_{0}\rfloor$, so that:

$\left\|S_{\varepsilon,n+1}\right\|\leq(1+Kt_{0})e^{2Kt_{0}}\frac{\omega^{\prime}(\varepsilon)}{2K\varepsilon},$

which converges to $0$ uniformly for $1\leq
n+1\leq\lfloor\varepsilon^{-1}t_{0}\rfloor$. In particular, if $\varepsilon$
is small enough (depending on $t_{0}$ but not on $n$), $S_{\varepsilon,n+1}$
belongs to $U$. This finishes the proof of our intermediate claim.

As a consequence, for all small enough $\varepsilon$ and all $t\in[0,t_{0}]$,
the matrix $M_{\varepsilon}(t)$ belongs to $U$ and satisfies the inequality:

$\left\|M_{\varepsilon}(t)-e^{-tR}\right\|\leq(1+Kt_{0})e^{2Kt_{0}}\frac{\omega^{\prime}(\varepsilon)}{2K\varepsilon}+\left\|e^{-tR}\right\|\max_{s\in[0,\varepsilon]}\left\|e^{sR}-\operatorname{Id}\right\|,$

where the right-hand side converges to $0$ when $\varepsilon$ vanishes. The
family of functions $(F_{\varepsilon})_{\varepsilon>0}$ thus converges to
$t\mapsto e^{-tR}$ uniformly on $[0,t_{0}]$. In addition, since the error term
only depends on $(R_{\varepsilon,n})_{\varepsilon>0,n\geq 0}$ through
$\omega$, so the convergence is uniform in
$(R_{\varepsilon,n})_{\varepsilon>0,n\geq 0}$ at fixed $\omega$.

Finally, let $T>0$. Divide the interval $[0,T]$ into a finite number of
subintervals of length less that $t_{0}$, taking an increasing sequence
$0=s_{0}<s_{1}<\ldots<s_{\ell}=T$ with $s_{i+1}-s_{i}<t_{0}$. On each such
subinterval, the intermediate claim yields:

$\lim_{\varepsilon\to
0}\sup_{s\in[s_{i},s_{i+1}]}\left\|M_{\varepsilon}(s)M_{\varepsilon}(s_{i})^{-1}-e^{-(s-s_{i})R}\right\|=0.$

The concatenation of these bounds for $0\leq i\leq\ell$ yields the uniform
convergence of $M_{\varepsilon}$ on $[0,T]$, and ends the proof of Lemma 2.8.
∎

######  Remark 2.9.

If the family $(R_{\varepsilon,n})_{\varepsilon>0,n\geq 0}$ does not depend on
$n$, which in our future application is the case for random walks, then the
conclusion of Lemma 2.8 follows directly from the identity

$M_{\varepsilon}(t)=e^{\lfloor\varepsilon^{-1}t\rfloor\ln\left(\operatorname{Id}-\varepsilon
R+R_{\varepsilon,0}\right)},$

and the fact that $\ln\left(\operatorname{Id}-\varepsilon
R+R_{\varepsilon,0}\right)=\operatorname{Id}-\varepsilon R+o(\varepsilon)$.

###  2.3 Cone contraction at time scale $\Theta(\varepsilon^{-1})$

In this subsection we prove Lemma 1.19. Let us recall briefly the strategy:
thanks to the stability estimates of Subsection 2.1, in particular Corollary
2.6, and to Lemma 2.8, we shall prove that, for all piecewise constant $f$,

$\lim_{\varepsilon\to
0}\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f=e^{-t({\vphantom{R}}^{\mathit{t}}{R})}f.$

Then, a coupling argument referenced at the start of the section (for similar
arguments, see e.g. [35, Chapter 5] in the context of Markov chains, and [10]
in the context of hyperbolic dynamics and standard pairs) yields tension: for
some constants $C$, $c>0$, if in addition $\int_{[I]}f\;{\rm d}\mu_{I}=0$,
then

$\left\|\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f\right\|_{\infty}\leq
Ce^{-ct}\left\|f\right\|_{\infty}.$

Integrating over $t\geq 0$ then relates
$(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}$, and thus
$Q_{\varepsilon}$, with $R_{0}^{-1}$.

######  Remark 2.10 (Perturbative argument and the spectrum of
$\mathcal{L}_{\varepsilon}$).

As already mentioned, another direction of attack for this problem is to try
and adapt the perturbative methods of G. Keller and C. Liverani [29], in
particular in the form used by D. Dolgopyat and P. Wright [17]. In this
setting, one views $\mathcal{L}_{\varepsilon}$ as a perturbation of

$\mathcal{L}_{[0]}^{\oplus|I|}:\left\\{\begin{array}[]{rcl}\mathcal{B}_{I}&\to&\mathcal{B}_{I}\\\
(f(\cdot,i))_{i\in I}&\mapsto&(\mathcal{L}_{[0]}(f(\cdot,i)))_{i\in
I}\end{array}\right.,$

whose spectrum
$\operatorname{Sp}(\mathcal{L}_{[0]}^{\oplus|I|}\curvearrowright\mathcal{B}_{I})$
is a $|I|$-fold copy of
$\operatorname{Sp}(\mathcal{L}_{[0]}\curvearrowright\mathcal{B})$.

This approach is likely to give a more accurate description of
$\mathcal{L}_{\varepsilon}$, although with no incidence on Theorem 0.1. For
instance, under Hypothesis 1.18, we can hope to get an eigendecomposition

$\mathcal{L}_{\varepsilon}=\mathcal{M}_{\varepsilon}+\mathcal{N}_{\varepsilon},$

with $\left\|\mathcal{M}_{\varepsilon}^{n}\right\|_{\mathcal{B}_{I}}\leq
Ce^{-cn}$ for some $C$, $c>0$ and all $\varepsilon>0$, and
$\mathcal{N}_{\varepsilon}$ of finite rank, and almost conjugated with
$(\operatorname{Id}-\varepsilon{\vphantom{R}}^{\mathit{t}}{R})$. In
particular, the spectrum of $\mathcal{N}_{\varepsilon}$ should be
$o(\varepsilon)$-close to that of
$(\operatorname{Id}-\varepsilon{\vphantom{R}}^{\mathit{t}}{R})$, with
eigenfunctions close to those of
$\Pi^{*}{\vphantom{R}}^{\mathit{t}}{R}\Pi_{*}$. If this holds, the bound of
Equation 2.14 below may be improved to

$\left\|\mathcal{L}_{\varepsilon}^{n}(f)\right\|_{\mathcal{B}_{I,0}}\leq
Ce^{-\rho\varepsilon n}\left\|f\right\|_{\mathcal{B}_{I,0}},$

with $\rho<\rho_{R}$ and $C$ depends only on $\rho$.

$1$$\Lambda^{-1}$$0$$\operatorname{Sp}(\mathcal{L}_{[0]}\curvearrowright\mathcal{B})$$0$$\rho_{R}$$\operatorname{Sp}(R\curvearrowright\mathbb{C}^{I})$$1$$\Lambda^{-1}$$0$$\operatorname{Sp}(\mathcal{L}_{\varepsilon}\curvearrowright\mathcal{B}_{I})$

Figure 4: Conjectural picture for the relationship between the spectra of
$\mathcal{L}_{[0]}$, $R$ and $\mathcal{L}_{\varepsilon}$ with $|I|=5$. The
spectrum of $\mathcal{L}_{\varepsilon}$ is a perturbation of magnitude
$\Theta(\varepsilon)$ of a five-fold copy of
$\operatorname{Sp}(\mathcal{L}_{[0]}\curvearrowright\mathcal{B})$, and the
spectrum of $R$ can be recovered by zooming in a window of size $\varepsilon$
around
$1\in\operatorname{Sp}(\mathcal{L}_{\varepsilon}\curvearrowright\mathcal{B}_{I})$,
then applying a central symmetry.

Let us finally mention the recent work of H. Tanaka [50], who recently
investigated such multidimensional spectral degenerescences in the context of
equilibrium measures of subshifts of finite type.

Our first step is to show that we can indeed apply Lemma 2.8 in our context.

######  Proposition 2.11.

Assume Hypotheses 1.2, 1.18 and 1.10.

For all $K\geq 0$, there exists a nonnegative function $\omega$ such that
$\omega(\varepsilon)=_{\varepsilon\to 0}o(\varepsilon)$, and with the
following property.

Let $(f_{\varepsilon})_{\varepsilon>0}$ be a family of functions from $[I]$ to
$\mathbb{R}_{+}$ such that $f_{\varepsilon}\in C_{K}(\varepsilon)$ for all
small enough $\varepsilon$. Then there exists a sequence of square matrices
$(R_{\varepsilon,n})_{\varepsilon>0,n\geq 0}$ such that

$\sup_{n\geq 0}\left\|R_{\varepsilon,n}\right\|\leq\omega(\varepsilon)$

and, for all $n\geq 0$,

$\Pi_{*}\mathcal{L}_{\varepsilon}^{n+1}f_{\varepsilon}=(\operatorname{Id}-\varepsilon({\vphantom{R}}^{\mathit{t}}{R})+R_{\varepsilon,n})(\Pi_{*}\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}).$

###### Proof.

Let $K\geq 0$ and $(f_{\varepsilon})_{\varepsilon>0}$ be as in the hypotheses
of Proposition 2.11. Without loss of generality, we assume that $K\geq K_{1}$,
where $K_{1}$ is given by Corollary 2.6. By Corollary 2.6, there exists
$\lambda_{1}\geq 1$ such that $\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}\in
C_{\lambda_{1}K}(\varepsilon)$ for all small enough $\varepsilon$. Since
$\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}\in
C_{\lambda_{1}K}(\varepsilon)$ and
$\left\|\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}=\left\|f_{\varepsilon}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$,

$\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}\right\|_{\infty}\leq\lambda_{1}K\varepsilon\left\|f_{\varepsilon}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}.$
(2.10)

Set
$g_{\varepsilon,n}:=(\operatorname{Id}-\Pi^{*}\Pi_{*})\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}$.
By construction, $\Pi_{*}g_{\varepsilon,n}=0$. For all $i$, $j\in I$,

$\left|\int_{[j]}g_{\varepsilon,n}\cdot\mathbf{1}_{[j]}\circ
T_{\varepsilon}\;{\rm
d}\mu\right|\leq\left\|g_{\varepsilon,n}\right\|_{\infty}\mu(x\in A:\
T_{\varepsilon}(x,i)\in[j])=\left\|g_{\varepsilon,n}\right\|_{\infty}P_{\varepsilon,ij}.$

Since $\mathcal{L}_{\varepsilon}$ preserves $\mu_{I}$, for all $i\in I$,

$\displaystyle\int_{[i]}\mathcal{L}_{\varepsilon}(g_{\varepsilon,n})\;{\rm
d}\mu$
$\displaystyle=|I|\int_{[I]}\mathbf{1}_{[i]}\mathcal{L}_{\varepsilon}(g_{\varepsilon,n})\;{\rm
d}\mu_{I}$
$\displaystyle=|I|\int_{[I]}g_{\varepsilon,n}\cdot\mathbf{1}_{[i]}\circ
T_{\varepsilon}\;{\rm d}\mu_{I}$ $\displaystyle=\sum_{j\in
I}\int_{[j]}g_{\varepsilon,n}\cdot\mathbf{1}_{[i]}\circ T_{\varepsilon}\;{\rm
d}\mu-\sum_{j\in I}\int_{[i]}g_{\varepsilon,n}\cdot\mathbf{1}_{[j]}\circ
T_{\varepsilon}\;{\rm d}\mu$ $\displaystyle=\int_{[i]}g_{\varepsilon,n}\;{\rm
d}\mu+\sum_{j\neq
i}\left(\int_{[j]}g_{\varepsilon,n}\cdot\mathbf{1}_{[i]}\circ
T_{\varepsilon}\;{\rm
d}\mu-\int_{[i]}g_{\varepsilon,n}\cdot\mathbf{1}_{[j]}\circ
T_{\varepsilon}\;{\rm d}\mu\right),$

so that, by Equation (2.10),

$\displaystyle\left|\int_{[i]}\mathcal{L}_{\varepsilon}(g_{\varepsilon,n})\;{\rm
d}\mu-\int_{[i]}g_{\varepsilon,n}\;{\rm d}\mu\right|$
$\displaystyle\leq\left\|g_{\varepsilon,n}\right\|_{\infty}\sum_{j\neq
i}\left(P_{\varepsilon,ij}+P_{\varepsilon,ji}\right)$
$\displaystyle\leq\lambda_{1}K\varepsilon\left\|\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}O(\varepsilon).$

Since $\Pi_{*}g_{\varepsilon,n}=0$, this later inequality can be read as
$\left\|\Pi_{*}\mathcal{L}_{\varepsilon}g_{\varepsilon,n}\right\|_{\infty}=KO(\varepsilon^{2})\left\|\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$.

Recall that
${\vphantom{P}}^{\mathit{t}}{P}_{\varepsilon}=\Pi_{*}\mathcal{L}_{\varepsilon}\Pi^{*}$.
We apply $\Pi_{*}\mathcal{L}_{\varepsilon}$ to $g_{\varepsilon,n}$:

$\Pi_{*}\mathcal{L}_{\varepsilon}^{n+1}f_{\varepsilon}={\vphantom{P}}^{\mathit{t}}{P}_{\varepsilon}\Pi_{*}\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}+\Pi_{*}\mathcal{L}_{\varepsilon}g_{\varepsilon,n}=(\operatorname{Id}-\varepsilon({\vphantom{R}}^{\mathit{t}}{R})+o(\varepsilon))\Pi_{*}\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}+KO(\varepsilon^{2})\left\|\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}.$

As $f_{\varepsilon}$ is nonnegative,
$\left\|\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}=\left\|\Pi_{*}\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}\right\|_{\ell^{1}(I)}$,
so that

$\Pi_{*}\mathcal{L}_{\varepsilon}^{n+1}f_{\varepsilon}=(\operatorname{Id}-\varepsilon({\vphantom{R}}^{\mathit{t}}{R})+o(\varepsilon)+KO(\varepsilon^{2}))\Pi_{*}\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon},$

and the error term $o(\varepsilon)+KO(\varepsilon^{2})$ only depends on $K$. ∎

Our proximate goal is to understand the sequence
$(\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon})_{n\geq 0}$ when
$f_{\varepsilon}$ has average $0$. In order to apply Proposition 2.11, we
shall write a function with average $0$ as the difference of two functions in
$C_{K}(\varepsilon)$. This can be efficiently formalized by introducing an
auxiliary family of norms on $\mathcal{B}_{I}$.

Let $K$, $\varepsilon>0$. Given a real function $f\in\mathcal{B}_{I}$, we
define its $(K,\varepsilon)$-norm as

$\left\|f\right\|_{K,\varepsilon}:=\inf_{\begin{subarray}{c}f_{+},f_{-}\in
C_{K}(\varepsilon)\\\
f=f_{+}-f_{-}\end{subarray}}\max\\{\left\|f_{+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})},\left\|f_{-}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\\}.$

For complex-valued $f\in\mathcal{B}_{I}$, we put
$\left\|f\right\|_{K,\varepsilon}:=\max\\{\left\|\Re
f\right\|_{K,\varepsilon},\left\|\Im f\right\|_{K,\varepsilon}\\}$. For
instance, if $f\in\mathcal{B}_{I,0}$ is constant on each $[i]$, then
$\left\|f\right\|_{K,\varepsilon}=\max\\{\left\|\Re
f\right\|_{\mathbb{L}^{1}([I],\mu_{I})},\left\|\Im
f\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\\}/2$.

######  Lemma 2.12.

For any $K$, $\varepsilon>0$, the $(K,\varepsilon)$-norm is equivalent to the
$\mathcal{B}_{I}$-norm.

###### Proof.

Let $K$, $\varepsilon>0$.

Let $f\in\mathcal{B}_{I}$ with $\left\|f\right\|_{K,\varepsilon}<+\infty$.
Assume that $f$ is real-valued. Let $\delta>0$, and write $f=f_{+}-f_{-}$ with
$\left\|f_{+}\right\|_{\mathbb{L}^{1}}$,
$\left\|f_{-}\right\|_{\mathbb{L}^{1}}\leq\left\|f\right\|_{K,\varepsilon}+\delta$,
and $f_{+}$, $f_{-}\in C_{K}(\varepsilon)$. Then
$\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})f_{+}\right\|_{\mathcal{B}_{I}}\leq
K\varepsilon\left\|f_{+}\right\|_{\mathbb{L}^{1}}$, so that, by Lemma 2.4,

$\left\|f_{+}\right\|_{\mathcal{B}_{I}}\leq(|I|+K\varepsilon)\left\|f_{+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\leq(|I|+K\varepsilon)(\left\|f\right\|_{K,\varepsilon}+\delta)$

and similarly for $f_{-}$. Hence, $\left\|f\right\|_{\mathcal{B}_{I}}\leq
2(|I|+K\varepsilon)(\left\|f\right\|_{K,\varepsilon}+\delta)$. Since this is
true for all $\delta>0$, we get $\left\|f\right\|_{\mathcal{B}_{I}}\leq
2(|I|+K\varepsilon)\left\|f\right\|_{K,\varepsilon}$. If $f$ is complex-
valued, by taking real and imaginary parts, we get

$\left\|f\right\|_{\mathcal{B}_{I}}\leq
4(|I|+K\varepsilon)\left\|f\right\|_{K,\varepsilon}.$ (2.11)

Let $f\in\mathcal{B}_{I}$ be real. Let $C:=1+2/(K\varepsilon)$. We choose
$f_{+}:=f+C\left\|f\right\|_{\mathcal{B}_{I}}$ and
$f_{-}:=C\left\|f\right\|_{\mathcal{B}_{I}}$. Then $f_{-}\in
C_{0}(\varepsilon)$ and

$\frac{2\left\|f\right\|_{\mathcal{B}_{I}}}{K\varepsilon}\leq
f_{+}\leq\frac{2(1+K\varepsilon)\left\|f\right\|_{\mathcal{B}_{I}}}{K\varepsilon}.$

We deduce from the lower bound on $f_{+}$:

$\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})f_{+}\right\|_{\mathcal{B}_{I}}=\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})f\right\|_{\mathcal{B}_{I}}\leq
2\left\|f\right\|_{\mathcal{B}_{I}}\leq
K\varepsilon\left\|f_{+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}.$

Hence $f_{+}\in C_{K}(\varepsilon)$. Using the upper bound on $f_{+}$,

$\left\|f_{+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\leq\frac{2(1+K\varepsilon)}{K\varepsilon}\left\|f\right\|_{\mathcal{B}_{I}}.$

Doing the same with the imaginary part of $f$ finally yields:

$\left\|f\right\|_{K,\varepsilon}\leq\frac{2(1+K\varepsilon)}{K\varepsilon}\left\|f\right\|_{\mathcal{B}_{I}}.\qed$

Thanks to Lemma 2.8, we now apply Proposition 2.11.

######  Corollary 2.13.

Assume Hypotheses 1.2, 1.18 and 1.10.

Let $K$, $C$, $T>0$.

Let $(f_{\varepsilon})_{\varepsilon>0}$ be a family of functions in
$\mathcal{B}_{I}$. Assume that
$\sup_{\varepsilon>0}\left\|f_{\varepsilon}\right\|_{K,\varepsilon}\leq C$.
Then

$\lim_{\varepsilon\to
0}\sup_{t\in[0,T]}\left\|\Pi_{*}\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}-e^{-t({\vphantom{R}}^{\mathit{t}}{R})}\Pi_{*}f_{\varepsilon}\right\|_{\ell^{1}(I)}=0,$
(2.12)

where the convergence is uniform for fixed $K$ and $C$.

###### Proof.

Let $K$, $C$, $T>0$. Let $(f_{\varepsilon})_{\varepsilon>0}$ be a family of
functions in $\mathcal{B}_{I}$ such that
$\sup_{\varepsilon>0}\left\|f_{\varepsilon}\right\|_{K,\varepsilon}\leq C$.

Let $(\varepsilon_{k})$ be a positive sequence converging to $0$. For all $k$,
there exist $f_{k,+}$, $f_{k,-}$, $f_{k,i+}$, $f_{k,i-}$ in
$C_{K}(\varepsilon)$ such that
$f_{\varepsilon_{k}}=f_{k,+}-f_{k,-}+if_{k,i+}-if_{k,i-}$ and
$\left\|f_{k,+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\leq 2C$, and likewise
for $f_{k,-}$, $f_{k,i+}$, $f_{k,i-}$.

Note that

$\displaystyle\sup_{t\in[0,T]}$
$\displaystyle\left\|\Pi_{*}\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon_{k}^{-1}t\rfloor}f_{k,+}-e^{-t({\vphantom{R}}^{\mathit{t}}{R})}\Pi_{*}f_{k,+}\right\|_{\ell^{1}(I)}$
$\displaystyle\leq\sup_{t\in[0,T]}\left\|\prod_{n=0}^{\lfloor\varepsilon^{-1}t\rfloor-1}(\operatorname{Id}-\varepsilon_{k}({\vphantom{R}}^{\mathit{t}}{R})+R_{\varepsilon_{k},n})-e^{-t({\vphantom{R}}^{\mathit{t}}{R})}\right\|_{\ell^{1}(I)\to\ell^{1}(I)}\left\|\Pi_{*}f_{k,+}\right\|_{\ell^{1}(I)},$

where $(R_{\varepsilon_{k},n})_{k,n\geq 0}$ is a sequence of matrices obtained
by Proposition 2.11. The quantity
$\left\|\Pi_{*}f_{k,+}\right\|_{\ell^{1}(I)}$ is equal to
$|I|\cdot\left\|f_{k,+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$, and thus is
bounded by $2C|I|$. By Lemma 2.8,

$\lim_{k\to+\infty}\sup_{t\in[0,T]}\left\|\Pi_{*}\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon_{k}^{-1}t\rfloor}f_{k,+}-e^{-t({\vphantom{R}}^{\mathit{t}}{R})}\Pi_{*}f_{k,+}\right\|_{\ell^{1}(I)}=0,$

where the convergence is uniform in $C$. The same holds for $f_{k,-}$,
$f_{k,i+}$, $f_{k,i-}$. Summing all four limits yields

$\lim_{k\to+\infty}\sup_{t\in[0,T]}\left\|\Pi_{*}\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon_{k}^{-1}t\rfloor}f_{\varepsilon_{k}}-e^{-t({\vphantom{R}}^{\mathit{t}}{R})}\Pi_{*}f_{\varepsilon_{k}}\right\|_{\ell^{1}(I)}=0.$

Since the subsequence $(\varepsilon_{k})$ is arbitrary, this finishes the
proof. ∎

Corollary 2.13 gives us convergence of
$\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}$
to $e^{-t({\vphantom{R}}^{\mathit{t}}{R})}\Pi_{*}f_{\varepsilon}$, in a
relatively weak sense, but uniformly on all compacts in $t$. As announced, we
now implement a coupling argument to show that
$(\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon})_{t\geq
0,\varepsilon>0}$ is exponentially tight.

######  Proposition 2.14.

Assume Hypotheses 1.2, 1.18 and 1.10.

For any $\rho\in(0,\rho_{R})$ and $K\geq 0$, there exist constants $C$,
$\varepsilon_{0}>0$ such that, for all $\varepsilon\leq\varepsilon_{0}$ and
$n\geq 0$, the operator norm of $\mathcal{L}_{\varepsilon}^{n}$ acting on
$(\mathcal{B}_{I,0},\left\|\cdot\right\|_{K,\varepsilon})$ is no larger than
$Ce^{-\rho\varepsilon n}$.

###### Proof.

Let $\rho^{\prime}\in(\rho,\rho_{R})$. Since $R$ is an irreducible bi-L-
matrix, so is ${\vphantom{R}}^{\mathit{t}}{R}$. By Lemma 1.9, there exists a
constant $C$ such that, for all $t\geq 0$,

$\left\|e^{-t({\vphantom{R}}^{\mathit{t}}{R})}-\pi\right\|_{\ell^{1}(I)\to\ell^{1}(I)}\leq
Ce^{-\rho^{\prime}t},$

where $\pi(F)=|I|^{-1}\left(\sum_{I}F\right)\mathbf{1}$ is the eigenprojection
associated with the eigenvalue $0$ of ${\vphantom{R}}^{\mathit{t}}{R}$.

Let $\sigma\in(0,1)$, to be fixed at the end of this proof. Let $K_{\sigma}$
and $n_{\sigma}$ be given by Lemma 2.5. Let $T>0$ be such that
$C|I|e^{-\rho^{\prime}T}\leq\sigma/3$. We assume without loss of generality
that $K\geq K_{\sigma}$.

A renormalization map

Given $f_{+}\in C_{K}(\varepsilon)$, we define :

$\Phi_{\sigma,\varepsilon}(f_{+}):=\sigma^{-1}\left(\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}T\rfloor}f_{+}-(1-\sigma)\int_{[I]}f_{+}\;{\rm
d}\mu_{I}\right).$

Our intermediate goal is to prove that $\Phi_{\sigma,\varepsilon}$ maps
$C_{K}(\varepsilon)$ into itself if $\varepsilon$ is small enough.

Let $C^{\prime}>0$. Let $(\varepsilon_{k})_{k\geq 0}$ be a sequence of
positive numbers converging to $0$, and $(f_{k,+})_{k\geq 0}\in
C_{K}(\varepsilon_{k})$ with
$\left\|f_{k,+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\leq C^{\prime}$. By
Corollary 2.13,

$\lim_{k\to+\infty}\sup_{t\in[0,T]}\left\|\Pi_{*}\mathcal{L}_{\varepsilon_{k}}^{\lfloor\varepsilon_{k}^{-1}t\rfloor}f_{k,+}-e^{-t({\vphantom{R}}^{\mathit{t}}{R})}\Pi_{*}f_{k,+}\right\|_{\ell^{1}(I)}=0,$

uniformly for fixed $K$ and $C^{\prime}$. By our choice of $T$,

$e^{-T({\vphantom{R}}^{\mathit{t}}{R})}\Pi_{*}f_{k,+}\geq\pi(\Pi_{*}f_{k,+})-Ce^{-\rho^{\prime}T}\left\|\Pi_{*}f_{k,+}\right\|_{\ell^{1}(I)}\geq\left(1-\frac{\sigma}{3}\right)\int_{[I]}f_{k,+}\;{\rm
d}\mu_{I}.$

Hence, if $k$ is large enough, then

$\Pi_{*}\mathcal{L}_{\varepsilon_{k}}^{\lfloor\varepsilon_{k}^{-1}T\rfloor}f_{k,+}\geq\left(1-\frac{2\sigma}{3}\right)\int_{[I]}f_{k,+}\;{\rm
d}\mu_{I}.$

If $k$ is large enough, then $\varepsilon_{k}^{-1}T\geq n_{\sigma}$, so that
$\mathcal{L}_{\varepsilon_{k}}^{\lfloor\varepsilon_{k}^{-1}T\rfloor}f_{k,+}\in
C_{\sigma K}(\varepsilon_{k})$. This has two consequences. The first is that,
if $k$ is large enough,

$\displaystyle\mathcal{L}_{\varepsilon_{k}}^{\lfloor\varepsilon_{k}^{-1}T\rfloor}f_{k,+}$
$\displaystyle\geq\Pi_{*}\mathcal{L}_{\varepsilon_{k}}^{\lfloor\varepsilon_{k}^{-1}T\rfloor}f_{k,+}-\sigma
K\varepsilon_{k}\int_{[I]}f_{k,+}\;{\rm d}\mu_{I}$
$\displaystyle\geq(1-\sigma)\int_{[I]}f_{k,+}\;{\rm d}\mu_{I}.$

In particular, $\Phi_{\sigma,\varepsilon_{k}}(f_{k,+})\geq 0$. The second
consequence is that, if $k$ is large enough, then

$\displaystyle\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})\Phi_{\sigma,\varepsilon_{k}}(f_{k,+})\right\|_{\mathcal{B}_{I}}$
$\displaystyle=\sigma^{-1}\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})\mathcal{L}_{\varepsilon_{k}}^{\lfloor\varepsilon_{k}^{-1}T\rfloor}f_{k,+}\right\|_{\mathcal{B}_{I}}$
$\displaystyle\leq\sigma^{-1}\sigma
K\varepsilon_{k}\left\|\mathcal{L}_{\varepsilon_{k}}^{\lfloor\varepsilon_{k}^{-1}T\rfloor}f_{k,+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$
$\displaystyle=K\varepsilon_{k}\left\|\Phi_{\sigma,\varepsilon_{k}}(f_{k,+})\right\|_{\mathbb{L}^{1}([I],\mu_{I})}.$

Hence, $\Phi_{\sigma,\varepsilon_{k}}(f_{k,+})\in C_{K}(\varepsilon_{k})$.
Since $\Phi_{\sigma,\varepsilon_{k}}$ is linear,
$\Phi_{\sigma,\varepsilon_{k}}(C_{K}(\varepsilon_{k}))\subset
C_{K}(\varepsilon_{k})$ for all large enough $k$. Finally, since
$(\varepsilon_{k})_{k\geq 0}$ is arbitrary,
$\Phi_{\sigma,\varepsilon}(C_{K}(\varepsilon))\subset C_{K}(\varepsilon)$ for
all small enough $\varepsilon$.

Bound on the operator norm

Let $\varepsilon$ be small enough that $\Phi_{\sigma,\varepsilon}$ preserves
$C_{K}(\varepsilon)$. Let $f\in\mathcal{B}_{I,0}$, which we assume without
loss of generality to be real. Let $f_{+}$, $f_{-}\in C_{K}(\varepsilon)$ be
such that $f=f_{+}-f_{-}$ and
$\left\|f_{+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$,
$\left\|f_{-}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\leq
2\left\|f\right\|_{K,\varepsilon}$. Then

$\displaystyle\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}T\rfloor}f$
$\displaystyle=\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}T\rfloor}f_{+}-\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}T\rfloor}f_{-}$
$\displaystyle=\left(\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}T\rfloor}f_{+}-(1-\sigma)\int_{[I]}f_{+}\;{\rm
d}\mu_{I}\right)-\left(\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}T\rfloor}f_{+}-(1-\sigma)\int_{[I]}f_{+}\;{\rm
d}\mu_{I}\right)$
$\displaystyle=\sigma\left(\Phi_{\sigma,\varepsilon}(f_{+})-\Phi_{\sigma,\varepsilon}(f_{-})\right).$

By recursion, we get, for all $m\geq 0$:

$\mathcal{L}_{\varepsilon}^{m\lfloor\varepsilon^{-1}T\rfloor}f=\sigma^{m}\left(\Phi_{\sigma,\varepsilon}^{m}(f_{+})-\Phi_{\sigma,\varepsilon}^{m}(f_{-})\right).$

The functions $\Phi_{\sigma,\varepsilon}^{m}(f_{+})$ and
$\Phi_{\sigma,\varepsilon}^{m}(f_{-})$ belong to $C_{K}(\varepsilon)$ and have
the same integral as $f_{+}$.

Let $k\geq 0$. By Corollary 2.6, without loss of generality, there exists a
constant $\lambda_{1}\geq 1$ such that, and all $\varepsilon>0$, $k\geq 0$,
the functions
$\mathcal{L}_{\varepsilon}^{k}\Phi_{\sigma,\varepsilon}^{m}(f_{+})$ and
$\mathcal{L}_{\varepsilon}^{k}\Phi_{\sigma,\varepsilon}^{m}(f_{-})$ belong to
$C_{\lambda_{1}K}(\varepsilon)$. Hence, setting

$\tilde{f}_{+}:=(\lambda_{1}-1)\left\|f_{+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}+\mathcal{L}_{\varepsilon}^{k}\Phi_{\sigma,\varepsilon}^{m}(f_{+})$

and likewise for $f_{-}$,

$\mathcal{L}_{\varepsilon}^{m\lfloor\varepsilon^{-1}T\rfloor+k}f=\sigma^{m}(\tilde{f}_{+}-\tilde{f}_{-}).$

By construction,

$\displaystyle\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})\tilde{f}_{+}\right\|_{\mathcal{B}_{I}}$
$\displaystyle=\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})\mathcal{L}_{\varepsilon}^{k}\Phi_{\sigma,\varepsilon}^{m}(f_{+})\right\|_{\mathcal{B}_{I}}$
$\displaystyle\leq\lambda_{1}K\varepsilon\left\|\mathcal{L}_{\varepsilon}^{k}\Phi_{\sigma,\varepsilon}^{m}(f_{+})\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$
$\displaystyle=\lambda_{1}K\varepsilon\left\|f_{+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$
$\displaystyle=K\varepsilon\left\|\tilde{f}_{+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}.$

Since the same estimate holds for $\tilde{f}_{-}$, the functions
$\tilde{f}_{+}$, $\tilde{f}_{-}$ belong to $C_{K}(\varepsilon)$ and their
$\mathbb{L}^{1}$-norm is bounded by
$\lambda_{1}\left\|f_{+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\leq
2\lambda_{1}\left\|f\right\|_{K,\varepsilon}$. Hence,

$\left\|\mathcal{L}_{\varepsilon}^{m\lfloor\varepsilon^{-1}T\rfloor+k}f\right\|_{K,\varepsilon}\leq
2\lambda_{1}\sigma^{m}\left\|f\right\|_{K,\varepsilon}.$ (2.13)

Let $n\geq 0$. Take $m:=\lfloor\varepsilon T^{-1}n\rfloor$ and
$k:=n-m\lfloor\varepsilon^{-1}T\rfloor$. Equation (2.13) implies :

$\left\|\mathcal{L}_{\varepsilon}^{n}f\right\|_{K,\varepsilon}\leq
2\lambda_{1}\sigma^{\lfloor\varepsilon
T^{-1}n\rfloor}\left\|f\right\|_{K,\varepsilon}.$

Finally, recall that the only constraint on $T$ is that
$C|I|e^{-\rho^{\prime}T}\leq\sigma/3$. Hence, we can take
$T:=\rho^{\prime-1}|\ln(\sigma/(3C|I|))|$, so that:

$\displaystyle\left\|\mathcal{L}_{\varepsilon}^{n}f\right\|_{K,\varepsilon}$
$\displaystyle\leq
2\lambda_{1}e^{-|\ln(\sigma)|\lfloor\rho^{\prime}\varepsilon|\ln(\sigma/(3C|I|))|^{-1}n\rfloor}\left\|f\right\|_{K,\varepsilon}$
$\displaystyle\leq
2\lambda_{1}\sigma^{-1}e^{-\rho^{\prime}\frac{\ln(\sigma)}{\ln(\sigma)-\ln(3C|I|)}\varepsilon
n}\left\|f\right\|_{K,\varepsilon}.$

Choose $\sigma$ small enough that

$\rho^{\prime}\frac{\ln(\sigma)}{\ln(\sigma)-\ln(3C|I|)}\leq\rho\quad;$

then

$\left\|\mathcal{L}_{\varepsilon}^{n}f\right\|_{K,\varepsilon}\leq
2\lambda_{1}\sigma^{-1}e^{-\rho\varepsilon
n}\left\|f\right\|_{K,\varepsilon}.\qed$

######  Remark 2.15 (Bound in operator norm).

The bound on the operator norm of $\mathcal{L}_{\varepsilon}^{n}$ in
Proposition 2.14 is given for the norm $\left\|\cdot\right\|_{K,\varepsilon}$
For the original Lipschitz norm $\left\|\cdot\right\|_{\mathcal{B}_{I}}$,
which does not depend on $\varepsilon$, for any $\rho<\rho_{R}$, the bounds in
the proof of Lemma 2.12 yield a constant $C$ such that:

$\left\|\mathcal{L}_{\varepsilon}^{n}\right\|_{\mathcal{B}_{I,0}\to\mathcal{B}_{I,0}}\leq\frac{C}{\varepsilon}e^{-\rho\varepsilon
n}.$ (2.14)

While not necessary for the remainder of this article, notice that Lemma 2.12
and Proposition 2.14 give an explicit lower bound on the spectral gap of
$\mathcal{L}_{\varepsilon}$. By ergodicity of $([I],\mu_{I},T_{\varepsilon})$,
the operator $\mathcal{L}_{\varepsilon}$ has $1$ as a simple eigenvalue,
corresponding to constant eigenfunctions. Define the _spectral gap_ of
$\mathcal{L}_{\varepsilon}$ acting on $\mathcal{B}_{I}$ as, equivalently, one
minus the spectral radius of $\mathcal{L}_{\varepsilon}$ acting on
$\mathcal{B}_{I,0}$, that is

$\min\\{1-|\lambda|:\
\lambda\in\operatorname{Sp}(\mathcal{L}_{\varepsilon}\curvearrowright\mathcal{B}_{I}),\
\lambda\neq 1\\}.$

######  Corollary 2.16.

Assume Hypotheses 1.2, 1.18 and 1.10.

When $\varepsilon$ goes to $0$, the spectral gap of
$\mathcal{L}_{\varepsilon}$ acting on $\mathcal{B}_{I}$ is at least
$\rho_{R}\varepsilon(1-o(1))$.

Integrating over $t\geq 0$, we can now relate $S$ with $R_{0}^{-1}$.

######  Corollary 2.17.

Assume Hypotheses 1.2, 1.18 and 1.10.

Let $K\geq 0$. Let $(f_{\varepsilon})_{\varepsilon>0}$ be a family of
functions in $\mathcal{B}_{I,0}$. Assume that
$\sup_{\varepsilon>0}\left\|f_{\varepsilon}\right\|_{K,\varepsilon}<+\infty$.
Then, in $\mathcal{B}_{I,0}$,

$\sum_{n=0}^{+\infty}\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}=\frac{1}{\varepsilon}\Pi^{*}({\vphantom{R}}^{\mathit{t}}{R}_{0}^{-1})\Pi_{*}f_{\varepsilon}+o(\varepsilon^{-1}).$
(2.15)

###### Proof.

Changing the time scale by a factor of $\varepsilon^{-1}$, we get:

$\sum_{n=0}^{+\infty}\mathcal{L}_{\varepsilon}^{n}f_{\varepsilon}=\frac{1}{\varepsilon}\int_{0}^{+\infty}\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}\;{\rm
d}t.$

By Corollary 2.13, on all compact subsets of $\mathbb{R}_{+}$,

$\lim_{\varepsilon\to
0}\sup_{t\in[0,T]}\left\|\Pi_{*}\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}-e^{-t({\vphantom{R}}^{\mathit{t}}{R})}\Pi_{*}f_{\varepsilon}\right\|_{\ell^{1}(I)}=0.$

By Proposition 2.14, there exists a constant $C$ such that
$\left\|\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}\right\|_{K,\varepsilon}\leq
C\left\|f\right\|_{K,\varepsilon}$. We write
$\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}=f_{+}-f_{-}$
with $f_{+}$, $f_{-}\in C_{K}(\varepsilon)$ and
$\left\|f_{+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}$,
$\left\|f_{-}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\leq
2\left\|f\right\|_{K,\varepsilon}$. Up to taking a larger value of $K$, the
function $\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{+}$
belongs to $C_{\lambda_{1}K}(\varepsilon)$ for all $t$ by Corollary 2.6.
Hence,

$\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{+}\right\|_{\mathcal{B}_{I}}\leq\lambda_{1}K\varepsilon\left\|\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{+}\right\|_{\mathbb{L}^{1}([I],\mu_{I})}\leq
2\lambda_{1}K\varepsilon\left\|f\right\|_{K,\varepsilon}.$

The function $f_{-}$ satisfies a similar estimate. It follows that

$\left\|(\operatorname{Id}-\Pi^{*}\Pi_{*})\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}\right\|_{\mathcal{B}_{I}}\leq
4\lambda_{1}K\varepsilon\left\|f\right\|_{K,\varepsilon}.$

Hence,

$\lim_{\varepsilon\to
0}\sup_{t\in[0,T]}\left\|\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}-\Pi^{*}e^{-t({\vphantom{R}}^{\mathit{t}}{R})}\Pi_{*}f_{\varepsilon}\right\|_{\mathcal{B}_{I}}=0.$
(2.16)

In addition, by Proposition 2.14 again, there exist constants $C$, $\rho>0$
such that the operator norm of $\mathcal{L}_{\varepsilon}^{n}$ acting on
$(\mathcal{B}_{I,0},\left\|\cdot\right\|_{K,\varepsilon})$ is no larger than
$Ce^{-\rho\varepsilon n}$. Using Equation (2.11),

$\left\|\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}\right\|_{\mathcal{B}_{I}}\leq
4(1+K\varepsilon)\left\|\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}\right\|_{K,\varepsilon}\leq
4(1+K\varepsilon)Ce^{\rho\varepsilon}e^{-\rho
t}\left\|f\right\|_{K,\varepsilon}.$

This gives the tightness needed for the convergence of integrals:

$\lim_{\varepsilon\to
0}\left\|\int_{0}^{+\infty}\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}\;{\rm
d}t-\int_{0}^{+\infty}\Pi^{*}e^{-t({\vphantom{R}}^{\mathit{t}}{R})}\Pi_{*}f_{\varepsilon}\;{\rm
d}t\right\|_{\mathcal{B}_{I}}=0.$

Finally, denoting by ${\vphantom{R}}^{\mathit{t}}{R}_{0}$ the action of
${\vphantom{R}}^{\mathit{t}}{R}$ on $\mathbb{C}_{0}^{I}$, to which
$\Pi_{*}f_{\varepsilon}$ belongs,

$\int_{0}^{+\infty}\Pi^{*}e^{-t({\vphantom{R}}^{\mathit{t}}{R})}\Pi_{*}f_{\varepsilon}\;{\rm
d}t=\Pi^{*}\left(\int_{0}^{+\infty}e^{-t({\vphantom{R}}^{\mathit{t}}{R}_{0})}\;{\rm
d}t\right)\Pi_{*}f_{\varepsilon}=\Pi^{*}({\vphantom{R}}^{\mathit{t}}{R}_{0}^{-1})\Pi_{*}f_{\varepsilon}.$

We get the claim by multiplying both side by $\varepsilon^{-1}$:

$\left\|\frac{1}{\varepsilon}\int_{0}^{+\infty}\mathcal{L}_{\varepsilon}^{\lfloor\varepsilon^{-1}t\rfloor}f_{\varepsilon}\;{\rm
d}t-\frac{1}{\varepsilon}\Pi^{*}({\vphantom{R}}^{\mathit{t}}{R}_{0}^{-1})\Pi_{*}f_{\varepsilon}\right\|_{\mathcal{B}_{I}}=o(\varepsilon^{-1}).\qed$

We now finish the proof of Lemma 1.19.

###### Proof of Lemma 1.19.

Given $f\in\mathbb{C}_{0}^{I}$, the quantity
$\left\|\Pi^{*}f\right\|_{K,\varepsilon}$ is bounded jointly in $K$ and
$\varepsilon$, and Corollary 2.17 applies. By Lemma 1.14,
${\vphantom{R}}^{\mathit{t}}{R}_{0}^{-1}$ is irreducible in the sense of
Definition 1.13. This yields, for all $f$, $g\in\mathbb{C}_{0}^{I}$,

$\displaystyle\varepsilon^{-1}\langle
f,S(g)\rangle_{\ell^{2}(I)}+o(\varepsilon^{-1})$ $\displaystyle=\langle
f,Q_{\varepsilon}(g)\rangle_{\ell^{2}(I)}$
$\displaystyle=\langle\Pi_{*}(\operatorname{Id}-\mathcal{L}_{\varepsilon})^{-1}\Pi^{*}(f),g\rangle_{\ell^{2}(I)}$
$\displaystyle=\varepsilon^{-1}\langle\Pi_{*}\Pi^{*}({\vphantom{R}}^{\mathit{t}}{R}_{0}^{-1})\Pi_{*}\Pi^{*}(f),g\rangle_{\ell^{2}(I)}+o(\varepsilon^{-1})$
$\displaystyle=\varepsilon^{-1}\langle
f,R_{0}^{-1}(g)\rangle_{\ell^{2}(I)}+o(\varepsilon^{-1}),$

whence $S=R_{0}^{-1}$. ∎

##  3 The ergodic case

All the work in Section 2 was done under Hypothesis 1.18, that is, that the
first return map $(A,\mu,T_{[0]})$ is mixing. This hypothesis is inconvenient,
as we would like to work only with conditions involving the initial data: the
system $(A,\mu,T)$ and the function $F$.

If $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ is recurrent and
ergodic, then $(A,\mu,T_{[0]})$ is well-defined and ergodic. Hence, ergodicity
comes freely, and we would like to use this property instead of the stronger
mixing hypothesis. This is the goal of this section.

###  3.1 Structure of extensions with non-mixing first return maps

Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be a Markov
$\mathbb{Z}^{d}$-extension of a Gibbs-Markov map $(A,\alpha,d,\mu,T)$ with
jump function $F$. Assume that
$([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ is ergodic and recurrent.
As the induced map $(A,\mu,T_{[0]})$ is well-defined and ergodic, by Corollary
A.10, there exists an integer $M\geq 1$ and a partition
$A=\bigsqcup_{\ell\in\mathbb{Z}_{/M\mathbb{Z}}}A_{\ell}$ such that each
$A_{\ell}$ is $\alpha^{*}$-measurable, $T_{[0]}(A_{\ell})\subset A_{\ell+1}$
for all $\ell$, and $T_{[0]}^{M}:A_{\ell}\to A_{\ell}$ is mixing for all
$\ell$.

At first, we show the constraints that the existence of a non-trivial period
for $(A,\mu,T_{[0]})$ impose on the whole extension
$([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$. We introduce a colouration
of $[\mathbb{Z}^{d}]$.

######  Definition 3.1 (Color of a site).

Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be a Markov
$\mathbb{Z}^{d}$-extension of a Gibbs-Markov map $(A,\alpha,d,\mu,T)$. Assume
that $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ is ergodic and
recurrent. Let $A=\bigsqcup_{\ell\in\mathbb{Z}_{/M\mathbb{Z}}}A_{\ell}$ be the
decomposition of $(A,\mu,T_{[0]})$ associated with its period $M$. For
$\widetilde{\mu}$-almost every $(x,p)\in[\mathbb{Z}^{d}]$, let

$\widetilde{\varphi}_{[0]}(x,p):=\inf\\{n\geq 1:\
\widetilde{T}^{n}(x,p)\in[0]\\}.$ (3.1)

The _colour_ of $(x,p)$ is defined as the element
$\ell\in\mathbb{Z}_{/M\mathbb{Z}}$ such that
$\widetilde{T}^{\widetilde{\varphi}_{[0]}(x,p)}(x,p)\in A_{\ell}$. We shall
denote it by $C(x,p)$.

Interestingly, the structural constraints we get are different depending on
whether $d=1$ or $d=2$, the later case being simpler. As the following
proposition states, in the $d=2$ case, all sites have the same colour, up to a
neighbourhood of $[0]$. In the $d=1$ case, we may need two colours. This
difference comes from the fact that the plane minus a point is connected,
while the line minus a point has two connected components.

######  Proposition 3.2.

Let $([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ be a Markov
$\mathbb{Z}^{d}$-extension of a Gibbs-Markov map $(A,\alpha,d,\mu,T)$ with
jump function $F$. Assume that
$([\mathbb{Z}^{d}],\widetilde{\mu},\widetilde{T})$ is ergodic and recurrent.

If $d=2$, or $d=1$ and
$\left\|F\right\|_{\mathbb{L}^{\infty}(A,\mu)}=+\infty$, then there exists
$R>0$ and $\ell\in\mathbb{Z}_{/M\mathbb{Z}}$ such that $C(x,p)=\ell$ whenever
$\left\|p\right\|\geq R$.

If $d=1$ and $\left\|F\right\|_{\mathbb{L}^{\infty}(A,\mu)}<+\infty$, then
there exists $R>0$ and $\ell_{-}$, $\ell_{+}\in\mathbb{Z}_{/M\mathbb{Z}}$ (not
necessarily distinct) such that $C(x,p)=\ell_{-}$ whenever $p\leq-R$ and
$C(x,p)=\ell_{+}$ whenever $p\geq R$.

###### Proof.

We work under the hypotheses of Proposition 3.2. First, let us note that
$C\circ\widetilde{T}(x,p)=C(x,p)$ for all
$(x,p)\in[\mathbb{Z}^{d}]\setminus[0]$.

By ergodicity and recurrence, $[\mathbb{Z}^{d}]$ is swept by the forward
images of $[0]$, that is,

$[\mathbb{Z}^{d}]=\bigcup_{x\in
A}\bigsqcup_{k=0}^{\varphi_{[0]}(x)-1}\widetilde{T}^{k}(x,0).$

Let $\ell\in\mathbb{Z}_{/M\mathbb{Z}}$. The property that
$T_{[0]}(A_{\ell-1})=A_{\ell}$ implies that, for all $x\in A_{\ell-1}$ and all
$0\leq k<\varphi_{[0]}(x)$,

$\widetilde{T}^{\widetilde{\varphi}_{[0]}(\widetilde{T}^{k}(x,0))}\circ\widetilde{T}^{k}(x,0)=\widetilde{T}^{\varphi_{[0]}(x)-k}\circ\widetilde{T}^{k}(x,0)=T_{[0]}(x)\in
A_{\ell},$

so that $C(\widetilde{T}^{k}(x,0))=\ell$. Doing this for all $\ell$ yields
$\\{C=\ell\\}=\bigcup_{x\in
A_{\ell-1}}\bigsqcup_{k=0}^{\varphi_{[0]}(x)-1}\widetilde{T}^{k}(x,0)$: modulo
a negligible subset, the points of colour $\ell$ are exactly the forward
images of $A_{\ell-1}$ up to time $\varphi_{[0]}-1$.

Let $\ell\in\mathbb{Z}_{/M\mathbb{Z}}$. We can also write the later fact as

$\\{C=\ell\\}=\bigcup_{k=0}^{+\infty}\widetilde{T}^{k}(A_{\ell-1}\cap\\{\varphi_{[0]}>k\\}).$

Since $\varphi_{[0]}$ is a stopping time, $\\{\varphi_{[0]}>k\\}$ is
$\sigma(\alpha^{(k)})$-measurable. The set $A_{\ell-1}$ is also
$\sigma(\alpha^{*})$-measurable by Corollary A.10, and thus also
$\sigma(\alpha^{(k)})$-measurable. The function $S_{k}F$ is constant on each
cylinder of length $k$, so that we can write it as a function of the cylinder.
Hence, setting $B(p,\ell):=\\{x\in A:\ C(x,p)=\ell\\}$, we have

$B(p,\ell)=\bigcup_{k=0}^{+\infty}\bigsqcup_{\begin{subarray}{c}a\in\alpha^{(k)}\\\
a\subset A_{\ell-1}\cap\\{\varphi_{[0]}>k\\}\\\
S_{k}(a)=p\end{subarray}}T^{k}(a).$ (3.2)

By the big image property of Gibbs-Markov maps, for all $k\geq 0$ and nonempty
$a\in\alpha^{(k)}$, we have
$\mu(T^{k}(a))\geq\delta_{0}:=\inf_{a\in\alpha}\mu(T(a))>0$. Each subset
$T^{k}(a)$ in Equation (3.2) has measure at least $\delta_{0}$. Hence, for any
$p\in\mathbb{Z}^{d}$ and $\ell\in\mathbb{Z}_{/M\mathbb{Z}}$, we have a
dichotomy:

  * •

either $\mu(B(p,\ell))=0$;

  * •

or $\mu(B(p,\ell))\geq\delta_{0}>0$.

In other words, on each site, either a colour is absent, or it is present in a
quantity of at least $\delta_{0}$.

Let $p\in\mathbb{Z}^{d}\setminus\\{0\\}$ and $\ell$ such that
$\mu(B(p,\ell))>0$. From the expression in Equation (3.2), the set $B(p,\ell)$
is $\sigma(\alpha^{*})$-measurable, so that