# Fourier decay of equilibrium states for bunched attractors

Gaétan Leclerc

###### Abstract

Let $M$ be a closed manifold, and let $f:M\rightarrow M$ be a $C^{2+\alpha}$
Axiom A diffeomorphism. Suppose that $f$ has an attractor $\Omega$ with
codimension 1 stable lamination. Under a generic nonlinearity condition and a
suitable bunching condition, we prove polynomial Fourier decay in the unstable
direction for a large class of invariant measures on $\Omega$. Our result
applies in particular for the measure of maximal entropy. We construct in the
appendix an explicit solenoid that satisfies the nonlinearity and bunching
assumption.

##  1 Introduction

###  1.1 On fractals and chaotic dynamical systems

It is known since the work of Fatou and Julia that fractals appears as natural
invariant sets in various dynamical systems, for example as attractors or
repellers. The geometric properties of such invariant subsets depends
intimately with the nature of the map $f$, and it is tempting to study the way
they relate to each other. One way to relate the geometry of invariant subsets
$K$ to the nature of the dynamics is by the mean of invariant measures with
support $K$. In the context of hyperbolic dynamics, a natural choice is given
by the so-called equilibrium states, a family of probability measures that
encompasses the measure of maximal entropy or the SRB measure.

If we fix some probability measure with support $K\subset\mathbb{R}^{d}$, the
Fourier transform being a linear bijection
$\mathcal{S}^{\prime}(\mathbb{R}^{d})\rightarrow\mathcal{S}^{\prime}(\mathbb{R}^{d})$,
the information contained in $\mu$ are the same that the one that are
contained in

$\widehat{\mu}(\xi):=\int_{K}e^{i\xi\cdot x}d\mu(x).$

In a sense, studying $\mu$ or studying $\widehat{\mu}$ should brings the same
quantities of information about the dynamics and the geometry of the support.
Yet for equilibrium states, even though much is known on the measure in its
spatial representation, we have a very poor understanding of its Fourier
transform in the general setting. One of the most basic question one can ask
about $\widehat{\mu}$ is its behavior when $\xi$ goes to infinity. Does
$\widehat{\mu}(\xi)\longrightarrow 0$, and if yes, at which rate ? From the
geometric point of view, this question is linked to the notion of Fourier
dimension of a set. Recall that for a set $K\subset\mathbb{R}^{d}$, we call
the Fourier dimension of $K$ the quantity

$\dim_{F}(K):=\sup\left\\{\alpha\in[0,d]\ |\
\exists\mu\in\mathcal{P}(K),\exists C>0,\forall\xi\in\mathbb{R}^{d},\
|\widehat{\mu}(\xi)|\leq C(1+|\xi|)^{-\alpha/2}\right\\}$

where $\mathcal{P}(K)$ denotes the set of probability measures supported on
$K$. The Fourier dimension of a set can be thought as a way to quantify
additive chaos in the set $K$. It is always less than $\dim_{H}(E)$, the
Hausdorff dimension of $K$.

In a probabilistic setting, the work of Salem [Sa51], Kahane [Ka66], Bluhm
[Bl96] (to cite a few) suggest that sets satisfying $\dim_{H}K=\dim_{F}K$ are,
in some sense, generic. But except from some specific arithmetic constructions
(see [Kau81], [QR03], [Ha17] and [FH20]), explicit examples of such sets are
hard to find. Even (nontrivial) explicit sets with positive Fourier dimension
are difficult to construct.

###  1.2 Recent development and main results

The first explicit and natural fractal with positive Fourier dimension (in the
sense that it is an invariant set for some chaotic dynamical system) is due to
Bourgain and Dyatlov in [BD17]. They prove that the limit set of a non-
elementary Fuschian Schottky group, seen as a subset of $\mathbb{R}$, has
positive Fourier dimension (notice that such sets are Cantor sets). More
precisely, Patterson-Sullivan measures exhibit polynomial Fourier decay in
this context.

This paper introduced a new method to prove Fourier decay of invariant
measures based on previous results of Bourgain from additive combinatorics,
called "sum-product phenomenons" (see [Bo10], or [Gr09] for an accessible
introduction). A concrete example of such sum-product result is given in the
following theorem.

######  Theorem 1.1 ([BD17]).

For all $\delta>0$, there exists $\varepsilon_{1},\varepsilon_{2}>0$ and
$k\in\mathbb{N}$ such that the following holds. Let $\mathcal{Z}$ be a finite
set and $\zeta:\mathcal{Z}\rightarrow[1/2,2]$ a map. Let $\eta>1$ be large
enough. Assume that for all
$\sigma\in\left[\eta^{-1},\eta^{-\varepsilon_{1}}\right]$,

$\\#\left\\{(\mathbf{b},\mathbf{c})\in\mathcal{Z}^{2}\ ,\
|\zeta(\mathbf{b})-\zeta(\mathbf{c})|<\sigma\right\\}\leq\\#\mathcal{Z}^{2}\sigma^{\delta}.\quad(*)$

Then

$\Big{|}{\\#\mathcal{Z}^{-k}}\sum_{(\mathbf{b}_{1},\dots,\mathbf{b}_{k})\in\mathcal{Z}^{k}}e^{i\eta\zeta(\mathbf{b}_{1})\dots\zeta(\mathbf{b}_{k})}\Big{|}\leq\eta^{-\varepsilon_{2}}.$

Intuitively, the idea is that multiplications may spread the phase so that
cancellations happens in the sum. The idea to prove Fourier decay, then, is to
use the autosimilarity of equilibrium states to break the Fourier transform
into a sum of exponentials on which this sum-product phenomenon applies. One
of the difficulty in the strategy being to actually prove the non
concentration hypothesis $(*)$ made on the phase.

The result was generalized in 2019 with the work of Li, Naud and Pan [LNP19]
in the context of limits sets for general Kleinian Schottky groups. In this
context, the limit set is still a Cantor set, but in the complex plane.

The method was generalized further in 2020 by Sahlsten and Stevens [SS20] for
very general Cantor sets in the real line. They prove that under a
_nonlinearity condition_ on the underlying dynamics (in this context,
expanding maps on the Cantor set), equilibrium states exhibit polynomial
Fourier decay, and hence the Fourier dimension of those Cantor sets is
positive. An important remark is that, in the context of the triadic Cantor
set $C$, the underlying dynamics is the shift map $x\mapsto 3x\mod 1$, which
is linear, and the Fourier dimension of $C$ is zero.

The strategy was pushed past the case of Cantor sets by the author in 2021
[Le21] who studied the case of Julia sets for hyperbolic rational maps in the
complex plane. The result is the following: when the Julia set is not included
in a circle, any equilibrium state associated to $C^{1}$ potentials enjoys
polynomial Fourier decay. The use of Markov partitions, and the control over
the dynamics provided by the conformal setting, allowed us to adapt the
previous method in this case. Notice that the condition made on the Julia set
is a form of nonlinearity condition, as it implies for example that the
dynamics is not conjugated (via a Möbius transformation) to $z\mapsto z^{d}$,
which is linear on the circle.

In this paper we explore the method in the context of attractors for
codimension one Axiom A diffeomorphisms. Our main goal was to investigate the
Fourier dimension of such attractors in the nonlinear setting. By the nature
of the existing procedure, Fourier decay of equilibrium state couldn’t be
achieved, but we get a result of polynomial Fourier decay _in the unstable
direction_ for nonlinear dynamics, as long as a suitable bunching condition is
satisfied.

Even though this bunching condition seems quite restrictive, we think that the
result should hold for attractors of generic codimension one Axiom A
diffeomorphisms, as the nonlinearity condition is generic, and as we only use
the bunching condition for one argument (to be able to use some Dolgopyat’s
estimates). On the other hand, the bunching condition has the advantage to
allow one to easily construct explicit examples of attractors on which our
theorem applies. We prove the genericity of the nonlinearity condition in the
appendix A and we give an example of such a construction in the appendix B.

It should be stressed that if one succeed to relax the bunching condition,
then Fourier decay in the unstable direction for equilibrium state would hold
for generic codimension one Anosov diffeomorphisms. In dimension 2, replacing
$f$ by $f^{-1}$ would then give Fourier decay in the stable direction, which
should be enough to prove that equilibrium states for generic Anosov
diffeomorphisms on surfaces exhibit Fourier decay.

Let us precise our setting. Let $M$ be a closed manifold, let $f:M\rightarrow
M$ be a $C^{2+\alpha}$ Axiom A diffeomorphism (that if, $f$ is $C^{2}$, and
($d^{2}f)$ is $\alpha$-Hölder for some $\alpha>0$). Suppose that $f$ has an
attractor $\Omega$, and suppose that its stable lamination is codimension 1
(in particular, it is $C^{1+\alpha}$). The nonlinearity condition is defined
by putting an independance condition on some Lyapunov exponents.

######  Definition 1.1.

Define $\Omega_{\text{per}}:=\\{x\in\Omega\ \text{periodic}\\}$. For any
$x\in\Omega_{\text{per}}$ with minimal period $n$, define its local unstable
Lyapunov exponent $\widehat{\lambda}(x)$ by the formula

$\widehat{\lambda}(x)=\frac{1}{n}\sum_{k=0}^{n-1}\ln|\partial_{u}f(f^{k}(x))|$

where $|\partial_{u}f|$ is the absolute value of the derivative of $f$ in the
unstable direction.
We say that $\Omega$ satisfy the nonlinearity condition (NL) if

$\text{dim}_{\mathbb{Q}}\text{Vect}_{\mathbb{Q}}\widehat{\lambda}\left(\Omega_{\text{per}}\right)=\infty.$

######  Definition 1.2.

We introduce our bunching condition: see for example [Ha97], [GRH21], or
[ABV14], [HP69] for similar conditions in the case of flows. We say that the
bunching condition (B) is satisfied if

$\forall p\in\Omega,\ |\partial_{u}f(p)|\cdot\|{{(df)_{p}}_{|E^{s}_{p}}}\|<1,$

where the norm on $(df)_{|E^{s}}$ is the operator norm induced by the
riemannian metric on $M$. It implies that $\Omega$ is a proper attractor. In
this case, by continuity of $df$ and compacity of $\Omega$, there exists some
$\alpha>0$ such that

$\forall p\in\Omega,\
|\partial_{u}f(p)|\cdot\|{{(df)_{p}}_{|E^{s}_{p}}}\|^{1/(1+\alpha)}\leq 1.$

In this case, the stable bunching parameter $b^{s}$ (see [GRH21], and [Ha97]
for the pointwise version) satisfies

$b^{s}(p)=1+\frac{\ln\|{{(df)_{p}}_{|E^{s}_{p}}}\|^{-1}}{\ln|\partial_{u}f(p)|}\geq
2+\alpha.$

This bunching condition implies that the stable lamination is $C^{2+\alpha}$.
Also, notice that in dimension 2, the bunching condition becomes

$\forall p\in\Omega,\ |\det(df)_{p}|<1.$

That is, we only need the dynamical system $(\Omega,f)$ to be dissipative.

######  Theorem 1.2.

Let $M$ be a closed manifold. Let $f:M\rightarrow M$ be a $C^{2+\alpha}$ Axiom
A diffeomorphism. Suppose that $f$ has an attractor $\Omega$ with codimension
1 stable lamination. Moreover, suppose that $f$ satisfies the nonlinearity
condition (NL) and the bunching condition (B). Then the following holds.

Let $\mu$ be an equilibrium state for some Hölder potential. For any
$C^{\alpha}$ map $\chi:\Omega\rightarrow\mathbb{R}$ and for any $C^{1+\alpha}$
map $\phi:\Omega\rightarrow\mathbb{R}$ such that $\underset{\text{supp}\
\chi}{\inf}|\partial_{u}\phi|>0$, we have

$\exists C>1,\exists\varepsilon>0,\ \forall\xi\in\mathbb{R},\
\left|\int_{\Omega}e^{i\xi\phi}\chi\ d\mu\right|\leq
C(1+|\xi|)^{-\varepsilon}.$

######  Corollary 1.3.

Every attractor for nonlinear and dissipative Axiom A diffeomorphism in
dimension 2 exhibit polynomial Fourier decay of equilibrium states in the
unstable direction.

###  1.3 Strategy of the proof

The goal is to find a way to adapt the existing method for expanding maps in
the case of Axiom A diffeomorphisms. For this, we use the standard
construction using Markov partitions and then reduce the problem to the one-
dimensional case.

  * •

In section 2, we collect facts about thermodynamic formalism in the context of
Axiom A diffeomorphisms and recall classic constructions. The section 2.7 is
devoted to a preliminary regularity result for equilibrium states in our
context. In the section 2.8 we state a large deviation result about Birkhoff
sums.

  * •

In the section 3 we use the large deviations to derive order of magnitude for
some dynamically-related quantities.

  * •

The proof of Theorem 1.2 begins in the section 4. Using the invariance of the
equilibrium state by the dynamics, we first reduce the Fourier transform to an
integral on a union of local unstable manifolds, thus reducing the problem to
a one dimensional expanding map.

  * •

We use a transfer operator to carefully approximate the integral by a sum of
exponentials. We then use a generalized version of Theorem 1.1.

  * •

We prove the non-concentration hypothesis that is needed to conclude in
section 6.

###  1.4 Acknowledgments

I would like to thank my PhD supervisor, Frederic Naud, for numerous helpful
conversations and for pointing out various helping references, especially on
the topic of total non-linearity and Dolgopyat’s estimates. I would also like
to thank Sébastien Gouëzel and Christophe Dupont for some interesting
discussions at the seminar of ergodic theory in Rennes.

##  2 Thermodynamic formalism for Axiom A diffeomorphisms

###  2.1 Axiom A diffeomorphisms and basic sets

We recall standard results about Axiom A diffeomorphisms. An introduction to
the topic can be found in [BS02]. A more in-depth study can be found in
[KH95]. For an introduction to the thermodynamic formalism of Axiom A
diffeomorphisms, we suggest the classic lectures notes of Bowen [Bo75].

######  Definition 2.1.

Let $f:M\rightarrow M$ be a diffeomorphism of a closed $\mathcal{C}^{\infty}$
riemannian manifold $M$. A compact set $\Lambda\subset M$ is said to be
hyperbolic for $f$ if $f(\Lambda)=\Lambda$, and if for each $x\in\Lambda$, the
tangent space $T_{x}M$ can be written as a direct sum

$T_{x}M=E^{s}_{x}\oplus E^{u}_{x}$

of subspaces such that

  1. 1.

$\forall x\in\Lambda$, $(df)_{x}(E^{s}_{x})=E^{s}_{f(x)}$ and
$(df)_{x}(E^{u}_{x})=E^{u}_{f(x)}$

  2. 2.

$\exists C>0,\ \exists\kappa\in(0,1),\ \forall x\in\Lambda,$

$\forall v\in E^{s}_{x},\ \forall n\geq 0,\ \|(df^{n})_{x}(v)\|\leq
C\kappa^{n}\|v\|$

and

$\forall v\in E^{u}_{x},\ \forall n\geq 0,\ \|(df^{n})_{x}(v)\|\geq
C^{-1}\kappa^{-n}\|v\|.$

It then follows that $E_{x}^{s}$ and $E_{x}^{u}$ are continuous sub-bundles of
$T_{x}M$.

######  Remark 2.1.

We can always choose the metric so that $C=1$ in the previous definition: this
is called an adapted metric (or a Mather metric) and we will fix one from now
on. See [BS02] for a quick proof.

######  Definition 2.2.

A point $x\in M$ is called non-wandering if, for any open neighborhood $U$ of
$x$,

$U\cap\bigcup_{n>0}f^{n}(U)\neq\emptyset.$

We denote the set of all non-wandering points by $\Omega(f)$.

It is easy to check that the non-wandering set of $f$ is a compact invariant
subset of $M$. Also, any periodic point of $f$ can be seen to lie in
$\Omega(f)$. The definition of Axiom A diffeomorphisms is chosen so that the
dynamical system $(f,\Omega(f))$ exhibit a chaotic behavior similar to the one
found in symbolic dynamics. Namely:

######  Definition 2.3.

A diffeomorphism $f:M\rightarrow M$ is said to be Axiom A if

  * •

$\Omega(f)$ is a hyperbolic set for $f$,

  * •

$\Omega(f)=\overline{\\{x\in M\ |\ \exists n>0,\ f^{n}(x)=x\\}}.$

In general, the non-wandering set of an Axiom A diffeomorphism can be written
as the union of smaller invariant compact sets. Those are the sets on which we
usually work.

######  Theorem 2.1.

One can write $\Omega(f)=\Omega_{1}\cup\dots\Omega_{k}$, where $\Omega_{i}$
are nonempty compact disjoint sets, such that

  * •

$f(\Omega_{i})=\Omega_{i}$, and $f_{|\Omega_{i}}$ is topologically transitive,

  * •

$\Omega_{i}=X_{1,i}\cup\dots\cup X_{r_{i},i}$ where the $X_{j,i}$ are disjoint
compact sets, $f(X_{j,i})=X_{j+1,i}$ ($X_{r_{i}+1,i}=X_{1,i}$) and
$f^{n_{i}}_{|X_{j,i}}$ are all topologically mixing.

The sets $\Omega_{i}$ are called basic sets.

######  Remark 2.2.

Notice that any basic set $\Omega$ that satisfies the nonlinearity hypothesis
(NL) is necessarily a perfect set. Indeed, if $p\in\Omega$ were an isolated
point, then by transitivity $p$ should have dense orbit in $\Omega$, but at
the same time be a periodic point. In would imply that $\Omega$ is composed of
a unique periodic orbit, which is forbidden by the nonlinearity hypothesis.

For this section, we fix a basic set (not necessarily an attractor)
$\Omega\subset M$ for an Axiom A diffeomorphism $f$.

###  2.2 Equilibrium states

######  Definition 2.4.

We say that a map $\varphi:X\subset M\rightarrow\mathbb{R}$ is Hölder if there
exists $\alpha\in(0,1)$ and $C>0$ such that $\forall x,y\in X,\
|\varphi(x)-\varphi(y)|\ \leq Cd(x,y)^{\alpha}$, where $d$ is the natural
geodesic distance induced by the metric on $M$. We note $C^{\alpha}(X)$ the
set of $\alpha$-Hölder maps on $X$. If $\alpha>1$, we denote by
$C^{\alpha}(M)$ the set of functions that are $\lfloor\alpha\rfloor$ times
differentiable, with $(\alpha-\lfloor\alpha\rfloor)$-Hölder derivatives.

######  Remark 2.3.

A theorem by McShane [Mc34] proves that any $\alpha$-Hölder map defined on a
subset of $M$ can always be extended to an $\alpha$-Hölder map on all $M$.
Hence, even if for some definitions the potentials $\varphi$ need only to be
defined on $\Omega$, we will always be able to consider them as maps in
$C^{\alpha}(M)$ if necessary.

######  Definition 2.5.

[Bo75], [Ru78] Let $\psi:\Omega\rightarrow\mathbb{R}$ be a Hölder potential.
Define the pressure of $\psi$ by

$P(\psi):=\sup_{\mu\in\mathcal{P}_{f}(\Omega)}\left\\{h_{f}(\mu)+\int_{\Omega}\psi
d\mu\right\\},$

where $\mathcal{P}_{f}(\Omega)$ is the compact set of all probability measures
supported on $\Omega$ that are $f$-invariant, and where $h_{f}(\mu)$ is the
entropy of $\mu$. There exists a unique measure $\mu_{\psi}\in M_{f}(\Omega)$
such that

$P(\psi)=h_{f}(\mu_{\psi})+\int_{\Omega}\psi d\mu_{\psi}.$

This measure has support equal to $\Omega$, is ergodic on $(\Omega,f)$, and is
called the equilibrium state associated to $\psi$.

Two particular choices of potentials stands among the others. The first one is
the constant potential $\psi:=C$. In this case, the equilibrium measure is the
measure of maximal entropy (which is known to be linked with the repartition
of periodic orbits, see for example [Be16]).

Another natural choice is the geometric potential

$\psi:=-\log|\det(df)_{|u}|,$

where $\det(df)_{|u}$ stands for the determinant of the linear map
${(df)_{x}}:E^{u}_{x}\rightarrow E^{u}_{f(x)}$, which is well defined up to a
sign since the $E^{u}$ are equipped with a scalar product. In the case where
$\Omega$ is an attractor, that is, if there exists a neighborhood $U$ of
$\Omega$ such that

$\bigcap_{n\geq 0}f^{n}(U)=\Omega,$

then the associated equilibrium measure is called a SRB measure. In this case,
for any continuous function $g:U\rightarrow\mathbb{R}$ and for lebesgue almost
all $x\in U$,

$\frac{1}{n}\sum_{k=0}^{n-1}g(f^{k}(x))\underset{n\rightarrow\infty}{\longrightarrow}\int_{\Omega}g\
d\mu_{SRB},$

which allows us to think of this measure as the physical equilibrium state of
the dynamical system. See [Yo02] for more details on SRB measures, and also
the last chapter of [Bo75].

###  2.3 Stable/unstable laminations, bracket and holonomies

In this subsection, we will recall some results about the existence of
stable/unstable laminations, some regularity results in our particular case,
and the consequence on the regularity of the holonomies.

######  Definition 2.6.

Let $x\in\Omega$. For $\varepsilon>0$ small enough, we define the local stable
and unstable manifold at $x$ by

$W^{s}_{\epsilon}(x):=\\{y\in M\ |\ \forall n\geq 0,\
d(f^{n}(x),f^{n}(y))\leq\varepsilon\\},$ $W^{u}_{\varepsilon}(x):=\\{y\in M\
|\ \forall n\leq 0,\ d(f^{n}(x),f^{n}(y))\leq\varepsilon\\}.$

We also define the global stable and unstable manifolds at $x$ by

$W^{s}(x):=\\{y\in M\ |\
d(f^{n}(x),f^{n}(y))\underset{n\rightarrow+\infty}{\longrightarrow}0\\},$
$W^{u}(x):=\\{y\in M\ |\
d(f^{n}(x),f^{n}(y))\underset{n\rightarrow-\infty}{\longrightarrow}0\\}.$

######  Theorem 2.2 ([BS02], [KH95], [Bo75]).

Let $f$ be a $\mathcal{C}^{r}$ Axiom A diffeomorphism and let $\Omega$ be a
basic set. For $\varepsilon>0$ small enough and for $x\in\Omega$:

  * •

$W^{s}_{\varepsilon}(x)$ and $W^{u}_{\varepsilon}(x)$ are $\mathcal{C}^{r}$
embedded disks,

  * •

$\forall y\in\Omega\cap W^{s}_{\varepsilon}(x),\
T_{y}W^{s}_{\varepsilon}(x)=E^{s}_{y}$,

  * •

$\forall y\in\Omega\cap W^{u}_{\varepsilon}(x),\
T_{y}W^{u}_{\varepsilon}(x)=E^{u}_{y}$,

  * •

$f(W^{s}_{\varepsilon}(x))\subset\subset W^{s}_{\varepsilon}(f(x))$ and
$f(W^{u}_{\varepsilon}(x))\supset\supset W^{u}_{\varepsilon}(f(x))$ where
$\subset\subset$ means compactly included,

  * •

$\forall y\in W^{s}_{\varepsilon}(x),\ \forall n\geq 0,\
d^{s}(f^{n}(x),f^{n}(y))\leq\kappa^{n}d^{s}(x,y)$ where $d^{s}$ denotes the
geodesic distance on the submanifold $W^{s}_{\varepsilon}$,

  * •

$\forall y\in W^{u}_{\varepsilon}(x),\ \forall n\geq 0,\
d^{u}(f^{-n}(x),f^{-n}(y))\leq\kappa^{n}d^{u}(x,y)$ where $d^{u}$ denotes the
geodesic distance on the submanifold $W^{u}_{\varepsilon}$.

Moreover

$\bigcup_{n\geq 0}f^{-n}(W^{s}_{\varepsilon}(f^{n}(x)))=W^{s}(x)$

and

$\bigcup_{n\geq 0}f^{n}(W^{u}_{\varepsilon}(f^{-n}(x)))=W^{u}(x),$

and so the global stable and unstable manifolds are injectively immersed
manifolds in $M$.

The family $(W^{s}_{\varepsilon}(x),W^{u}_{\varepsilon}(x))_{x\in\Omega}$
forms two transverse continuous laminations, which allows us to define the so-
called bracket and holonomies maps.

######  Definition 2.7 ([Bo75], [KH95], [BS02]).

For $\varepsilon>0$ small enough, there exists $\delta>0$ such that
$W^{s}_{\varepsilon}(x)\cap W^{u}_{\varepsilon}(y)$ consists of a single point
$[x,y]$ whenever $x,y\in\Omega$ and $d(x,y)<\delta$. In this case,
$[x,y]\in\Omega$, and the map

$[\cdot,\cdot]:\\{(x,y)\in\Omega\times\Omega\ ,\
d(x,y)<\delta\\}\longrightarrow\Omega$

is continuous. Moreover, there exists $C>0$ such that

$d^{s}([x,y],x)\leq Cd(x,y),\quad\text{and}\quad d^{u}([x,y],y)\leq Cd(x,y).$

######  Definition 2.8.

Fix $x,y$ two close enough points in $\Omega$ lying in the same local stable
manifold. Let $U^{u}\subset W^{u}_{\varepsilon}(x)\cap\Omega$ be a small open
neighborhood of $x$ relatively to $W^{u}_{\varepsilon}(x)\cap\Omega$. The map

$\pi_{x,y}:U^{u}\subset W^{u}_{\varepsilon}(x)\cap\Omega\longrightarrow
W^{u}_{\varepsilon}(y)\cap\Omega.$

defined by $\pi_{x,y}(z):=[z,y]$ is called a stable holonomy map. One can
define an unstable holonomy map similarly on pieces of local stable manifolds
intersected with $\Omega$. Since the stable and unstable laminations are
Hölder regular in the general case, those honolomy maps are only Hölder
regular in general ([KH95], theorem 19.1.6).

To work, we will need a bit of regularity on the unstable holonomies.
Fortunately for us, in the particular case where the stable lamination are
codimension 1, we have some regularity results: this is Theorem 1 page 25 of
[Ha89], and Theorem 19.1.11 in [KH95] for Anosov diffeomorphisms. For general
hyperbolic sets, the proof is done is [PR02].

######  Theorem 2.3.

Let $f$ be an Axiom A diffeomorphism, and let $\Omega$ be a basic set. Suppose
that $f$ has codimension one stable laminations, that is, $\dim E^{u}_{x}=1$
for all $x\in\Omega$. Then the stable lamination is $C^{1+\alpha}$ for some
$\alpha>0$. In particular, the stable holonomies maps are $C^{1+\alpha}$
diffeomorphisms.

######  Remark 2.4.

Here, the holonomies being $C^{1+\alpha}$ means that the map $\pi_{x,y}$
extends to small curves $W^{u}_{\delta}(x)\longrightarrow
W^{u}_{\varepsilon}(y)$, and that the extended map is a $C^{1+\alpha}$
diffeomorphism.

From now on, we will work under the assumption that $f$ has codimension one
stable laminations. This is always true if $\dim M=2$, or if $\dim M=3$ by
exchanging $f$ by $f^{-1}$ if necessary. We fix a Hölder potential
$\psi:\Omega\rightarrow\mathbb{R}$ and its associated equilibrium state $\mu$.

###  2.4 Markov partitions

In this subsection, we will construct the topological space on which we will
work in this paper: the hypothesis made on the dimension of the unstable
lamination will allow us to approximate the dynamics by a dynamical system on
a finite disjoint union of smooth curves. For this we need to recall some
results about Markov partitions.

######  Definition 2.9.

A set $R\subset\Omega$ is called a _rectangle_ if

$\forall x,y\in R,\ [x,y]\in R.$

A rectangle is called proper if $\overline{\text{int}_{\Omega}(R)}=R$. If
$x\in R$, and if $\text{diam}(R)$ is small enough with respect to
$\varepsilon$, we define

$W^{s}(x,R):=W^{s}_{\varepsilon}(x)\cap R\quad\text{and}\quad
W^{u}(x,R):=W^{u}_{\varepsilon}(x)\cap R.$

######  Remark 2.5.

Notice that a rectangle isn’t always connected, and might even have an
infinite number of connected component, even if $\Omega$ is itself connected.
This technicality is noticed in [Pe19], at the first paragraph of subsection
3.3, and is an obstruction to the existence of finite Markov partition with
connected elements.

######  Definition 2.10.

A Markov partition of $\Omega$ is a finite covering
$\\{R_{a}\\}_{a\in\mathcal{A}}$ of $\Omega$ by proper rectangles such that

  * •

$\text{int}_{\Omega}R_{a}\cap\text{int}_{\Omega}R_{b}=\emptyset$ if $a\neq b$

  * •

$f\left(W^{u}(x,R_{a})\right)\supset W^{u}(f(x),R_{b})$ and
$f\left(W^{s}(x,R_{a})\right)\subset W^{s}(f(x),R_{b})$
when $x\in\text{int}_{\Omega}R_{a}\cap
f^{-1}\left(\text{int}_{\Omega}R_{a}\right)$.

######  Theorem 2.4 ([Bo75],[KH95]).

Let $\Omega$ be a basic set for an Axiom A diffeomorphism $f$. Then $\Omega$
has Markov partitions of arbitrary small diameter.

From now on, we fix once and for all a Markov partition
$\\{R_{a}\\}_{a\in\mathcal{A}}$ of $\Omega$ with small enough diameter.
Remember that, since $\Omega$ is not an isolated cycle, it is a perfect set.
In particular, $\text{diam}R_{a}>0$ for all $a\in\mathcal{A}$.

######  Definition 2.11.

We fix for the rest of this paper some periodic points
$x_{a}\in\text{int}_{\Omega}R_{a}$ for all $a\in\mathcal{A}$ (this is possible
by density of such points in $\Omega$). By periodicity, $x_{a}\notin
W^{u}(x_{b})$ when $a\neq b$.

######  Definition 2.12.

We set, for all $a\in\mathcal{A}$,

$S_{a}:=W^{s}(x_{a},R_{a})\quad\text{and}\quad U_{a}:=W^{u}(x_{a},R_{a}).$

They are closed sets included in $R_{a}$, and are defined so that
$[U_{a},S_{a}]=R_{a}$. They will allow us to decompose the dynamics into a
stable and an unstable part. Notice that the decomposition is unique:

$\forall x\in R_{a},\exists!(y,z)\in U_{a}\times S_{a},\ x=[y,z].$

The intuition of the construction to come is that, after a large enough number
of iterates, $f^{n}$ can be approximated by a map that is only defined on the
$(U_{a})_{a\in\mathcal{A}}$.

###  2.5 A factor dynamics

In this section we construct what will take the role of the shift map in our
context. The construction is inspired by the symbolic case and already appear
in the work of Dolgopyat [Do98].

######  Notations 2.6.

Let $a$ and $b$ be two letters in $\mathcal{A}$. We note $a\rightarrow b$ if
$f(\text{int}_{\Omega}R_{a})\cap\text{int}_{\Omega}R_{b}\neq\emptyset$.

######  Definition 2.13.

We define

$\mathcal{R}:=\bigsqcup_{a\in A}R_{a},\quad\mathcal{S}:=\bigsqcup_{a\in
A}S_{a},\quad\mathcal{U}:=\bigsqcup_{a\in A}U_{a}$

where $\bigsqcup$ denote a formal disjoint union. we also define

$\mathcal{R}^{(0)}:=\bigsqcup_{a\in\mathcal{A}}\text{int}_{\Omega}R_{a}\subset\mathcal{R}$

and

$\mathcal{R}^{(1)}:=\bigsqcup_{a\rightarrow b}(\text{int}_{\Omega}R_{a})\cap
f^{-1}(\text{int}_{\Omega}R_{b})\subset\mathcal{R}^{(0)}$

So that the map $f:\Omega\rightarrow\Omega$ may be naturally seen as a map
$f:\mathcal{R}^{(1)}\longrightarrow\mathcal{R}^{(0)}$. We then define

$\mathcal{R}^{(k)}:=f^{-k}(\mathcal{R}^{(0)})$

and, finally, we denote the associated residual set by

$\widehat{\mathcal{R}}:=\bigcap_{k\geq 0}\mathcal{R}^{(k)}$

so that $f:\widehat{\mathcal{R}}\longrightarrow\widehat{\mathcal{R}}$. Seen as
a subset of $\Omega$, $\widehat{\mathcal{R}}$ as full measure, by ergodicity
of the equilibrium measure $\mu$. Hence $\mu$ can naturally be though as a
probability measure on $\widehat{\mathcal{R}}$.

######  Definition 2.14.

Let $\mathcal{R}/\mathcal{S}$ be the topological space defined by the
equivalence relation $x\sim y\Leftrightarrow\exists a\in\mathcal{A},\ y\in
W^{s}(x,R_{a})$ in $\mathcal{R}$. Let
$\pi:\mathcal{R}\rightarrow\mathcal{R}/\mathcal{S}$ denotes the natural
projection. The map $f:\widehat{\mathcal{R}}\rightarrow\widehat{\mathcal{R}}$
induces a factor map
$F:\widehat{\mathcal{R}}/\mathcal{S}\rightarrow\widehat{\mathcal{R}}/\mathcal{S}$.
Moreover, the measure $\nu:=\pi_{*}\mu$ is a $F$-invariant probability measure
on $\mathcal{R}/\mathcal{S}$.

###### Proof.

We just have to check that
$f:\mathcal{R}^{(1)}\longrightarrow\mathcal{R}^{(0)}$ satisfy

$fW^{s}(x,R_{a})\subset W^{s}(f(x),R_{b})$

for $x\in(\text{int}_{\Omega}R_{a})\cap f^{-1}(\text{int}_{\Omega}R_{b})$.
This is true by definition of Markov partitions. The induced map satisfy
$F\circ\pi=\pi\circ f$, and so $F_{*}\nu=\nu$. ∎

######  Remark 2.7.

There is a natural isomorphism $\mathcal{U}\simeq\mathcal{R}/\mathcal{S}$ that
is induced by the inclusion $\mathcal{U}\hookrightarrow\mathcal{R}$. This
allows us to identify all the precedent construction to a dynamical system on
$\mathcal{U}$. Namely:

  * •

The projection $\pi:\mathcal{R}\rightarrow\mathcal{R}/\mathcal{S}$ is
identified with

$\begin{array}[t]{lrcl}\pi:&\mathcal{R}&\longrightarrow&\quad\mathcal{U}\\\
&x\in R_{a}&\longmapsto&[x,x_{a}]\in U_{a}\end{array}$

  * •

The factor map
$F:\widehat{\mathcal{R}}/\mathcal{S}\rightarrow\widehat{\mathcal{R}}/\mathcal{S}$
is identified with

$\begin{array}[t]{lrcl}F:&\widehat{\mathcal{U}}&\longrightarrow&\quad\widehat{\mathcal{U}}\\\
&x\in R_{a}\cap f^{-1}(R_{b})&\longmapsto&[f(x),x_{b}]\in U_{b}\end{array}$

where $\widehat{\mathcal{U}}$ is defined similarly to $\widehat{\mathcal{R}}$,
but with $F$ replacing $f$ in the construction.

  * •

The measure $\nu$ is identified to the unique measure on $\mathcal{U}$ such
that:

$\forall h\in C^{0}(\mathcal{R},\mathbb{R})\ \text{S-constant},\
\int_{\mathcal{U}}hd\nu\ =\ \int_{\mathcal{R}}hd\mu$

where $S$-constant means that $\forall a\in\mathcal{A},\forall x,y\in R_{a},\
x\in W^{s}(y,R_{a})\Rightarrow h(x)=h(y)$.

######  Remark 2.8.

Since our centers $x_{a}$ are periodic, $x_{a}\in\widehat{\mathcal{R}}$, and
hence $x_{a}\in\widehat{\mathcal{U}}$.

######  Lemma 2.5.

Under the nonlinearity hypothesis, the set $\widehat{\mathcal{U}}$ is a
perfect set. In particular, $\text{diam}\ U_{a}>0$ for all $a\in\mathcal{A}$.

###### Proof.

First of all, we prove that $\widehat{\mathcal{U}}$ is infinite. By
hypothesis, there exists an infinite family of distinct periodic orbits for
$f$ in $\Omega$. Since $f$ is Axiom A, the set of periodic orbits for $f$ is
dense in $\Omega$. In particular, the set of $f$-periodic points in
$\mathcal{R}^{(0)}$ is infinite.

Let $x\neq y\in R_{a}$ be two $f$-periodic points with distinct periodic
orbits. Then $\pi(x)$ and $\pi(y)$ are $F$-periodic with distinct periodic
orbits. (In particular, they are in $\widehat{\mathcal{U}}$.) Indeed, if there
existed some $n_{0}$ and $m_{0}$ such that
$\pi(f^{n_{0}}(x))=\pi(f^{m_{0}}y)$, then by definition of $\pi$ we would have
$f^{n_{0}}(x)\in W^{s}(f^{m_{0}}y)$. Hence
$d(f^{n+n_{0}}(x),f^{n+m_{0}}(y))\underset{n\rightarrow+\infty}{\longrightarrow}0$,
a contradiction.

So $\widehat{\mathcal{U}}$ is an infinite set. Now we justify that
$F_{|\widehat{\mathcal{U}}}$ is topologically transitive.

We see by the Baire category theorem that $\widehat{\mathcal{R}}$ is a dense
subset of $\Omega$. Moreover,
$f(\widehat{\mathcal{R}})=\widehat{\mathcal{R}}$. It then follows from the
transitivity of $f_{|\Omega}$ that $f_{|\widehat{\mathcal{R}}}$ is
topologically transitive. Hence, $F_{|\widehat{\mathcal{U}}}$ is topologically
transitive, as a factor of $f_{|\widehat{\mathcal{R}}}$. We conclude the proof
using the argument of remark 2.2. ∎

######  Remark 2.9.

The fact that $\widehat{\mathcal{U}}$ is perfect, combined with the fact that
holonomies extend to $C^{1+\alpha}$ maps, allows us to consider for
$x\in\mathcal{R}$ the (absolute value of the) derivatives along the unstable
direction of the holonomies in a meaningful way (without having to chose an
extension). We denote them by $\partial_{u}\pi(x)$. (Formally,
$|\partial_{u}\pi(x)|:=|(d\pi)_{x}(\vec{n})|$ where $\vec{n}$ is a unit vector
in $E^{u}_{x}$ and where $|\cdot|$ is the norm on $E^{u}_{x}$.)

It is then known that those quantities are uniformly bounded [PR02]. In other
words:

$\exists c,C>0,\forall x\in\mathcal{R},\ c<|\partial_{u}\pi(x)|\leq C.$

######  Remark 2.10.

The map $F:\widehat{\mathcal{U}}\rightarrow\widehat{\mathcal{U}}$ is an
eventually expanding map.

Indeed, let $n\geq 0$. By Theorem 2.2, if $x,y\in\widehat{\mathcal{U}}$ are
close enough depending on $n$, then
$d(f^{n}(x),f^{n}(y))\geq\kappa^{-n}d(x,y)$ and $f^{n}(x),f^{n}(y)$ are still
in the same $R_{b}$. Since $F^{n}=\pi\circ f^{n}$, and since $\pi$ has bounded
derivatives, we get:

$\exists c>0,\ \forall n,\ \forall(x,y)\ \text{close enough},\
d^{u}(F^{n}x,F^{n}y)\geq c\kappa^{-n}d^{u}(x,y).$

###  2.6 A transfer operator

The dynamical system $(F,\widehat{\mathcal{U}},\nu)$ is constructed to behave
like a semi-shift, and so it is natural to search for a transfer operator
defined on $\mathcal{U}$ that leaves $\nu$ invariant. Recall that $\mu$ is the
equilibrium measure for some Hölder potential $\psi$ on $M$: the fact that
$\psi$ isn’t necessarily $S$-constant suggest that we will have to search for
another potential, cohomologous to $\psi$, that is $S$-invariant. Looking at
the symbolic case, we notice from the proof of the proposition 1.2 in [PP90]
that the potential $\varphi_{0}\in C^{\alpha}(\mathcal{R},\mathbb{R})$ (take
$\alpha$ smaller if necessary) defined by the formula

$\varphi_{0}(x):=\psi(\pi(x))+\sum_{n=0}^{\infty}\left(\psi(f^{n+1}\pi
x)-\psi(f^{n}\pi fx)\right)\quad\quad\quad$

might be interesting to consider. First of all, recall what is a transfer
operator.

######  Definition 2.15.

For some Hölder potential $\varphi:\mathcal{U}\rightarrow\mathbb{R}$, define
the associated _transfer operator_
$\mathcal{L}_{\varphi}:C^{0}(\mathcal{U}^{(0)},\mathbb{C})\longrightarrow
C^{0}(\mathcal{U}^{(1)},\mathbb{C})$ by

$\forall x\in\mathcal{U}^{(0)},\ \mathcal{L}_{\varphi}h(x):=\sum_{y\in
F^{-1}(x)}e^{\varphi(y)}h(y).$

Iterating $\mathcal{L}_{\varphi}$ gives

$\forall x\in\mathcal{U}^{(0)},\ \mathcal{L}_{\varphi}^{n}h(x)=\sum_{y\in
F^{-n}(x)}e^{S_{n}\varphi(y)}h(y),$

where $S_{n}\varphi(z):=\sum_{k=0}^{n-1}\varphi(F^{k}(z))$ is a Birkhoff sum.

By duality, $\mathcal{L}_{\varphi}$ also acts on the set of measures on
$\mathcal{U}$. If $m$ is a measure on $\mathcal{U}^{(1)}$, then
$\mathcal{L}_{\varphi}^{*}m$ is the unique measure on $\mathcal{U}^{(0)}$ such
that

$\forall h\in C^{0}(\mathcal{U}^{(1)},\mathbb{C}),\ \int_{\mathcal{U}}h\
d\mathcal{L}_{\varphi}^{*}m=\int_{\mathcal{U}}\mathcal{L}_{\varphi}h\ dm.$

######  Remark 2.11.

We may rewrite the definition by highlighting the role of the inverse branches
of $F$. For some $a\rightarrow b$, we see that $F:U_{ab}^{(1)}\rightarrow
U_{b}^{(0)}$ is (the restriction of) a diffeomorphism. We denote by
$g_{ab}:U_{b}\rightarrow U_{ab}$ its local inverse, that can be defined by the
formula

$\forall x\in U_{b},\ g_{ab}(x):=f^{-1}([x,f(x_{a})]).$

Then the transfer operator can be rewritten as follow:

$\forall x\in U_{b},\ \mathcal{L}_{\varphi}h(x)=\sum_{a\rightarrow
b}e^{\varphi(g_{ab}(x))}h(g_{ab}(x)).$

######  Theorem 2.6.

There exists a Hölder function $h:\mathcal{U}\rightarrow\mathbb{R}$ such that
$\mathcal{L}_{\varphi_{0}}^{*}(h\nu)=e^{P(\psi)}h\nu$, where

$\varphi_{0}(x):=\psi(\pi(x))+\sum_{n=0}^{\infty}\left(\psi(f^{n+1}\pi
x)-\psi(f^{n}\pi fx)\right).$

###### Proof.

This fact follows from the symbolic setting, but a quick and clean proof can
also be achieved using the following geometrical results, extracted from
[Le00] and [Cl20]. See in particular Theorem 3.10 in [Cl20]. Define, when
$x\in W^{s}(y)$,

$\omega^{+}(x,y):=\sum_{n=0}^{\infty}\left(\psi(f^{n}(x))-\psi(f^{n}(y))\right),$

and when $x\in W^{u}(y)$,

$\omega^{-}(x,y):=\sum_{n=0}^{\infty}\left(\psi(f^{-n}(x))-\psi(f^{-n}(y))\right).$

There exists two families of (nonzero and) finite measures $m_{x}^{s}$,
$m_{x}^{u}$ indexed on $x\in\mathcal{R}^{(0)}$ such that
$\text{supp}(m_{x}^{u})\subset W^{u}(x,R_{a})$ and
$\text{supp}(m_{x}^{s})\subset W^{s}(x,R_{a})$ if $x\in R_{a}$, and that
satisfies the following properties:

  * •

If $\pi$ is a stable holonomy map between $W^{u}(y,R_{a})$ and
$W^{u}(\pi(y),R_{a})$, then

$\frac{d\left(\pi_{*}m_{y}^{u}\right)}{dm_{\pi(y)}^{u}}(\pi(x))=e^{\omega^{+}(x,\pi
x)}.$

and a similar formula holds for the $m_{y}^{s}$ with an unstable holonomy map.

  * •

The family of measures is $f$-conformal:

$\frac{d\left(f_{*}m_{y}^{u}\right)}{dm_{f(y)}^{u}}(f(x))=e^{\psi(x)-P(\psi)}$

and a similar formula holds for the $m^{s}_{y}$.

Finally, this family of measures is related to $\mu$ in the following way.
There exists positive constants $(c_{a})_{a\in\mathcal{A}}$ such that, for any
measurable map $g:M\rightarrow\mathbb{C}$, the following formula holds:

$\int_{M}gd\mu=\sum_{a\in\mathcal{A}}\int_{U_{a}}\int_{S_{a}}e^{\omega^{+}([x,y],x)+\omega^{-}([x,y],y)}g([x,y])c_{a}dm_{x_{a}}^{s}(y)dm_{x_{a}}^{u}(x).$

From this we can link $\nu$ and the family $m_{x_{a}}^{u}$ in the following
way. Define $h_{0}:\mathcal{U}\rightarrow\mathbb{R}$ by the formula

$\forall a\in\mathcal{A},\ \forall x\in U_{a},\
h_{0}(x):=c_{a}\int_{S_{a}}e^{\omega^{-}([x,y],y)+\omega^{+}([x,y],y)}dm_{x_{a}}^{s}(y)>0.$

Then, for any $S$-constant map $g:\mathcal{R}\rightarrow\mathbb{R}$, we have

$\int_{\mathcal{U}}gd\nu=\int_{M}gd\mu=\sum_{a\in\mathcal{A}}\int_{U_{a}}g(x)h_{0}(x)dm_{x_{a}}^{u}(x).$

It follows from the properties of the bracket and the fact that $\psi$ is
supposed Hölder that $h_{0}$ is Hölder. So $h:=h_{0}^{-1}$ is Hölder and
satisfy $h\nu=m^{u}$, where $m^{u}_{|U_{a}}:=m^{u}_{x_{a}}$. We check that
$m^{u}$ is an eigenmeasure for $\mathcal{L}_{\varphi_{0}}$. Fix $a\rightarrow
b\in\mathcal{A}$, and $G:U_{b}\rightarrow\mathbb{C}$ continuous. We have

$\int_{U_{b}}G(g_{ab}(x))dm_{x_{b}}^{u}(x)=\int_{U_{b}}G\left(f^{-1}[x,f(x_{a})]\right)dm_{x_{b}}^{u}(x)$
$=\int_{f(U_{ab})}G\left(f^{-1}x\right)e^{\omega^{+}(x,\pi
x)}dm_{f(x_{a})}^{u}(x)$ $=\int_{U_{ab}}G(x)e^{\omega^{+}(fx,\pi
fx)}e^{-\psi(x)+P(\psi)}dm^{u}_{x_{a}}(x).$
$=e^{P(\psi)}\int_{U_{ab}}G(x)e^{-\varphi_{0}(x)}dm_{x_{a}}^{u}.$

In particular, for any continuous $G:\mathcal{U}\rightarrow\mathbb{R}$:

$\int_{\mathcal{U}}\mathcal{L}_{\varphi_{0}}(G)dm^{u}=\sum_{b\in\mathcal{A}}\int_{U_{b}}\sum_{a\rightarrow
b}e^{\varphi_{0}(g_{ab}(x))}G(g_{ab}(x))dm^{u}_{x_{a}}$
$=e^{P(\psi)}\underset{a\rightarrow
b}{\sum_{a,b\in\mathcal{A}}}\int_{U_{ab}}G(x)dm^{u}_{x_{a}}=\int_{\mathcal{U}}Gdm^{u}.$

Which concludes the proof. ∎

######  Lemma 2.7.

Define $\varphi:=\varphi_{0}-\ln h\circ F+\ln h-P(\psi)$. Then
$\varphi:\mathcal{U}\rightarrow\mathbb{R}$ is Hölder and normalized, that is:
$\mathcal{L}_{\varphi}^{*}\nu=\nu$, $\mathcal{L}_{\varphi}1=1$ on
$\widehat{\mathcal{U}}$. Moreover, $\varphi\leq 0$ on $\widehat{\mathcal{U}}$,
and there exists an integer $N\geq 0$ such that $S_{N}\varphi<0$ on
$\widehat{\mathcal{U}}$.

###### Proof.

It follows from the definition of $\varphi$ that
$\mathcal{L}_{\varphi}^{*}\nu=\nu$. Then, we see that
$\mathcal{L}_{\varphi}1=1$ on $L^{2}(\nu)$. Indeed, for any $L^{2}(\nu)$ map
$G$, we have

$\int_{\mathcal{U}}\mathcal{L}_{\varphi}(1)Gd\nu=\int_{\mathcal{U}}\mathcal{L}_{\varphi}(G\circ
F)d\nu=\int_{\mathcal{U}}G\circ Fd\nu=\int_{\mathcal{U}}Gd\nu.$

Since $\mathcal{L}_{\varphi}1$ and $1$ are continuous functions on
$\widehat{\mathcal{U}}$, and since $\nu$ has full support, it follows that
$\mathcal{L}_{\varphi}1=1$ on all $\widehat{\mathcal{U}}$, and even on
$\mathcal{U}$ by density. We check that $\varphi$ is eventually negative on
the residual set. First of all, the equality $1=\mathcal{L}_{\varphi}(1)$
implies that $\varphi\leq 0$ on $\widehat{\mathcal{U}}$. Then, since $F$ is
uniformly expanding, there exists some integer $N$ such that $F^{-N}(x)$
always contains at least two distinct points, for any $x$. The equality
$1=\mathcal{L}_{\varphi}^{N}(1)$ then allows us to conclude that
$S_{N}\varphi<0$ on $F^{-N}(\widehat{\mathcal{U}})=\widehat{\mathcal{U}}$. ∎

###  2.7 Some regularity for $\nu$

In this subsection, we introduce the usual symbolic formalism, recall the
Gibbs estimates for $\nu$, and prove an upper regularity result for $\nu$. In
particular, we prove that $\nu$ don’t have atoms under our non-linearity
condition. Notice that if $\Omega$ were an isolated cycle, then the measure of
maximal entropy would be the Dirac on the cycle, which is fully discrete, so
this is not completely trivial.

######  Notations 2.12.

For $n\geq 1$, a word $\mathbf{a}=a_{1}\dots a_{n}\in\mathcal{A}^{n}$ is said
to be admissible if $a_{1}\rightarrow a_{2}\rightarrow\dots\rightarrow a_{n}$.
We define:

  * •

$\mathcal{W}_{n}:=\\{\mathbf{a}\in\mathcal{A}^{n}\ |\ \mathbf{a}\ \text{is
admissible}\ \\}$.

  * •

For $\mathbf{a}\in\mathcal{W}_{n}$, define
$g_{\mathbf{a}}:=g_{a_{1}a_{2}}g_{a_{2}a_{3}}\dots
g_{a_{n-1}a_{n}}:U_{a_{n}}\longrightarrow U_{a_{1}}$.

  * •

For $\mathbf{a}\in\mathcal{W}_{n}$, define
$U_{\mathbf{a}}:=g_{\mathbf{a}}\left(U_{a_{n}}\right)$,
$U_{\mathbf{a}}:=g_{\mathbf{a}}\left(U_{a_{n}}\right)$, and
$\widehat{U}_{\mathbf{a}}:=g_{\mathbf{a}}\left(\widehat{\mathcal{U}}\cap
U_{a_{n}}\right)$

  * •

For $\mathbf{a}\in\mathcal{W}_{n}$, set
$x_{\mathbf{a}}:=g_{\mathbf{a}}\left(x_{a_{n}}\right)\in\widehat{U}_{\mathbf{a}}$.

######  Remark 2.13.

Since $F$ is eventually expanding, the maps $g_{\mathbf{a}}$ are eventually
contracting as $n$ becomes large. As $\mathcal{U}$ is included in a finite
union of unstable curve, that are one dimensional riemannian manifolds, it
makes sense to consider absolute value of the derivatives of $F$ and
$g_{\mathbf{a}}$, and we will do it from now on. For points in $\mathcal{U}$,
this correspond to the absolute value of the derivative in the unstable
direction.

Since the holonomies are uniformly bounded, and by hyperbolicity of $f$, we
see that

$\forall x\in\widehat{\mathcal{U}},|\partial_{u}F^{n}(x)|\geq c\kappa^{-n}.$

Consequently,

$\forall\mathbf{a}\in\mathcal{W}_{n},\ \forall x\in U_{a_{n}},\
|g_{\mathbf{a}}^{\prime}(x)|\leq c^{-1}\kappa^{n}.$

Moreover, using remark 2.10, we see that

$\forall\mathbf{a}\in\mathcal{W}_{n},\text{diam}(U_{\textbf{a}})=\text{diam}\
g_{\mathbf{a}}\left(U_{a_{n}}\right)\leq c^{-1}\kappa^{n}.$

A consequence for our potential is that it has exponentially vanishing
variations. Namely, since $\varphi$ is Hölder, the following holds:

$\exists C>0,\ \forall n\geq 1,\ \forall\mathbf{a}\in\mathcal{W}_{n},\ \forall
x,y\in U_{\mathbf{a}},\ |\varphi(x)-\varphi(y)|\leq C\kappa^{\alpha n}.$

######  Remark 2.14.

For a fixed $n$, the family $(U_{\mathbf{a}})_{\mathbf{a}\in W_{n}}$ is a
partition of $\mathcal{U}$ (modulo a boundary set of zero measure). In
particular, for any continuous map $g:\mathcal{U}\rightarrow\mathbb{C}$, we
can write

$\int_{\mathcal{U}}gd\nu=\sum_{\mathbf{a}\in\mathcal{W}_{n}}\int_{U_{\mathbf{a}}}gd\nu=\sum_{\mathbf{a}\in\mathcal{W}_{n}}\int_{U_{\mathbf{a}}}gd\nu.$

######  Lemma 2.8 (Gibbs estimates, [PP90]).

$\exists C_{0}>1,\ \forall n\geq 1,\ \forall\mathbf{a}\in\mathcal{W}_{n},\
\forall x\in U_{\mathbf{a}},\ \
C_{0}^{-1}e^{S_{n}\varphi(x)}\leq\nu(U_{\mathbf{a}})\leq
C_{0}e^{S_{n}\varphi(x)}$

###### Proof.

We have

$\int_{\mathcal{U}}e^{-\varphi}\mathbb{1}_{U_{a_{1}\dots
a_{n}}}d\nu_{\varphi}=\int_{\mathcal{U}}\mathcal{L}_{\varphi}\left(e^{-\varphi}\mathbb{1}_{U_{a_{1}\dots
a_{n}}}\right)d\nu=\int_{\mathcal{U}}\mathbb{1}_{U_{a_{2}\dots
a_{n}}}d\nu=\nu(U_{a_{2}\dots a_{n}}).$

Moreover, since $\varphi$ has exponentially decreasing variations, we can
write that:

$\forall x\in U_{a_{1}\dots a_{n}},\ e^{-\varphi(x)-C\kappa^{\alpha
n}}\nu(U_{a_{1}\dots
a_{n}})\leq\int_{\mathcal{U}}e^{-\varphi}\mathbb{1}_{U_{a_{1}\dots
a_{n}}}d\nu\leq e^{-\varphi(x)+C\kappa^{\alpha n}}\nu(U_{a_{1}\dots a_{n}})$

And so

$\forall x\in U_{a_{1}\dots a_{n}},\ e^{-\varphi(x)-C\kappa^{\alpha
n}}\leq\frac{\nu(U_{a_{2}\dots a_{n}})}{\nu(U_{a_{1}\dots a_{n}})}\leq
e^{-\varphi(x)+C\kappa^{\alpha n}}.$

Multiplying those inequalities gives us the desired relation. ∎

######  Lemma 2.9.

The measure $\nu$ is upper regular, that is, there exists $C>0$ and
$\delta_{\text{up}}>0$ such that

$\forall x\in\mathcal{U},\ \forall r>0,\ \nu(B(x,r))\leq
Cr^{\delta_{\text{up}}}.$

Where $B(x,r)$ is a ball of center $x$ and radius $r$ in $\mathcal{U}$.

###### Proof.

Let $x\in\mathcal{U}$, and let $r>0$ be small enough. If
$B(x,r)\cap\widehat{\mathcal{U}}=\emptyset$ then $\nu(B(x,r))=0$ since
$\widehat{\mathcal{U}}$ has full measure, and the proof is done. If
$B(x,r)\cap\widehat{\mathcal{U}}\neq\emptyset$, then let $\tilde{x}\in
B(x,r)\cap\widehat{\mathcal{U}}$. We then see that
$\nu(B(x,r))\leq\nu(B(\tilde{x},2r)$.

Recall that $F$ has bounded derivative on $\mathcal{U}$:
$|F^{\prime}|_{\infty}<\infty$. In particular, there exists a constant
$\kappa_{1}\in(0,1)$ such that
$|(F^{n})^{\prime}|_{\infty}\leq\kappa_{1}^{-n}$. Hence, $\forall n\geq 1,\
\forall\mathbf{a}\in\mathcal{W}_{n},\
\text{inf}|g_{\mathbf{a}}^{\prime}|>\kappa_{1}^{n}$. In particular,
$\text{diam}(U_{\mathbf{a}})>\kappa_{1}^{n}\eta_{0}$, where
$\eta_{0}:=\min_{a\in\mathcal{A}}\text{diam}(U_{a})>0$ (This is where we use
the fact that $\widehat{\mathcal{U}}$ is perfect).

Recall also from the lemma 2.7 that there exists an integer $N$ such that
$S_{N}\varphi<0$ on $\widehat{\mathcal{U}}$. We then let $n(r)$ be the unique
integer $n$ such that

$\eta_{0}\kappa_{1}^{N(n+1)}<2r\leq\eta_{0}\kappa_{1}^{Nn}.$

Then, by construction,
$2r\leq\text{diam}(U_{\mathbf{a}}),\forall\mathbf{a}\in\mathcal{W}_{n(r)}$.
Since everything is one dimensional, it follows that $B(\tilde{x},2r)$ is
contained in at most two sets of the form $U_{\mathbf{a}}$,
$\mathbf{a}\in\mathcal{W}_{n}$. In particular, by the Gibbs estimates, there
exists eventually a $y\in\widehat{\mathcal{U}}$ such that

$\nu\left(B(x,r)\right)\leq
C_{0}e^{S_{Nn(r)}\varphi(x)}+C_{0}e^{S_{Nn(r)}\varphi(y)}.$

But then notice that
$S_{nN}\varphi(x)<-n\inf_{\widehat{\mathcal{U}}}|S_{N}\varphi|$, and similarly
for $S_{nN}\varphi(y)$. Hence:

$\nu\left(B(x,r)\right)\leq
2C_{0}e^{-n(r)\inf_{\widehat{\mathcal{U}}}|S_{N}\varphi|}\leq
Cr^{\delta_{\text{up}}}$

for some $C>0$ and $\delta_{\text{up}}>0$. ∎

In particular, it follows that $\nu$ don’t have atoms, which is some good news
since we are willing to prove Fourier decay. Moreover, we get a regularity
result on $\mu$ (that holds more generally if one replace $(NL)$ by
$\widehat{\mathcal{U}}$ is perfect).

######  Corollary 2.10.

Let $f$ be an Axiom A diffeomorphism, and let $\Omega$ be a basic set with
codimension one stable foliation. Suppose that $\Omega$ satisfies (NL). Then
any equilibrium state $\mu$ associated to Hölder potential
$\psi:\Omega\rightarrow\mathbb{R}$ is upper regular.

###### Proof.

Let $x\in\Omega$ and $r>0$ be small enough. Write
$B(x,r)=\bigcup_{a\in\mathcal{A}}B(x,r)\cap R_{a}$. Notice that $B(x,r)\cap
R_{a}\subset\\{y\in R_{a}\ |\ \exists z\in B(x,r)\cap R_{a},\
\pi(y)=\pi(z)\\}$, and so:

$\mu(B(x,r))=\sum_{a\in\mathcal{A}}\mu\left(B(x,r)\cap
R_{a}\right)\leq\sum_{a\in\mathcal{A}}\nu\left(\pi\left(B(x,r)\cap
R_{a}\right)\right)\leq|\mathcal{A}|C\|\partial_{u}\pi\|_{\infty,\mathcal{R}}^{\delta_{up}}(2r)^{\delta_{up}}\leq\tilde{C}r^{\delta_{up}}.$

using the previous lemma, and using the fact that
$\text{diam}_{u}\left(\pi\left(B(x,r)\cap R_{a}\right)\right)\leq
r\|\partial_{u}\pi\|_{\infty,\mathcal{R}}$. ∎

###  2.8  Large deviations

We finish this preliminary section by recalling some large deviation results.
There exists a large bibliography on the subject, see for example [Ki90].
Large deviations in the context of Fourier decay for some 1-dimensional shift
was used in [JS16] and in [SS20]. A simple proof in the context of Julia sets
using the pressure function can be found in the appendix of [Le21].

######  Theorem 2.11 ([LQZ03]).

Let $g:\Omega\rightarrow\mathbb{C}$ be any continuous map. Then, for all
$\varepsilon>0$, there exists $n_{0}(\varepsilon)$ and
$\delta_{0}(\varepsilon)>0$ such that

$\forall n\geq n_{0}(\varepsilon),\ \mu\left(\left\\{x\in\Omega,\
\left|\frac{1}{n}\sum_{k=0}^{n-1}g(f^{k}(x))-\int_{\Omega}gd\mu\right|\geq\varepsilon\right\\}\right)\leq
e^{-\delta_{0}(\varepsilon)n}$

Notice that applying Theorem 2.11 to a $S$-constant function immediately gives
the same statement for $(\nu,\widehat{\mathcal{U}},F)$. We apply it to some
special cases.

######  Definition 2.16.

Define $\tau_{f}:=\ln|\partial_{u}f(x)|$ (so that $-\tau_{f}$ is the geometric
potential) and $\tau_{F}:=\ln|F^{\prime}(x)|$. Then, define the associated
global Lyapunov exponents by:

$\lambda:=\int_{\Omega}\tau_{f}d\mu>0\quad,\quad\Lambda:=\int_{\mathcal{U}}\tau_{F}d\nu>0$

Also, define the dimension of $\nu$ by the formula

$\delta:=-\frac{1}{\Lambda}\int_{\mathcal{U}}\varphi d\nu>0.$

A consequence of Theorem 2.11 is the following:

######  Corollary 2.12 ([SS20], [Le21]).

For every $\varepsilon>0$, there exists $n_{1}(\varepsilon)$ and
$\delta_{1}(\varepsilon)>0$ such that

$\forall n\geq n_{1}(\varepsilon),\
\nu\left(\left\\{x\in\widehat{\mathcal{U}}\ ,\
\left|\frac{1}{n}S_{n}\tau_{F}(x)-\Lambda\right|\geq\varepsilon\text{ or
}\left|\frac{S_{n}\varphi(x)}{S_{n}\tau_{F}(x)}+\delta\right|\geq\varepsilon\right\\}\right)\leq
e^{-\delta_{1}(\varepsilon)n}.$

Those quantitative results tells us that, for most $x\in\Omega$,
$|\partial_{u}f^{n}(x)|$ have order of magnitude $\exp(\lambda n)$, and
$|(F^{n})^{\prime}(x)|$ have order of magnitude $\exp(\Lambda n)$. Studying
those two quantities is therefore central for us. An important (and natural)
remark is the following.

######  Lemma 2.13.

The global unstable Lyapunov exponents $\lambda$ and $\Lambda$ are equal. In
other words,

$\int_{\Omega}\ln|\partial_{u}f|d\mu=\int_{\mathcal{U}}\ln|F^{\prime}|d\nu.$

###### Proof.

The fact that $\pi\circ f=F\circ\pi$ implies that

$\tau_{f}=\tau_{F}(\pi(x))+\ln|\partial_{u}\pi(f(x))|-\ln|\partial_{u}\pi(x)|.$

In particular, $\tau_{f}$ and $\tau_{F}\circ\pi$ are $f$-cohomologous. This
implies the desired equality, by definition of the measure $\nu$, and by
$f$-invariance of $\mu$. ∎

##  3 Computing some orders of magnitudes

Starting now, we begin to do analysis. For this, it will be useful to work on
connected sets. Hence, we make from now on the assumption that $\Omega$ is an
attractor. A useful result in this case is the following.

######  Lemma 3.1.

Suppose that $\Omega$ is an attractor for $f$. Let $x\in\Omega$. Then
$W^{u}(x)\subset\Omega.$

In particular, in this setting, $(U_{a})_{a\in\mathcal{A}}$ is a finite
collection of compact connected smooth curves.

###### Proof.

Let $x\in\Omega$ and $y\in W^{u}(x)$. Since $\Omega$ is an attractor, there
exists $U$ an open neighborhood of $\Omega$ such that $\bigcap_{n\geq
0}f^{n}(U)=\Omega$. Since
$d(f^{-n}(x),f^{-n}(y))\underset{n\rightarrow+\infty}{\longrightarrow}0$,
there exists $N\geq 0$ so that $\tilde{y}:=f^{-N}(y)\in U$, and for all $n\geq
0$, $f^{-n}(\tilde{y})\in U$. We can then write
$\tilde{y}=f^{n}(f^{-n}(\tilde{y}))\in f^{n}(U)$ for all $n\geq 0$. Hence
$\tilde{y}\in\Omega$, and so $y\in\Omega$. ∎

This allows us to see holonomy maps as genuine $C^{1+\alpha}$ diffeomorphisms
between curves, and to see $F$ as a smooth 1-dimensional map. It defines a
piecewise expanding map in the sense of [DV21].

The goal of this section is to use the large deviations to compute orders of
magnitude for dynamical quantities. Once this preparatory step is done, we
will be able to study oscillatory integrals involving $\mu$. First of all, we
recall some useful symbolic notations.

  * •

Recall that $\mathcal{W}_{n}$ stands for the set of admissible words of length
$n$. For $\mathbf{a}=a_{1}\dots a_{n}a_{n+1}\in\mathcal{W}_{n+1}$, define
$\mathbf{a}^{\prime}:=a_{1}\dots a_{n}\in\mathcal{W}_{n}$.

  * •

Let $\mathbf{a}\in\mathcal{W}_{n+1}$ and $\mathbf{b}\in\mathcal{W}_{m+1}$.
Denote $\mathbf{a}\rightsquigarrow\mathbf{b}$ if $a_{n+1}=b_{1}$. In this
case, $\mathbf{a}^{\prime}\mathbf{b}\in\mathcal{W}_{n+m+1}$.

  * •

Let $\mathbf{a}\in\mathcal{W}_{n+1}$. We denote by $b(\mathbf{a})$ the last
letter of $\mathbf{a}$.

With those notations, we may rewrite the formula for the iterate of our
transfer operator. For any continuous function
$h:\mathcal{U}\longrightarrow\mathbb{C}$, we have

$\forall b\in\mathcal{A},\ \forall x\in U_{b},\
\mathcal{L}_{\varphi}^{n}h(x)=\underset{\mathbf{a}\rightsquigarrow
b}{\sum_{\mathbf{a}\in\mathcal{W}_{n+1}}}e^{S_{n}\varphi(g_{\mathbf{a}}(x))}h(g_{\mathbf{a}(x)})=\underset{\mathbf{a}\rightsquigarrow
b}{\sum_{\mathbf{a}\in\mathcal{W}_{n+1}}}h(g_{\mathbf{a}}(x))w_{\mathbf{a}}(x).$

where

$w_{\mathbf{a}}(x):=e^{S_{n}\varphi(g_{\mathbf{a}}(x))}\simeq\nu(U_{\mathbf{a}}).$

Iterating $\mathcal{L}_{\varphi}^{n}$ again leads us to the formula

$\forall x\in U_{b},\
\mathcal{L}_{\varphi}^{nk}h(x)=\sum_{\mathbf{a}_{1}\rightsquigarrow\dots\rightsquigarrow\mathbf{a_{k}}\rightsquigarrow
b}h(g_{\mathbf{a}_{1}^{\prime}\dots\mathbf{a}_{k-1}^{\prime}\mathbf{a}_{k}}(x))w_{\mathbf{a}_{1}^{\prime}\dots\mathbf{a}_{k-1}^{\prime}\mathbf{a}_{k}}(x).$

######  Definition 3.1.

Let, for small $\varepsilon>0$ and $n\geq 1$,

$B_{n}(\varepsilon):=\left\\{x\in\mathcal{R}\
,\left|\frac{1}{n}S_{n}\tau_{F}(\pi(x))-\lambda\right|<\varepsilon\text{ and
}\left|\frac{S_{n}\varphi(\pi(x))}{S_{n}\tau_{F}(\pi(x))}+\delta\right|<\varepsilon\right\\}$
$\text{and}\ C_{n}(\varepsilon):=\left\\{x\in{\Omega}\ ,\ \
\left|\frac{1}{n}\sum_{k=0}^{n-1}\tau_{f}(f^{k}(x))-\lambda\right|<\varepsilon\right\\}.$

Finally, let $A_{n}(\varepsilon):=B_{n}(\varepsilon)\cap C_{n}(\varepsilon)$.
Then $\mu(A_{n}(\varepsilon))\geq 1-e^{-\delta_{2}(\varepsilon)n}$ for $n$
large enough depending on $\varepsilon$ and for some
$\delta_{2}(\varepsilon)>0$, by Theorem 2.11 and Corollary 2.12.

######  Notations 3.1.

To simplify the reading, when two quantities dependent of $n$ satisfy
$b_{n}\leq Ca_{n}$ for some constant $C$, we denote it by $a_{n}\lesssim
b_{n}$. If $a_{n}\lesssim b_{n}\lesssim c_{n}$, we denote it by $a_{n}\simeq
b_{n}$. If there exists $c,C$ and $\beta$, independent of $n$ and
$\varepsilon$, such that $ce^{-\varepsilon\beta n}a_{n}\leq b_{n}\leq
Ce^{\varepsilon\beta n}a_{n}$, we denote it by $a_{n}\sim b_{n}$. Throughout
the text $\beta$ will be allowed to change from line to line. It correspond to
some positive constant.

Eventually, we will chose $\varepsilon$ small enough such that this
exponentially growing term gets absorbed by the other leading terms, so we can
neglect it.

######  Lemma 3.2.

Let $\mathbf{a}\in\mathcal{W}_{n+1}(\varepsilon)$ be such that
$U_{\mathbf{a}}\cap A_{n}(\varepsilon)\neq\emptyset$. Then:

  * •

uniformly on $x\in U_{\mathbf{a}}$, $|\partial_{u}f^{n}(x)|\sim e^{n\lambda}.$

  * •

uniformly on $x\in U_{b(\mathbf{a})},\ |g_{\mathbf{a}}^{\prime}(x)|\sim
e^{-n\lambda}$

  * •

$\mathrm{diam}(\widehat{U}_{\mathbf{a}}),\ \mathrm{diam}(U_{\mathbf{a}})\sim
e^{-n\lambda}$

  * •

uniformly on $x\in U_{\mathbf{a}},\ w_{\mathbf{a}}(x)\sim e^{-\delta\lambda
n}$

  * •

$\nu(\widehat{U}_{\textbf{a}})\sim e^{-\delta\lambda n}$

######  Remark 3.2.

The definition of $\delta$ is chosen so that
$\mu(U_{\mathbf{a}})\sim\text{diam}(U_{\mathbf{a}})^{\delta}$ when
$U_{\mathbf{a}}\cap A_{n}(\varepsilon)\neq\emptyset$. Beware that the
diameters are for the distance $d^{u}$.

###### Proof.

The first thing to notice is that, since any Hölder map have exponentially
vanishing variations, Birkhoff sums are easy to control when $n$ becomes
large. In particular, the estimates that are true on a piece of
$U_{\mathbf{a}}$ will extends on $U_{\mathbf{a}}$. For example, let $x\in
U_{\mathbf{a}}$, and let $y_{\mathbf{a}}\in U_{\mathbf{a}}\cap
A_{n}(\varepsilon)$. Since $\tau_{f}$ has exponentially vanishing variations,
we can write

$\left|\frac{1}{n}\sum_{k=0}^{n-1}\tau_{f}(f^{k}(x))-\lambda\right|\leq\left|\frac{1}{n}\sum_{k=0}^{n-1}\tau_{f}(f^{k}(x))-\frac{1}{n}\sum_{k=0}^{n-1}\tau_{f}(f^{k}(y_{\mathbf{a}}))\right|+\left|\frac{1}{n}\sum_{k=0}^{n-1}\tau_{f}(f^{k}(y_{\mathbf{a}}))-\lambda\right|.$

Since $f^{k}(U_{\mathbf{a}})\subset R_{a_{k+1}\dots a_{n}}$ is diffeomorphic
to $U_{a_{k+1}\dots a_{n}}$ by the mean of the holonomy map $\pi$ which have
bounded differential, we have

$\frac{1}{n}\sum_{k=0}^{n-1}\left|\tau_{f}(f^{k}(x))-\tau_{f}(f^{k}(y_{\mathbf{a}}))\right|\leq\frac{1}{n}\sum_{k=0}^{n-1}|\tau_{f}|_{\alpha}\text{diam}\left(f^{k}(U_{\mathbf{a}})\right)^{\alpha}$
$\lesssim\frac{1}{n}\sum_{k=0}^{n-1}\text{diam}(U_{a_{k+1}\dots
a_{n}})^{\alpha}\lesssim\frac{1}{n}\sum_{k=0}^{n-1}\kappa^{\alpha
k}\leq\frac{C}{n}$

for some constant $C$. In particular we see that for $n$ large enough
depending on $\varepsilon$, the estimate

$\forall x\in U_{\mathbf{a}},\
\left|\frac{1}{n}\sum_{k=0}^{n-1}\tau_{f}(f^{k}(x))-\lambda\right|<2\varepsilon$

holds. Similarly the estimates related to $B_{n}(\varepsilon)$ extends on
$U_{\mathbf{a}}$ replacing $\varepsilon$ with $2\varepsilon$ as long as $n$ is
large enough. Once this is known, the estimates for $\partial_{u}(f^{n})$,
$g_{\mathbf{a}}^{\prime}$ and $w_{\mathbf{a}}$ are easy, and the estimates for
the diameters is a direct consequence of our 1-dimensional setting and the
mean value theorem. The estimates for $\nu$ follows from the Gibbs estimates.
∎

######  Definition 3.2.

For some fixed set $\omega\subset\mathcal{U}$ such that $\nu(\omega)>0$,
define the set of ($\omega$-localized) $\varepsilon$-regular words by

$\mathcal{R}_{n+1}(\omega):=\left\\{\mathbf{a}\in\mathcal{W}_{n+1}\ |\
\omega\cap U_{\mathbf{a}}\cap A_{n}(\varepsilon)\neq\emptyset\right\\}.$

When no localization is asked, define
$\mathcal{R}_{n+1}:=\mathcal{R}_{n+1}(M)$. Then, define the set of
($\omega$-localized) $\varepsilon$-regular $k$-blocks by

$\mathcal{R}_{n+1}^{k}(\omega)=\left\\{\mathbf{A}=\mathbf{a}_{1}^{\prime}\dots\mathbf{a}_{k-1}^{\prime}\mathbf{a}_{k}\in\mathcal{W}_{nk+1}\
|\ \forall i\geq 2,\ \mathbf{a}_{i}\in\mathcal{R}_{n+1},\ \text{and}\
\mathbf{a}_{1}\in\mathcal{R}_{n+1}(\omega)\right\\}.$

Finally, define the associated geometric points to be

$R_{n+1}^{k}(\omega):=\bigcup_{\mathbf{A}\in\mathcal{R}_{n+1}^{k}(\omega)}U_{\mathbf{A}}.$

The sets $\mathcal{R}_{n+1}^{k}$, $R_{n+1}^{k}$ denotes the regular blocks and
associated geometric points when $\omega=M$.

######  Lemma 3.3.

For $\omega$ and $k\geq 1$ fixed, and for $n$ large enough depending on
$\varepsilon$,

$\\#\mathcal{R}_{n+1}^{k}(\omega)\sim e^{k\delta\lambda n}.$

###### Proof.

First of all, notice that $A_{n}(\varepsilon)\subset R_{n+1}$, so that
$\nu(\mathcal{U}\setminus R_{n+1})\leq\nu(\mathcal{U}\setminus
A_{n})\leq\nu(\mathcal{U}\setminus B_{n})+\nu(\mathcal{U}\setminus C_{n})$. To
control the term in $C_{n}$, we notice that, by the same argument than in
Lemma 3.1:

$\forall x\in R_{a},\ x\in C_{n}(\varepsilon)\Rightarrow[x,S_{a}]\subset
C_{n}(2\varepsilon)$

as soon as $n$ is large enough. Hence, by the definition of the measure $\nu$:

$\nu(\mathcal{U}\setminus C_{n})=\mu\left(\mathcal{R}\setminus[\mathcal{U}\cap
C_{n},\mathcal{S}]\right)\leq\mu(\mathcal{R}\setminus
C_{n}(\varepsilon/2))\lesssim e^{-\delta_{1}n}.$

This proves that $\nu(R_{n+1})\geq 1-e^{-\delta_{0}n}$ for $n$ large enough
depending on $\varepsilon$ and for some $\delta_{0}(\varepsilon)>0$.
Since $R_{n+1}(\omega)=R_{n+1}\cap\omega$, it follows that
$\nu(\omega)-e^{-\delta_{0}n}\leq\nu(R_{n+1}(\omega))\leq\nu(\omega)$.

Then, define
$\tilde{R}_{n+1}:=\bigsqcup_{\mathbf{a}\in\mathcal{R}_{n+1}}\text{int}_{\mathcal{U}}\left(U_{\mathbf{a}}\right)$.
From the point of view of the measure $\nu$, it is indistinguishable from
$R_{n+1}$. First, we prove that

$\omega\cap\bigcap_{i=0}^{k-1}F^{-ni}\left(\tilde{R}_{n+1}\right)\subset
R_{n+1}^{k}(\omega).$

Let $x\in\omega\cap\bigcap_{i=0}^{k-1}F^{-ni}\left(\tilde{R}_{n+1}\right)$.
Since there exists
$\mathbf{A}=\mathbf{a}_{1}^{\prime}\dots\mathbf{a}_{k-1}^{\prime}\mathbf{a}_{k}\in\mathcal{W}_{kn+1}$
such that $x\in U_{\mathbf{A}}$, we see that for any $i$ we can write
$F^{ni}(x)\in
U_{\mathbf{a}_{1+i}^{\prime}\dots\mathbf{a}_{k}}\cap\tilde{R}_{n+1}$. So there
exists $\mathbf{b}_{i+1}\in\mathcal{R}_{n+1}$ such that
$U_{\mathbf{a}_{1+i}^{\prime}\dots\mathbf{a}_{k}}\cap\text{int}_{\mathcal{U}}U_{\mathbf{b}_{i+1}}\neq\emptyset$.
Then $\mathbf{b}_{i+1}=\mathbf{a}_{i+1}$, for all $i$, which implies that
$\mathbf{A}\in\mathcal{R}_{n+1}^{k}$. Moreover, since $\emptyset\neq\omega\cap
U_{\mathbf{A}}\subset\omega\cap U_{\mathbf{a}_{1}}$, $\mathbf{a}_{1}\in
R_{n+1}(\omega)$, and so $\mathbf{A}\in\mathcal{R}_{n+1}^{k}(\omega)$. Now
that the inclusion is proved, we see that

$\nu\left(\omega\setminus R_{n+1}^{k}(\omega)\right)\leq\nu(\omega\setminus
R_{n+1}(\omega))+\sum_{i=1}^{k-1}\nu\left(F^{-ni}\left(\mathcal{U}\setminus\tilde{R}_{n+1}\right)\right)=\nu(\omega\setminus
R_{n+1}(\omega))+(k-1)\nu(\mathcal{U}\setminus R_{n+1}),$

and so
$\nu(R_{n+1}^{k}(\omega))\geq\nu(\omega)-ke^{-\delta_{1}(\varepsilon)n}$ for
$n$ large enough depending on $\varepsilon$.

Now we may prove the cardinality estimate: since
$\nu(R_{n+1}^{k}(\omega))=\sum_{\mathbf{A}\in
R_{n+1}^{k}(\omega)}\nu(U_{\mathbf{A}})$, we have

$\\#\left(\mathcal{R}_{n+1}^{k}(\omega)\right)e^{-\varepsilon\beta
n}e^{-\delta\lambda
kn}\lesssim\nu\left(R_{n+1}^{k}(\omega)\right)\lesssim\\#\left(\mathcal{R}_{n+1}^{k}(\omega)\right)e^{\varepsilon\beta
n}e^{-\delta\lambda kn}$

and so

$e^{-\varepsilon\beta n}e^{\delta\lambda
kn}\left(\nu(\omega)-ke^{-\delta_{1}(\varepsilon)n}\right)\lesssim\\#\mathcal{R}_{n+1}^{k}(\omega)\lesssim
e^{\varepsilon\beta n}e^{\delta\lambda kn}.$

∎

##  4 Reduction to sums of exponential

We can finally start the proof of our main theorem. The goal is to follow the
strategy developed in [SS20] and [Le21] (which generalized the original method
developed in [BD17] and [LNP19]). To do so, we will reduce our oscillatory
integral over $\Omega$ with respect to $\mu$ to an oscillatory integral over
$\mathcal{U}$ with respect to $\nu$. The fact that $\nu$ is invariant by a
transfer operator related to an expanding map will allows us to get the same
final reduction as in those two papers: everything will boils down to a sum of
exponential that we will be able to control thanks to a combinatorial theorem
of Bourgain.

Before going on, recall the setting: we have fixed a $C^{2+\alpha}$ Axiom A
diffeomorphism $f:M\rightarrow M$ and an attractor $\Omega$ on which $f$ has
codimension 1 stable lamination and that satisfies our generic nonlinearity
condition (NL). The bunching condition (B) will only be used in section 6. We
have fixed a Hölder potential $\psi:\Omega\rightarrow\mathbb{R}$ and its
associated equilibrium state $\mu$. We denote by $(F,\mathcal{U},\nu)$ the
expanding dynamical system in factor defined in section 2. The measure $\nu$
is invariant by a transfer operator $\mathcal{L}_{\varphi}$, where $\varphi$
is some normalized and $\alpha$-Hölder potential. The Hölder exponent $\alpha$
is fixed for the rest of the paper.

Six quantities will be at play: $\xi$, $K$, $n$, $k$, $\varepsilon_{0}$ and
$\varepsilon$. We will think of $k$, $\varepsilon_{0}$ and $\varepsilon$ as
being fixed. The constant $k$ is fixed using Corollary 5.3. The constant
$\varepsilon_{0}>0$ will be fixed at at the end of the paper, in Lemma 6.5.
The constant $\varepsilon>0$ is chosen at the very end of the proof to be
small in front of every other constant that might appear. The only variable is
$\xi$. We relate it to $n$ and $K$ by the formulas

$n:=\left\lfloor\frac{\ln|\xi|}{(2k+1)\lambda+\varepsilon_{0}}\right\rfloor\quad\text{and}\quad
K:=\left\lfloor\frac{((2k+1)\lambda+2\varepsilon_{0})n}{\alpha|\ln\kappa|}\right\rfloor.$

######  Definition 4.1.

Let $\chi\in C^{\alpha}(M,\mathbb{C})$ (with support of positive measure). Let
$\phi\in C^{1+\alpha}(M,\mathbb{R})$. Suppose that there exists a constant
$C_{\phi,\chi}>1$ such that

  * •

$\|\chi\|_{C^{\alpha}}+\|\phi\|_{C^{1+\alpha}}<C_{\phi,\chi}$

  * •

$\forall x\in\text{supp}\ \chi,\ |\partial_{u}\phi(x)|>C_{\phi,\chi}^{-1}$.

Then define, for $\xi\in\mathbb{R}$:

$\widehat{\mu}(\xi):=\int_{M}e^{i\xi\phi}\chi\ d\mu.$

Since the condition on $\partial_{u}\phi$ is open, it holds on a small open
neighborhood $\tilde{\omega}\supset\text{supp}\chi$. Choosing a small enough
Markov partition allows us to choose $\tilde{\omega}$ as a nonempty union of
rectangles $R_{a}$. It follows that $\omega:=\tilde{\omega}\cap\mathcal{U}$
have positive measure, and hence we can use the results of Lemma 3.2.

######  Remark 4.1.

The constant $C$ in Theorem 1.2 will depends only on the riemannian manifold
$M$, the diffeomorphism $f$, the equilibrium state $\mu$, the Hölder exponent
$\alpha$ and the constant $C_{\phi,\chi}>1$.

######  Notations 4.2.

We recall a final set of notations, inspired from [BD17]. For a fixed $n$ and
$k$, denote:

  * •

$\textbf{A}=(\textbf{a}_{0},\dots,\textbf{a}_{k})\in\mathcal{W}_{n+1}^{k+1}\
,\ \textbf{B}=(\textbf{b}_{1},\dots,\textbf{b}_{k})\in\mathcal{W}_{n+1}^{k}$.

  * •

We write $\textbf{A}\leftrightarrow\textbf{B}$ iff
$\textbf{a}_{j-1}\rightsquigarrow\textbf{b}_{j}\rightsquigarrow\textbf{a}_{j}$
for all $j=1,\dots k$.

  * •

If $\textbf{A}\leftrightarrow\textbf{B}$, then we define the words
$\textbf{A}*\textbf{B}:=\textbf{a}_{0}^{\prime}\textbf{b}_{1}^{\prime}\textbf{a}_{1}^{\prime}\textbf{b}_{2}^{\prime}\dots\textbf{a}_{k-1}^{\prime}\textbf{b}_{k}^{\prime}\textbf{a}_{k}$
and
$\textbf{A}\\#\textbf{B}:=\textbf{a}_{0}^{\prime}\textbf{b}_{1}^{\prime}\textbf{a}_{1}^{\prime}\textbf{b}_{2}^{\prime}\dots\textbf{a}_{k-1}^{\prime}\textbf{b}_{k}$.

  * •

Denote by $b(\textbf{A})\in\mathcal{A}$ the last letter of $\textbf{a}_{k}$.

We prove the following reduction.

######  Proposition 4.1.

Define $J_{n}:=\\{e^{\varepsilon_{0}n/2}\leq|\eta|\leq
e^{2\varepsilon_{0}n}\\}$ and

$\zeta_{j,\mathbf{A}}(\mathbf{b}):=e^{2\lambda
n}|g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}^{\prime}(x_{\mathbf{a}_{j}})|.$

There exists some constant $\beta>1$ such that for $n$ large enough depending
on $\varepsilon$:

$|\widehat{\mu}(\xi)|^{2}\lesssim e^{\varepsilon\beta
n}e^{-\lambda\delta(2k+1)n}\sum_{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}\sup_{\eta\in
J_{n}}\Bigg{|}\underset{\mathbf{A}\leftrightarrow\mathbf{B}}{\sum_{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}}e^{i\eta\zeta_{1,\mathbf{A}}(\mathbf{b}_{1})\dots\zeta_{k,\mathbf{A}}(\mathbf{b}_{k})}\Bigg{|}$
$\quad\quad\quad\quad\quad\quad\quad\quad+e^{-\varepsilon_{0}n}+e^{-\delta_{1}(\varepsilon)n}+e^{\varepsilon\beta
n}\left(e^{-\lambda\alpha n}+\kappa^{\alpha
n}+e^{-(\alpha\lambda-\varepsilon_{0})n}+e^{-\varepsilon_{0}\delta_{up}n/2}\right).$

Once proposition 4.1 is established, if we manage to prove that the sum of
exponentials enjoys exponential decay in $n$, then choosing $\varepsilon$
small enough will allow us to see that $|\widehat{\mu}(\xi)|^{2}$ enjoys
polynomial decay in $\xi$, and our main theorem will be proved. We prove
Proposition 4.1 through a succession of lemmas.

######  Lemma 4.2.

For $n$ large enough there exists $C>0$ that depends only on $C_{\psi,\chi}$
such that

$\left|\int_{\Omega}e^{i\xi\phi}\chi d\mu-\int_{\mathcal{U}}e^{i\xi\phi\circ
f^{K}}\chi\circ f^{K}d\nu\right|\leq Ce^{-\varepsilon_{0}n}.$

###### Proof.

Let $h(x):=e^{i\xi\phi(x)}\chi(x)$. Then, since $\mu$ is $f$ invariant,

$\widehat{\mu}=\int_{\Omega}h\circ f^{K}d\mu.$

We then check that $h\circ f^{K}$ is close to a $S$-constant function as $K$
grows larger. First of all, notice that since $\phi$ is $C^{1}$,
$e^{i\xi\phi}$ also is. In particular it is Lipschitz with constant
$|\xi|C_{\phi,\chi}$. Since $M$ is a closed riemannian manifold, it follows
that $e^{i\xi\phi}$ is $\alpha$-Hölder on $M$ with constant
$|\xi|C_{\phi,\chi}\text{diam}(M)^{1-\alpha}$. Since $\chi$ is $\alpha$-Holder
too, with constant $C_{\phi,\chi}$, the product $h$ is also locally
$\alpha$-Hölder, with constant
$C_{\phi,\chi}+C_{\phi,\chi}^{2}|\xi|\text{diam}(M)^{1-\alpha}$.

Let $x\in\mathcal{R}_{a}$. By definition, $\pi(x)\in U_{a}$ is in $W^{s}(x)$.
Then

$|h(f^{K}(x))-h(f^{K}(\pi(x)))|\leq|h|_{\alpha}d(f^{K}(x),f^{K}(\pi(x)))^{\alpha}\leq|h|_{\alpha}\kappa^{\alpha
K}$
$\leq\left(C_{\phi,\chi}+C_{\phi,\chi}^{2}|\xi|\text{diam}(M)^{1-\alpha}\right)\kappa^{\alpha
K}.$

Since $|\xi|\kappa^{\alpha K}\lesssim e^{-\varepsilon_{0}n}$, there exists a
constant $C>1$ that depends only on $C_{\psi,\chi}$ such that

$\|h\circ f^{K}-h\circ f^{K}\circ\pi\|_{\infty,\mathcal{R}}\leq
Ce^{-\varepsilon_{0}n}.$

The desired estimates follows from the definition of $\nu$. ∎

Now we are ready to adapt the existing strategy for one dimensional expanding
maps. From this point, we will follow [SS20] and [Le21]. Notice however that
we use an additional word $\mathbf{C}$ to cancel the distortions induced by
$f^{K}$.

######  Lemma 4.3.

$\left|\int_{\mathcal{U}}e^{i\xi\phi\circ f^{K}}\chi\circ
f^{K}d\nu\right|^{2}\lesssim\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\quad\quad\quad\Bigg{|}\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}{\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}}{\sum}}}\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))}\chi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))w_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)d\nu(x)\Bigg{|}^{2}+e^{-\delta_{1}(\varepsilon)n}$

###### Proof.

Since $\nu$ is invariant by $\mathcal{L}_{\varphi}$, we can write

$\int_{\mathcal{U}}h\circ
f^{K}d\nu=\sum_{\mathbf{C}\in\mathcal{W}_{K+1}}\int_{U_{b(\mathbf{C})}}h\left(f^{K}(g_{\mathbf{C}}(x))\right)w_{\mathbf{C}}(x)d\nu(x)$

If we look at the part of the sum where $\mathbf{C}$ is not a regular word, we
get by the Gibbs estimates:

$\Bigg{|}\sum_{\mathbf{C}\notin\mathcal{R}_{K+1}}\int_{U_{b(\mathbf{A})}}h\left(f^{K}(g_{\mathbf{C}}(x))\right)w_{\mathbf{C}}(x)d\nu(x)\Bigg{|}\lesssim\sum_{\mathbf{B}\notin\mathcal{R}_{K}}\nu(U_{\mathbf{C}})\lesssim\nu(\mathcal{U}\setminus
R_{K})$

which decays exponentially in $n$ by Lemma 3.2. Then, we iterate our transfer
operator again on the main sum, to get the following term:

$\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}{\underset{\mathbf{B}\in\mathcal{W}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{W}_{n+1}^{k+1}}{\sum}}}\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))}\chi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))w_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)d\nu(x).$

Looking at the part of the sum where words are not regular again, we see by
the Gibbs estimates again that

$\Bigg{|}\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\text{or
}\mathbf{B}\notin\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\notin\mathcal{R}_{n+1}^{k+1}}{\sum_{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}}}\int_{U_{b(\mathbf{A})}}h\left(f^{K}(g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))\right)w_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)d\nu(x)\Bigg{|}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\quad\quad\quad\quad\quad\quad\quad\quad\lesssim\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\text{or
}\mathbf{B}\notin\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\notin\mathcal{R}_{n+1}^{k+1}}{\sum_{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}}}\nu(U_{\mathbf{C}})\nu(U_{\mathbf{A}*\mathbf{B}})\lesssim\nu(\mathcal{U}\setminus
R_{n}^{2k+1}),$

and the desired estimate follows from Lemma 3.2. ∎

######  Lemma 4.4.

Define
$\chi_{\mathbf{C}}(\mathbf{a}_{0}):=\chi(f^{K}g_{\mathbf{C}}(x_{\mathbf{a}_{0}}))$.
There exists some constant $\beta>0$ such that, for $n$ large enough:

$\Bigg{|}\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}{\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}}{\sum}}}\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))}w_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)\chi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))d\nu(x)\Bigg{|}^{2}\quad\quad\quad\quad\quad\quad\quad$
$\quad\quad\quad\quad\lesssim\Bigg{|}\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}{\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum}}}\chi_{\mathbf{C}}(\mathbf{a}_{0})\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)))}w_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)d\nu(x)\Bigg{|}^{2}+e^{\varepsilon\beta
n}e^{-\lambda\alpha n}.$

###### Proof.

Our first move is to get rid of terms in the sum where
$\chi(f^{K}g_{\mathbf{C}}g_{\mathbf{A}*\mathbf{B}})=0$. To this end, notice
that $f^{K}g_{\mathbf{C}}$ is an holonomy map that sends $U_{b(\mathbf{C})}$
in $R_{b(\mathbf{C})}$. It doesn’t really play a role for this question. The
only word that matter here is $\mathbf{a}_{0}$ : if it isn’t in
$R_{n+1}(\omega)$, then by definition it implies that $\omega\cap
U_{\mathbf{a}_{0}}=\emptyset$. It follows that $\tilde{\omega}\cap
R_{\mathbf{a}_{0}}=\emptyset$, and so
$\chi(f^{K}g_{\mathbf{C}}g_{\mathbf{A}*\mathbf{B}})=0$.

Hence our main term is equal to the same sum, but where we have restricted
$\mathbf{A}$ in $\mathcal{R}_{n+1}(\omega)$.

Next, denote
$\chi_{\mathbf{C}}(\mathbf{a}_{0}):=\chi(f^{K}g_{\mathbf{C}}(x_{\mathbf{a}_{0}})).$
The orders of magnitude of Lemma 3.1 and Remark 3.3 combined gives us

$\left|\chi(f^{K}g_{\mathbf{C}}g_{\mathbf{A}*\mathbf{B}}(x))-\chi_{\mathbf{C}}(\mathbf{a}_{0})\right|\leq
C_{\phi,\chi}d\left(f^{K}g_{\mathbf{C}}g_{\mathbf{A}*\mathbf{B}}(x),f^{K}g_{\mathbf{C}}(x_{\mathbf{a}_{0}})\right)^{\alpha}$
$\leq
C_{\phi,\chi}\|\partial_{u}(f^{K})\|_{\infty,U_{\mathbf{C}}}^{\alpha}\|g_{\mathbf{C}}^{\prime}\|_{\infty,U_{b(\mathbf{C})}}^{\alpha}\text{diam}(U_{\mathbf{a}_{0}})^{\alpha}\lesssim
e^{\varepsilon\beta n}e^{-\lambda\alpha n}.$

Hence, by the Gibbs estimates

${\Bigg{|}{\sum_{\mathbf{A},\mathbf{B},\mathbf{C}}}\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})})}\chi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})})w_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}d\nu-\sum_{\mathbf{A},\mathbf{B},\mathbf{C}}\chi_{\mathbf{C}}(\mathbf{a}_{0})\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})})}w_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}d\nu\Bigg{|}}$
$\lesssim e^{\varepsilon\beta n}e^{-\lambda\alpha
n}\sum_{\mathbf{A},\mathbf{B},\mathbf{C}}\nu(U_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})})\lesssim
e^{\varepsilon\beta n}e^{-\lambda\alpha n}.$

∎

######  Lemma 4.5.

There exists some constant $\beta>0$ such that, for $n$ large enough:

$\Bigg{|}\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}{\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum}}}\chi_{\mathbf{C}}(\mathbf{a}_{0})\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)))}w_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)d\nu(x)\Bigg{|}^{2}\quad\quad\quad\quad\quad\quad\quad$
$\quad\quad\quad\quad\lesssim e^{-(2k-1)\lambda\delta n}e^{-\lambda\delta
K}\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}{\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum}}}\Bigg{|}\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}(g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)))}w_{\mathbf{a}_{k}}(x)d\nu(x)\Bigg{|}^{2}+e^{\varepsilon\beta
n}\kappa^{\alpha n}.$

###### Proof.

Notice that $w_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)$ and
$w_{\mathbf{a}_{k}}(x)$ are related by

$w_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)=w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(g_{\mathbf{a}_{k}}(x))w_{\mathbf{a}_{k}}(x).$

Moreover:

$\frac{w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(g_{\mathbf{a}_{k}}(x))}{w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(x_{\mathbf{a}_{k}})}=\exp\left(S_{K+2nk}\varphi(g_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(g_{\mathbf{a}_{k}}(x)))-S_{K+2kn}\varphi(g_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(x_{\mathbf{a}_{k}}))\right)$

with

$\left|S_{K+2nk}\varphi(g_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(g_{\mathbf{a}_{k}}(x))-S_{K+2kn}\varphi(g_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(x_{\mathbf{a}_{k}}))\right|\lesssim\sum_{j=0}^{K+2nk-1}\kappa^{\alpha(K+n(2k+1)-j)}\lesssim\kappa^{\alpha
n}$

since $\varphi$ is $\alpha$-Hölder. Hence, there exists some constant $C>0$
such that

$e^{-C\kappa^{\alpha
n}}w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(x_{\mathbf{a}_{k}})\leq
w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(g_{\mathbf{a}_{k}}(x))\leq
e^{C\kappa^{\alpha
n}}w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(x_{\mathbf{a}_{k}})$

Which gives:

$\left|w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(g_{\mathbf{a}_{k}}(x))-w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(x_{\mathbf{a}_{k}})\right|\leq\max\left|e^{\pm
C\kappa^{\alpha
n}}-1\right|w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(x_{\mathbf{a}_{k}})\lesssim\kappa^{\alpha
n}w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(x_{\mathbf{a}_{k}}).$

Hence

$\Bigg{|}\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}{\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum}}}\chi_{\mathbf{C}}(\mathbf{a}_{0})\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))}\left(w_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x)-w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(x_{\mathbf{a}_{k}})w_{\mathbf{a}_{k}}(x)\right)d\nu(x)\Bigg{|}$

$\lesssim\kappa^{\alpha
n}\sum_{\mathbf{A},\mathbf{B},\mathbf{C}}\int_{U_{b(\mathbf{A})}}w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(x_{\mathbf{a}_{k}})w_{\mathbf{a}_{k}}(x)d\nu(x)\
\lesssim e^{\varepsilon\beta n}\kappa^{\alpha n}.$

by the Gibbs estimates. Moreover, by Cauchy-Schwartz and by the orders of
magnitude of Lemma 3.1,

$\Bigg{|}\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}{\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum}}}w_{\mathbf{C}^{\prime}(\mathbf{A}\\#\mathbf{B})}(x_{\mathbf{a}_{k}})\chi_{\mathbf{C}}(\mathbf{a}_{0})\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))}w_{\mathbf{a}_{k}}(x)d\nu(x)\Bigg{|}^{2}$
$\lesssim e^{\varepsilon\beta n}e^{-\lambda\delta(2k-1)n}e^{-\lambda\delta
K}\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}{\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum}}}\left|\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))}w_{\mathbf{a}_{k}}(x)d\nu(x)\right|^{2},$

where one could increase $\beta$ if necessary.∎

######  Lemma 4.6.

Define

$\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}(x,y):=\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))-\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(y)).$

There exists some constant $\beta>0$ such that, for $n$ large enough:

$e^{-(2k-1)\lambda\delta n}e^{-\lambda\delta
K}\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}\leftrightarrow\mathbf{B}}{\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum}}}\Bigg{|}\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))}w_{\mathbf{a}_{k}}(x)d\nu(x)\Bigg{|}^{2}$
$\lesssim e^{\varepsilon\beta n}e^{-\lambda\delta(2k+1)n}e^{-\lambda\delta
K}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}}}\iint_{U_{b(\mathbf{A})}^{2}}\Bigg{|}\underset{\mathbf{A}\leftrightarrow\mathbf{B}}{\sum_{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}}e^{i\xi\left|\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}\right|(x,y)}\Bigg{|}d\nu(x)d\nu(y).$

###### Proof.

We first open up the modulus squared:

$\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum_{\mathbf{A}\leftrightarrow\mathbf{B}}}}\Bigg{|}\int_{U_{b(\mathbf{A})}}e^{i\xi\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}(x))}w_{\mathbf{a}_{k}}(x)d\nu(x)\Bigg{|}^{2}$
$=\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum_{\mathbf{A}\leftrightarrow\mathbf{B}}}}\iint_{U_{b(\mathbf{A})}^{2}}w_{\mathbf{a}_{k}}(x)w_{\mathbf{a}_{k}}(y)e^{i\xi\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}(x,y)}d\nu(x)d\nu(y).$

Since this quantity is real, we get:

$=\underset{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum_{\mathbf{A}\leftrightarrow\mathbf{B}}}}\iint_{U_{b(\mathbf{A})}^{2}}w_{\mathbf{a}_{k}}(x)w_{\mathbf{a}_{k}}(y)\cos({\xi\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}(x,y)})d\nu(x)d\nu(y)$

$=\sum_{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}\iint_{U_{b(\mathbf{A})}^{2}}w_{\mathbf{a}_{k}}(x)w_{\mathbf{a}_{k}}(y)\underset{\mathbf{A}\leftrightarrow\mathbf{B}}{\sum_{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}}\cos\left(\xi|\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}|(x,y)\right)d\nu(x)d\nu(y),$

and then we conclude using the triangle inequality and the estimates of
section 3 as follow:

$\lesssim e^{\varepsilon\beta n}e^{-2\lambda\delta
n}\sum_{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}\iint_{U_{b(\mathbf{A})}^{2}}\Bigg{|}\underset{\mathbf{A}\leftrightarrow\mathbf{B}}{\sum_{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}}e^{i\xi\left|\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}\right|(x,y)}\Bigg{|}d\nu(x)d\nu(y).$

∎

######  Lemma 4.7.

Define

$\zeta_{j,\mathbf{A}}(\mathbf{b}):=e^{2\lambda
n}|g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}^{\prime}(x_{\mathbf{\mathbf{a}}_{j}})|$

and

$J_{n}:=\\{e^{\varepsilon_{0}n/2}\leq|\eta|\leq e^{2\varepsilon_{0}n}\\}.$

There exists $\beta>0$ such that, for $n$ large enough depending on
$\varepsilon$,

$e^{-\lambda\delta
K}\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}e^{-\lambda\delta(2k+1)n}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum}}\iint_{U_{b(\mathbf{A})}^{2}}\Bigg{|}\underset{\mathbf{A}\leftrightarrow\mathbf{B}}{\sum_{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}}e^{i\xi\left|\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}\right|(x,y)}\Bigg{|}d\nu(x)d\nu(y)\quad\quad\quad\quad\quad\quad$
$\quad\quad\quad\quad\quad\quad\lesssim
e^{-\lambda\delta(2k+1)n}\sum_{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}\sup_{\eta\in
J_{n}}\Bigg{|}\underset{\mathbf{A}\leftrightarrow\mathbf{B}}{\sum_{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}}e^{i\eta\zeta_{1,\mathbf{A}}(\mathbf{b}_{1})\dots\zeta_{k,\mathbf{A}}(\mathbf{b}_{k})}\Bigg{|}+e^{\varepsilon\beta
n}\left(e^{-(\alpha\lambda-\varepsilon_{0})n}+e^{-\varepsilon_{0}n\delta_{\text{up}}/2}\right).$

###### Proof.

Our goal is to carefully approximate
$\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}$ by a product of derivatives of
$g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}_{j}}$, and then to renormalize the
phase. Using arc length parameterization, our 1-dimensional setting allows us
to apply the mean value theorem: for all $x,y\in U_{b(\mathbf{A})}$, there
exists $z\in U_{b(\mathbf{A})}$ such that

$|\phi(f^{K}g_{\mathbf{C}}g_{\mathbf{A}*\mathbf{B}}(x))-\phi(f^{K}g_{\mathbf{C}}g_{\mathbf{A}*\mathbf{B}}(y))|=$
$|\partial_{u}\phi(f^{K}g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}z)||\partial_{u}f^{K}(g_{\mathbf{C}^{\prime}(\mathbf{A}*\mathbf{B})}z)||g_{\mathbf{C}}^{\prime}(g_{\mathbf{A}*\mathbf{B}}z)|\left[\prod_{j=1}^{k}|g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}_{j}}^{\prime}(g_{\mathbf{a}_{j}^{\prime}\mathbf{b}_{j+1}^{\prime}\dots\mathbf{a}_{k-1}^{\prime}\mathbf{b}_{k}}z))|\right]\
d^{u}(g_{\mathbf{a}_{k}}x,g_{\mathbf{a}_{k}}y).$

The estimates of section 3 gives

$\left|{\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}(x,y)}\right|\leq
C_{\phi,\chi}e^{\varepsilon\beta n}e^{-(2k+1)\lambda n}.$

We then relate $\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}$ to the
$\zeta_{j,\mathbf{A}}$. Set

$\tilde{\Delta}_{\mathbf{A},\mathbf{B},\mathbf{C}}(x,y):=|\partial_{u}\phi(f^{K}g_{\mathbf{C}^{\prime}}(x_{\mathbf{a}_{0}})||\partial_{u}f^{K}(g_{\mathbf{C}^{\prime}}(x_{\mathbf{a}_{0}}))||g_{\mathbf{C}^{\prime}}(x_{\mathbf{a}_{0}})|\left[\prod_{j=1}^{k}|g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}_{j}}^{\prime}(x_{\mathbf{a}_{j}}))|\right]\
d^{u}(g_{\mathbf{a}_{k}}x,g_{\mathbf{a}_{0}}y).$

Then, as before,

$\left|{\tilde{\Delta}_{\mathbf{A},\mathbf{B},\mathbf{C}}(x,y)}\right|\leq
C_{\phi,\chi}e^{\varepsilon\beta n}e^{-(2k+1)\lambda n}.$

Hence, using the fact that $|e^{s}-e^{t}|\leq e^{\max(s,t)}|s-t|$, we get

$\left|\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}(x,y)-\tilde{\Delta}_{\mathbf{A},\mathbf{B},\mathbf{C}}(x,y)\right|\leq
C_{\phi,\chi}e^{\varepsilon\beta n}e^{-(2k+1)\lambda
n}\left|\ln|\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}|(x,y)-\ln|\tilde{\Delta}_{\mathbf{A},\mathbf{B},\mathbf{C}}|(x,y)\right|.$

Moreover, using the estimates of section 3, and exponentially vanishing
variations of Hölder maps, we get:

  * •

$\left|\ln|\partial_{u}\phi(f^{K}g_{\mathbf{C}}(x_{\mathbf{a}_{0}})|-\ln|\partial_{u}\phi(f^{K}g_{\mathbf{C}}(x_{\mathbf{a}_{0}}))|\right|\lesssim
e^{\varepsilon\beta n}e^{-\alpha\lambda n}$

  * •

$\left|\ln|\partial_{u}f^{K}(g_{\mathbf{C}(\mathbf{A}*\mathbf{B})}z)|-\ln|\partial_{u}f^{K}(g_{\mathbf{C}^{\prime}}x_{\mathbf{a}_{0}})|\right|\lesssim
e^{\varepsilon\beta n}e^{-\alpha\lambda n}$

  * •

$\left|\ln|g_{\mathbf{C}}^{\prime}(g_{\mathbf{A}*\mathbf{B}}z)|-\ln|g_{\mathbf{C}}^{\prime}x_{\mathbf{a}_{0}}|\right|\lesssim
e^{\varepsilon\beta n}e^{-\alpha\lambda n}$

  * •

$\left|\ln|g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}_{j}}^{\prime}(g_{\mathbf{a}_{j}^{\prime}\mathbf{b}_{j+1}^{\prime}\dots\mathbf{a}_{k-1}^{\prime}\mathbf{b}_{k}}z))|-\ln|g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b_{j}}}^{\prime}(x_{\mathbf{a}_{j}})|\right|\lesssim
e^{\varepsilon\beta n}e^{-\alpha\lambda n}$

So that summing every estimates gives us

$\left|\ln|\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}|(x,y)-\ln|\tilde{\Delta}_{\mathbf{A},\mathbf{B},\mathbf{C}}|(x,y)\right|\lesssim
e^{\varepsilon\beta n}e^{-\alpha\lambda n}.$

Hence,

$\left|\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}(x,y)-\tilde{\Delta}_{\mathbf{A},\mathbf{B},\mathbf{C}}(x,y)\right|\lesssim
e^{\varepsilon\beta n}e^{-(2k+1+\alpha)\lambda n},$

which allows us to approximate our main integral as follows:

$e^{-(2k+1)\delta\lambda n-\delta\lambda
K}\Bigg{|}\underset{\mathbf{C}\rightsquigarrow\mathbf{A}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}}}\iint_{U_{b(\mathbf{A})}^{2}}\Big{|}\underset{\mathbf{A}\leftrightarrow\mathbf{B}}{\sum_{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}}e^{i\xi\left|\Delta_{\mathbf{A},\mathbf{B},\mathbf{C}}\right|}\Big{|}d\nu\otimes
d\nu-\underset{\mathbf{C}\rightsquigarrow\mathbf{A}}{\underset{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}{\sum_{\mathbf{C}\in\mathcal{R}_{K+1}}}}\iint_{U_{b(\mathbf{A})}^{2}}\Big{|}\underset{\mathbf{A}\leftrightarrow\mathbf{B}}{\sum_{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}}e^{i\xi\left|\tilde{\Delta}_{\mathbf{A},\mathbf{B},\mathbf{C}}\right|}\Big{|}d\nu\otimes
d\nu\Bigg{|}$

$\lesssim|\xi|e^{-(2k+1+\alpha)\lambda n}\lesssim
e^{-(\alpha\lambda-\varepsilon_{0})n},$

since $|\xi|\simeq e^{(2k+1)\lambda n}e^{\varepsilon_{0}n}$.

To conclude, we notice that
$|\xi||\tilde{\Delta}_{\mathbf{A},\mathbf{B},\mathbf{C}}|$ can be written as a
product like so:

$|\xi||\tilde{\Delta}_{\mathbf{A},\mathbf{B},\mathbf{C}}|(x,y)=\eta_{\mathbf{A},\mathbf{C}}(x,y)\zeta_{1,\mathbf{A}}(\mathbf{b}_{1})\dots\zeta_{k,\mathbf{A}}(\mathbf{b}_{k})$

where

$\eta_{\mathbf{A},\mathbf{C}}(x,y):=|\xi||\partial_{u}\phi(f^{K}g_{\mathbf{C}^{\prime}}(x_{\mathbf{a}_{0}})||\partial_{u}f^{K}(g_{\mathbf{C}^{\prime}}(x_{\mathbf{a}_{0}}))||g_{\mathbf{C}^{\prime}}(x_{\mathbf{a}_{0}})|e^{-2k\lambda
n}d^{u}(g_{\mathbf{a}_{k}}x,g_{\mathbf{a}_{0}}y).$

We estimate $\eta_{\mathbf{A},\mathbf{C}}$ using the hypothesis made on
$\partial_{u}\phi$, the estimates of section 3, and the mean value theorem, to
get

$C_{\phi,\chi}^{-1}e^{-\varepsilon\beta
n}e^{\varepsilon_{0}n}d^{u}(x,y)\leq\eta_{\mathbf{A},\mathbf{C}}(x,y)\leq
C_{\phi,\chi}e^{\varepsilon\beta n}e^{\varepsilon_{0}n}\leq
e^{2\varepsilon_{0}n}.$

Notice that for the lower inequality to hold, it was critical for
$\mathbf{a}_{0}$ to be in $\mathcal{R}_{n+1}(\omega)$, and not just in
$\mathcal{R}_{n+1}$.

We then see that $\eta_{\mathbf{A},\mathbf{C}}(x,y)\in J_{n}$ as soon as
$d^{u}(x,y)\geq C_{\phi,\chi}e^{\varepsilon\beta n-\varepsilon_{0}n/2}$. To
get rid of the part of the integral where $d^{u}(x,y)$ is too small, we use
the upper regularity of $\nu$, proved in Lemma 2.9. For all $y\in\mathcal{U}$,
the ball $B(y,C_{\phi,\chi}e^{\varepsilon\beta n-\varepsilon_{0}n/2})$ have
measure $\lesssim e^{-(\varepsilon_{0}/2-\beta\varepsilon)\delta_{up}n}$, so
that by integrating over $y$,

$\nu\otimes\nu\left(\\{(x,y)\in\mathcal{U}^{2}\ |\
|x-y|<C_{\phi,\chi}e^{\varepsilon\beta n-\varepsilon_{0}n/2}\\}\right)\lesssim
e^{-(\varepsilon_{0}/2-\beta\varepsilon)\delta_{up}n}$

as well. Hence we can cut the double integral in two, the part near the
diagonal which is controlled by the previous estimates, and the part far away
from the diagonal where $\eta_{\mathbf{A},\mathbf{C}}(x,y)\in J_{n}$.

Once this is done, the sum over $\mathbf{C}$ disappears, as there is no longer
a dependence over $\mathbf{C}$ in the phase. ∎

##  5 The sum product phenomenon

###  5.1 A key theorem

Our second move is to use a powerful theorem of Bourgain to control the sum of
exponential. This version is the Proposition 3.2 of [BD17]. This theorem was
generalized by J. Li in [Li18] and constitute the cornerstone of the method.

######  Proposition 5.1.

Fix $\gamma>0$. There exist $\varepsilon_{2}\in\ ]0,1[$ and
$k\in\mathbb{N}^{*}$ such that the following holds for $\eta\in\mathbb{R}$
with $|\eta|>1$. Let $C_{0}>1$ and let $\lambda_{1},\dots,\lambda_{k}$ be
Borel measures supported on the interval $[C_{0}^{-1},C_{0}]$ with total mass
less than $C_{0}$. Assume that each $\lambda_{j}$ satisfies the following non
concentration property:

$\forall\sigma\in[C_{0}|\eta|^{-1},C_{0}^{-1}|\eta|^{-\varepsilon_{2}}],\quad\lambda_{j}\otimes\lambda_{j}\left(\\{(x,y)\in\mathbb{R}^{2},\
|x-y|\leq\sigma\\}\right)\leq C_{0}\sigma^{\gamma}.$

Then there exists a constant $C_{1}$ depending only on $C_{0}$ and $\gamma$
such that

$\left|\int\exp(i\eta z_{1}\dots z_{k})d\lambda_{1}(z_{1})\dots
d\lambda_{k}(z_{k})\right|\leq C_{1}|\eta|^{-\varepsilon_{2}}$

Unfortunately, in our case the use of large deviations does not allow us to
apply it straightforwardly. To highlight the dependence of $C_{1}$ when
$C_{0}$ is permitted to grow gently, we prove the following proposition. The
proof is the same than in the complex case done in [Le21], but we include it
for completeness. Notice that the complex case does not immediately imply the
real case, as the _projective_ non concentration hypothesis is stronger than
our non concentration hypothesis.

######  Proposition 5.2.

Fix $0<\gamma<1$. There exist $\varepsilon_{1}>0$ and $k\in\mathbb{N}$ such
that the following holds for $\eta\in\mathbb{R}$ with $|\eta|$ large enough.
Let $1<R<|\eta|^{\varepsilon_{1}}$ and let $\lambda_{1},\dots,\lambda_{k}$ be
Borel measures supported on the interval $[R^{-1},R]$ with total mass less
than $R$. Assume that each $\lambda_{j}$ satisfies the following non
concentration property:

$\forall\sigma\in[|\eta|^{-2},|\eta|^{-\varepsilon_{1}}],\quad\lambda_{j}\otimes\lambda_{j}\left(\\{(x,y)\in\mathbb{R}^{2},\
|x-y|\leq\sigma\\}\right)\leq\sigma^{\gamma}.$

Then there exists a constant $c>0$ depending only on $\gamma$ such that

$\left|\int\exp(i\eta z_{1}\dots z_{k})d\lambda_{1}(z_{1})\dots
d\lambda_{k}(z_{k})\right|\leq c|\eta|^{-\varepsilon_{1}}$

###### Proof.

Fix $0<\gamma<1$, and let $\varepsilon_{2}$ and $k$ given by the previous
theorem. Choose $\varepsilon_{1}:=\frac{\varepsilon_{2}}{2(2k+1)}$. Let
$1<R<|\eta|^{\varepsilon_{1}}$, and let $\lambda_{1},\dots,\lambda_{k}$ be
measures that satisfy the hypothesis of Proposition 5.1. We are going to use a
dyadic decomposition.

Let $m:=\lfloor\log_{2}(R)\rfloor+1$. Then $\lambda_{j}$ is supported in the
interval $[2^{-m},2^{m}]$. Define, for $A$ a borel subset of $\mathbb{R}$ and
for $r=-m+1,\dots,m$:

$\lambda_{j,r}(A):=R^{-1}\lambda_{j}\left(2^{r}\left(A\cap[1/2,1[\right)\right)$

Those measures are all supported in $[1/2,1[$, and have total mass
$\lambda_{j,r}(\mathbb{R})\leq 1$.

Moreover, a non concentration property is satisfied by each $\lambda_{j,r}$.
If we fix some $r_{1},\dots,r_{k}$ between $-m+1$ and $m$ and define
${\eta}_{r_{1}\dots r_{k}}:=2^{r_{1}+\dots r_{k}}\eta$, then
$|\eta_{r_{1},\dots,r_{k}}|\geq(2R)^{-k}|\eta|>2^{-k}|\eta|^{1-k\varepsilon_{1}}>1$
if $\eta$ is large enough. Let
$\sigma\in[|\eta_{r_{1},\dots,r_{k}}|^{-1},|\eta_{r_{1},\dots,r_{k}}|^{-{\varepsilon_{2}}}]$.
Then

$\lambda_{j,r}\otimes\lambda_{j,r}\left(\\{(x,y)\in\mathbb{R}^{2},\
|x-y|\leq\sigma\\}\right)=\int_{\mathbb{R}}\lambda_{j,r}([x-\sigma,x+\sigma])d\lambda_{j,r}(x)$
$\leq
R^{-2}\int_{\mathbb{R}}\lambda_{j}\left([2^{r}x-2^{r}\sigma,2^{r}x+2^{r}\sigma]\right)d\lambda_{j}(2^{r}x)$
$=R^{-2}\lambda_{j}\otimes\lambda_{j}\left(\\{(x,y)\in\mathbb{R}^{2},\
|x-y|\leq 2^{r}\sigma\\}\right)$

Since
$2^{r}\sigma\in\left[2^{r}|\eta_{r_{1},\dots,r_{k}}|^{-1},2^{r}|\eta_{r_{1},\dots,r_{k}}|^{-\varepsilon_{2}}\right]\subset\left[(2R)^{-(k+1)}|\eta|^{-1},(2R)^{k+1}|\eta|^{-\varepsilon_{2}}\right]\subset\left[|\eta|^{-2},|\eta|^{\varepsilon_{1}}\right]$
if $|\eta|$ is large enough, we can use the non-concentration hypothesis
assumed for each $\lambda_{j}$ to get:

$\lambda_{j,r}\otimes\lambda_{j,r}\left(\\{(x,y)\in\mathbb{R}^{2},\
|x-y|\leq\sigma\\}\right)\leq
R^{-2}(2^{-r}\sigma)^{\gamma}\leq\sigma^{\gamma}.$

Hence, by the previous proposition, there exists a constant $C_{1}$ depending
only on $\gamma$ such that

$\left|\int\exp(i\eta_{r_{1}\dots r_{k}}z_{1}\dots
z_{k})d\lambda_{1,r_{1}}(z_{1})\dots d\lambda_{k,r_{k}}(z_{k})\right|\leq
C_{1}|\eta_{r_{1}\dots r_{k}}|^{-\varepsilon_{2}}.$

Finally, since

$\lambda_{j}(A)=R\sum_{r=-m+1}^{m}\lambda_{j,r}(2^{-r}A)$

we get that:

$\left|\int\exp(i\eta z_{1}\dots z_{k})d\lambda_{1}(z_{1})\dots
d\lambda_{k}(z_{k})\right|$ $\leq\sum_{r_{1},\dots
r_{k}}R^{k}\left|\int\exp(i\eta z_{1}\dots
z_{k})d\lambda_{1,r_{1}}(2^{-r_{1}}z_{1})\dots
d\lambda_{k,r_{k}}(2^{-r_{k}}z_{k})\right|$ $=\sum_{r_{1},\dots
r_{k}}R^{k}\left|\int\exp(i\eta_{r_{1}\dots r_{k}}z_{1}\dots
z_{k})d\lambda_{1,r_{1}}(z_{1})\dots d\lambda_{k,r_{k}}(z_{k})\right|$ $\leq
C_{1}(2m)^{k}R^{k}|\eta_{r_{1}\dots r_{k}}|^{-\varepsilon_{2}}\leq
4^{k}C_{1}m^{k}R^{2k}|\eta|^{-\varepsilon_{2}}$

Since $m\leq\log_{2}(R)+1$, and since $k$ depends only on $\gamma$, there
exists a constant $c$ that depends only on $\gamma$ such that
$4^{k}C_{1}m^{k}R^{2k}\leq cR^{2k+1}$ for any $R>1$. Finally,
$cR^{2k+1}|\eta|^{-\varepsilon_{2}}\leq|\eta|^{-\varepsilon_{1}}$. ∎

######  Corollary 5.3.

Fix $0<\gamma<1$. There exist $k\in\mathbb{N}^{*}$ and $\varepsilon_{1}>0$
depending only on $\gamma$ such that the following holds for
$\eta\in\mathbb{R}$ with $|\eta|$ large enough. Let
$1<R<|\eta|^{\varepsilon_{1}}$ , $N>1$ and
$\mathcal{Z}_{1},\dots,\mathcal{Z}_{k}$ be finite sets such that
$\\#\mathcal{Z}_{j}\leq RN$. Consider some maps
$\zeta_{j}:\mathcal{Z}_{j}\rightarrow\mathbb{R}$, $j=1,\dots,k$, such that,
for all $j$:

$\zeta_{j}(\mathcal{Z}_{j})\subset[R^{-1},R]$

and

$\forall\sigma\in[|\eta|^{-2},|\eta|^{-\varepsilon_{1}}],\quad\\#\\{\mathbf{b},\mathbf{c}\in\mathcal{Z}^{2}_{j},\
|\zeta_{j}(\mathbf{b})-\zeta_{j}(\mathbf{c})|\leq\sigma\\}\leq
N^{2}\sigma^{\gamma}.$

Then there exists a constant $c>0$ depending only on $\gamma$ such that

$\left|N^{-k}\sum_{\mathbf{b}_{1}\in\mathcal{Z}_{1},\dots,\mathbf{b}_{k}\in\mathcal{Z}_{k}}\exp\left(i\eta\zeta_{1}(\mathbf{b}_{1})\dots\zeta_{k}(\mathbf{b}_{k})\right)\right|\leq
c|\eta|^{-{\varepsilon_{1}}}$

###### Proof.

Define our measures as sums of dirac mass:

$\lambda_{j}:=\frac{1}{N}\sum_{\mathbf{b}\in\mathcal{Z}_{j}}\delta_{\zeta_{j}(\mathbf{b})}.$

We see that $\lambda_{j}$ is supported in $[R^{-1},R]$. The total mass is
bounded by

$\lambda_{j}(\mathbb{R})\leq N^{-1}\\#\mathcal{Z}_{j}\leq R.$

Then, if $\sigma\in[|\eta|^{-2},|\eta|^{-\varepsilon_{1}}]$, we have, for any
$a\in\mathbb{R}$:

$\lambda_{j}\otimes\lambda_{j}\left(\\{(x,y)\in\mathbb{R}^{2},\
|x-y|<\sigma\\}\right)=\frac{1}{N^{2}}\\#\left\\{\mathbf{b},\mathbf{c}\in\mathcal{Z}^{2}_{j},\
|\zeta_{j}(\mathbf{b})-\zeta_{j}(\mathbf{c})|\leq\sigma\right\\}\leq\sigma^{\gamma}.$

Hence, the previous theorem applies directly, and gives us the desired result.
∎

###  5.2 End of the proof assuming non concentration

We will use Corollary 5.3 on the maps $\zeta_{j,\mathbf{A}}$. Let’s carefully
define the framework.
For some fixed $\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)$, define for
$j=1,\dots,k$

$\mathcal{Z}_{j}:=\\{\mathbf{b}\in\mathcal{R}_{n+1},\mathbf{a}_{j-1}\rightsquigarrow\mathbf{b}\rightsquigarrow\mathbf{a}_{j}\
\\},$

so that the maps $\zeta_{j,\mathbf{A}}(\mathbf{b}):=e^{2\lambda
n}|g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}^{\prime}(x_{{\mathbf{a}}_{j}})|$
are defined on $\mathcal{Z}_{j}$. There exists a constant $\beta>0$ such that:

$\\#\mathcal{Z}_{j}\leq e^{\varepsilon\beta n}e^{\delta\lambda n}$

and

$\zeta_{j,\mathbf{A}}(\mathcal{Z}_{j})\subset\left[e^{-\varepsilon\beta
n},e^{\varepsilon\beta n}\right].$

Let $\gamma>0$ be small enough. The theorem 5.3 then fixes $k$ and some
$\varepsilon_{1}$. The goal is to apply Corollary 5.3 to the maps
$\zeta_{j,\mathbf{A}}$, for $N:=e^{\lambda\delta n}$, $R:=e^{\varepsilon\beta
n}$ and $\eta\in J_{n}$. Notice that choosing $\varepsilon$ small enough
ensures that $R<|\eta|^{\varepsilon_{1}}$, and taking $n$ large enough ensures
that $|\eta|$ is large. If we are able to prove the non concentration
hypothesis in this context, then Corollary 5.3 can be applied and we would be
able to conclude the proof of the main Theorem 1.2. Indeed, we already know
that

$|\widehat{\mu}(\xi)|^{2}\lesssim e^{\varepsilon\beta
n}e^{-\lambda\delta(2k+1)n}\sum_{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}\sup_{\eta\in
J_{n}}\Bigg{|}\underset{\mathbf{A}\leftrightarrow\mathbf{B}}{\sum_{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}}e^{i\eta\zeta_{1,\mathbf{A}}(\mathbf{b}_{1})\dots\zeta_{k,\mathbf{A}}(\mathbf{b}_{k})}\Bigg{|}$
$\quad\quad\quad\quad\quad\quad\quad\quad+e^{-\varepsilon_{0}n}+e^{-\delta_{1}(\varepsilon)n}+e^{\varepsilon\beta
n}\left(e^{-\lambda\alpha n}+\kappa^{\alpha
n}+e^{-(\alpha\lambda-\varepsilon_{0})n}+e^{-\varepsilon_{0}\delta_{up}n/2}\right)$

by Proposition 4.1. Since every error term already enjoys exponential decay in
$n$, we just have to deal with the sum of exponentials. By Corollary 5.3, we
can then write

$\sup_{\eta\in
J_{n}}\Bigg{|}\underset{\mathbf{A}\leftrightarrow\mathbf{B}}{\sum_{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}}e^{i\eta\zeta_{1,\mathbf{A}}(\mathbf{b}_{1})\dots\zeta_{k,\mathbf{A}}(\mathbf{b}_{k})}\Bigg{|}\leq
ce^{\lambda k\delta n}e^{-\varepsilon_{0}\varepsilon_{1}n/2},$

and hence we get

$e^{-\lambda\delta(2k+1)n}\sum_{\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)}\sup_{\eta\in
J_{n}}\Bigg{|}\underset{\mathbf{A}\leftrightarrow\mathbf{B}}{\sum_{\mathbf{B}\in\mathcal{R}_{n+1}^{k}}}e^{i\eta\zeta_{1,\mathbf{A}}(\mathbf{b}_{1})\dots\zeta_{k,\mathbf{A}}(\mathbf{b}_{k})}\Bigg{|}$
$\lesssim e^{\varepsilon\beta
n}e^{-\lambda\delta(2k+1)n}e^{\lambda\delta(k+1)n}e^{\lambda\delta
kn}e^{-\varepsilon_{0}\varepsilon_{1}n/2}\lesssim e^{\varepsilon\beta
n}e^{-\varepsilon_{0}\varepsilon_{1}n/2}.$

Now, we see that we can choose $\varepsilon$ small enough so that all terms
enjoy exponential decay in $n$, and since $|\xi|\simeq
e^{\left((2k+1)\lambda+\varepsilon_{0}\right)n}$, we have proved polynomial
decay of $|\widehat{\mu}|^{2}$.

##  6 The non-concentration hypothesis

The last part of this paper is devoted to the proof of the non-concentration
hypothesis that we just used. The strategy is the same than in [SS20] and
[Le21], but the theorem used to conclude will be more recent.

######  Definition 6.1.

For a given $\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}(\omega)$, define for
$j=1,\dots,k$

$\mathcal{Z}_{j}:=\\{\mathbf{b}\in\mathcal{R}_{n+1},\
\mathbf{a}_{j-1}\rightsquigarrow\mathbf{b}\rightsquigarrow\mathbf{a}_{j}\ \\}$

Then define

$\zeta_{j,\mathbf{A}}(\mathbf{b}):=e^{2\lambda
n}|g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}^{\prime}(x_{{\mathbf{a}}_{j}})|$

on $\mathcal{Z}_{j}$. The following is satisfied, for some fixed constant
$\beta>0$:

$\\#\mathcal{Z}_{j}\leq e^{\varepsilon\beta n}e^{\delta\lambda n}$

and

$\zeta_{j,\mathbf{A}}(\mathcal{Z}_{j})\subset\left[e^{-\varepsilon\beta
n},e^{\varepsilon\beta n}\right].$

We are going to prove the following fact, which will allow us to apply
Corollary for $\eta\in J_{n}$, $R:=e^{\varepsilon\beta n}$ and
$N:=e^{\lambda\delta n}$:

######  Proposition 6.1 (non concentration).

There exists $\gamma>0$, and we can choose $\varepsilon_{0}>0$, such that the
following holds. Let $\eta\in J_{n}$. Let
$\mathbf{A}\in\mathcal{R}_{n+1}^{k+1}$. Then, if $n$ is large enough,

$\forall\sigma\in[|\eta|^{-2},|\eta|^{-\varepsilon_{1}}],\quad\sup_{a\in\mathbb{R}}\
\\#\left\\{(\mathbf{b},\mathbf{c})\in\mathcal{Z}^{2}_{j},\
|\zeta_{j,\mathbf{A}}(\mathbf{b})-\zeta_{j,\mathbf{A}}(\mathbf{c})|\leq\sigma\right\\}\leq
N^{2}\sigma^{\gamma}.$

where $R:=e^{\varepsilon\beta n}$, $N:=e^{\lambda\delta n}$ and
$\varepsilon_{1},k$ are fixed by Corollary 5.3.

The proof of Proposition 6.1 is based on a uniform spectral gap for a family
of twisted transfer operators, as in [SS20], also known as Dolgopyat’s
estimates. Before introducing it, let us first reduce our non-concentration
estimate to a statement about Birkhoff sums.

######  Lemma 6.2.

If $\varepsilon_{0}$ and $\gamma$ are such that, for
$\sigma\in[e^{-5\varepsilon_{0}n},e^{-\varepsilon_{1}\varepsilon_{0}n/4}]$,

$\sup_{a\in\mathbb{R}}\\#\left\\{\mathbf{b}\in\mathcal{Z}_{j},\
S_{2n}{\tau}_{F}\left(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}})\right)\in[a-\sigma,a+\sigma]\right\\}\leq
N\sigma^{2\gamma},$

Then Proposition 6.1 is true.

###### Proof.

Suppose that the estimate is true. Let
$|\eta|\in[e^{\varepsilon_{0}n/2},e^{2\varepsilon_{0}n}]$, and then let
$\sigma\in[|\eta|^{-2},|\eta|^{-\varepsilon_{1}}]\subset[e^{-4\varepsilon_{0}n},e^{-\varepsilon_{0}\varepsilon_{1}n/2}].$
Let $a\in[R^{-1},R]$ (it is enough to conclude). Since for $n$ large enough

$\ln(a+\sigma)-\ln(a-\sigma)=\ln(1+\sigma a^{-1})-\ln(1-\sigma a^{-1})\leq
4\sigma a^{-1}\leq 4\sigma R,$

We find that

$\ln\left([a-\sigma,a+\sigma]\right)\subset[\ln a-4R\sigma,\ \ln a+4R\sigma].$

Hence:

$\\#\\{\mathbf{b}\in\mathcal{Z}_{j},\
{\zeta}_{j,\mathbf{A}}(\mathbf{b})\in[a-\sigma,a+\sigma]\\}\leq\\#\\{\mathbf{b}\in\mathcal{Z}_{j},\
\ln{\zeta}_{j,\mathbf{A}}(\mathbf{b})\in[\ln a-4R\sigma,\ln a+4R\sigma]\\}$
$=\\#\\{\mathbf{b}\in\mathcal{Z}_{j},\
S_{2n}{\tau}_{F}\left(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}})\right)\in\left[-\ln
a+2n\lambda-4\sigma R,-\ln a+2n\lambda+4\sigma R\right]\\}$ $\leq
N(4R\sigma)^{2\gamma}\leq N\sigma^{\gamma}$

since
$4R\sigma\in[e^{-5\varepsilon_{0}n},e^{-\varepsilon_{1}\varepsilon_{0}n/4}]$
for large enough $n$. Finally,

$\\#\left\\{(\mathbf{b},\mathbf{c})\in\mathcal{Z}^{2}_{j},\
|{\zeta}_{j,\mathbf{A}}(\mathbf{b})-{\zeta}_{j,\mathbf{A}}(\mathbf{c})|\leq\sigma\right\\}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\quad\quad\quad\quad\quad=\sum_{\mathbf{c}\in\mathcal{Z}_{j}}\left\\{\mathbf{b}\in\mathcal{Z}_{j},\
{\zeta}_{j,\mathbf{A}}(\mathbf{b})\in[\zeta_{j,\mathbf{A}}(\mathbf{c})-\sigma,\zeta_{j,\mathbf{A}}(\mathbf{c})+\sigma]\right\\}\leq
N^{2}\sigma^{\gamma},$

and so proposition 6.1 is true. ∎

To prove that the estimate of Lemma 6.2 is satisfied, we use the following
spectral gap for twisted transfer operators, established in [SS20] (Th. 5.1)
for full branched expanding maps. A similar statement can be found in [DV21]
(Th. 6.4) in a more general setting. This kind of theorem is not new and comes
from the early work of Dolgopyat [Do98]. This recent version is more adapted
in our context, as it deals explicitly with one dimensional shifts that
satisfies a nonlinearity condition that is easily seen to be true in our
context.

Unfortunately, the use of Dolgopyat’s estimates can only be checked if the
function $\tau_{F}$ is smooth enough (Hölder regular is not enough). This is
were we need our additional bunching assumption (B) to conclude: if our
attractor contract strongly enough in the stable direction, then the stable
lamination $W^{s}_{\varepsilon}(x)$ becomes $C^{2+\alpha}$. In particular, the
map $\pi$ is $C^{2+\alpha}$ in the unstable direction, and so $F$ is
$C^{2+\alpha}$ too. In this case, $\tau_{F}$ is $C^{1+\alpha}$, which is
enough to use Dolgopyat’s estimates.

######  Theorem 6.3.

Define, for $s\in\mathbb{C}$, a twisted transfer operator
$\mathcal{L}_{s}:C^{\alpha}(\mathcal{U},\mathbb{C})\rightarrow
C^{\alpha}(\mathcal{U},\mathbb{C})$ as follows:

$\forall x\in U_{b},\ \mathcal{L}_{s}h(x):=\sum_{a\rightarrow
b}e^{(\varphi+s\tau_{F})(g_{ab}(x))}h(g_{ab}(x))$

Iterating this transfer operator yields:

$\forall x\in U_{b},\
\mathcal{L}_{s}^{n}h(x)=\underset{\mathbf{a}\rightsquigarrow
b}{\sum_{\mathbf{a}\in\mathcal{W}_{n+1}}}w_{\mathbf{a}}(x)e^{sS_{n}{\tau}_{F}(g_{\mathbf{a}}(x))}h(g_{\mathbf{a}}(x)).$

Under our nonlinearity condition (NL) and the bunching condition (B), the
following holds. There exists $\rho\in\mathbb{N}$ such that, for any
$s\in\mathbb{C}$ such that $\text{Re}(s)=0$ and $|\text{Im}(s)|>\rho$,

$\forall h\in C^{\alpha}(\mathcal{U},\mathbb{C}),\ \forall n\geq 0,\
\|\mathcal{L}_{s}^{n}h\|_{\infty,\mathcal{U}}\leq\rho|\text{Im}(s)|^{\rho}e^{-n/\rho}\|h\|_{C^{\alpha}(\mathcal{U},\mathbb{C})}.$

It means that this twisted transfer operator is eventually contracting for
large $\text{Im}(s)$. This theorem will play another key role in this paper.

######  Remark 6.1.

This theorem is stated in [DV21] under a "total non linearity hypothesis" made
on $\tau_{F}$. The condition goes as follow: there exists no locally constant
map $\mathfrak{c}:\mathcal{U}^{(1)}\rightarrow\mathbb{R}$ and $\theta\in
C^{1}(\mathcal{U}^{(1)},\mathbb{R})$ such that

$\tau_{F}=\mathfrak{c}-\theta\circ F+\theta.$

This condition is satisfied in our setting. Indeed, suppose that some locally
constant $\mathfrak{c}:\mathcal{U}^{(1)}\rightarrow\mathbb{R}$ satisfy
$\tau_{F}=\mathfrak{c}+\theta\circ F-\theta$ for some $\mathcal{C}^{1}$ map
$\theta$. Then, recall that $\tau_{F}\circ\pi$ and $\tau_{f}$ are
$f$-cohomologous (Lemma 2.13). Hence, if $x\in\Omega_{per}$ is a $f$-periodic
point with period $n_{x}$, the unstable Lyapunov exponent of the associated
periodic orbit is

$\widehat{\lambda}(x)=\frac{1}{n_{x}}S_{n_{x}}\tau_{F}(\pi(x))\in\text{Span}_{\mathbb{Q}}\left(\mathfrak{c}(\mathcal{U})\right),$

which implies that

$\text{dim}_{\mathbb{Q}}\text{Vect}_{\mathbb{Q}}\widehat{\lambda}\left(\Omega_{\text{per}}\right)\leq\text{dim}_{\mathbb{Q}}\text{Span}_{\mathbb{Q}}\left(\mathfrak{c}(\mathcal{U})\right),$

and this is an obvious contradiction to the nonlinearity hypothesis (NL).

######  Lemma 6.4.

Define $\varepsilon_{0}:=\min\left(1/(5\rho(3+\rho)),\alpha\lambda/8\right)$.
Fix $\gamma:=1/4$, and let $\varepsilon_{1}$ and $k$ be fixed by Theorem 5.3.
For $\sigma\in[e^{-5\varepsilon_{0}n},e^{-\varepsilon_{1}\varepsilon_{0}n/4}]$
and if $n$ is large enough,

$\sup_{a\in\mathbb{R}}\\#\left\\{\mathbf{b}\in\mathcal{Z}_{j},\
S_{2n}{\tau}_{F}\left(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}})\right)\in[a-\sigma,a+\sigma]\right\\}\leq
N\sigma^{1/2}.$

###### Proof.

In the proof to come, all the $\simeq$ or $\lesssim$ will be uniform in $a$:
the only relevant information is $\sigma$. So fix
$\sigma\in[e^{-5\varepsilon_{0}n},e^{-\varepsilon_{1}\varepsilon_{0}n/4}]$,
and fix an interval of length $\sigma$, $[a-\sigma,a+\sigma]$. Then choose a
bump function $\chi$ such that $\chi=1$ on $[a-\sigma,a+\sigma]$,
$\text{supp}(\chi)\subset[a-2\sigma,a+2\sigma]$ and such that
$\|\chi\|_{L^{1}(\mathbb{R})}\simeq\sigma$. We can suppose that
$\|\chi^{(l)}\|_{L^{1}(\mathbb{R})}\simeq\sigma^{1-l}$.

Then, we can consider $h$, the $2\pi\mathbb{Z}$ periodic map obtained by
periodizing $\chi$. This will allows us to use Fourier series. By
construction, we see that $\mathbb{1}_{[a-\sigma,a+\sigma]}\leq h.$ Hence:

$\\#\left\\{\mathbf{b}\in\mathcal{Z}_{j},\
S_{2n}{\tau}_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))\in[a-\sigma,a+\sigma]\right\\}\leq\sum_{\mathbf{b}\in\mathcal{Z}_{j}}h\left(S_{2n}{\tau}_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))\right)$
$\leq
RN{\sum_{\mathbf{b}\in\mathcal{Z}_{j}}}w_{\mathbf{b}}(x_{\mathbf{a}_{j}})h\left(S_{2n}{\tau}_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))\right)$
$\leq
RN\underset{\mathbf{a}_{j-1}\rightsquigarrow\mathbf{b}\rightsquigarrow\mathbf{a}_{j}}{\sum_{\mathbf{b}\in\mathcal{W}_{n+1}}}w_{\mathbf{b}}(x_{\mathbf{a}_{j}})h\left(S_{2n}\tau_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))\right).$

Then we develop $h$ using Fourier series. We have:

$\forall x\in\mathbb{R},\
h(x)=\sum_{\mathfrak{n}\in\mathbb{Z}}c_{\mathfrak{n}}(h)e^{i\mathfrak{n}x},$

where

$c_{\mathfrak{n}}(h):=(2\pi)^{-1}\int_{a-\pi}^{a+\pi}h(x)e^{-i\mathfrak{n}x}dx.$

Notice that

$\mathfrak{n}^{l}|c_{\mathfrak{n}}(h)|\simeq|c_{\mathfrak{n}}(h^{(l)})|\lesssim\sigma^{1-l}.$

Plugging
$S_{2n}\tau_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))$
in this expression yields

$h\left(S_{2n}\tau_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))\right)=\sum_{\mathfrak{n}\in\mathbb{Z}}c_{\mathfrak{n}}(h)e^{i\mathfrak{n}S_{2n}\tau_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))},$

and so

$\\#\left\\{\mathbf{b}\in\mathcal{Z}_{j},\
S_{2n}{\tau}_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))\in[a-\sigma,a+\sigma]\right\\}$
$\leq
RN\sum_{\mathfrak{n}\in\mathbb{Z}}c_{\mathfrak{n}}(h)\underset{\mathbf{a}_{j-1}\rightsquigarrow\mathbf{b}\rightsquigarrow\mathbf{a}_{j}}{\sum_{\mathbf{b}\in\mathcal{W}_{n+1}}}w_{\mathbf{b}}(x_{\mathbf{a}_{j}})e^{i\mathfrak{n}S_{2n}\tau_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))}.$

For any block $\mathbf{A}\in\mathcal{W}_{n+1}$, integer $j\in\\{1,\dots,k\\}$
and integer $\mathfrak{n}\in\mathbb{Z}$, define
$\mathfrak{h}_{\mathbf{A},j}^{\mathfrak{n}}\in
C^{\alpha}(\mathcal{U},\mathbb{R})$ by

$\forall x\in U_{b(\mathbf{a})},\
\mathfrak{h}_{\mathbf{A},j}^{\mathfrak{n}}(x):=\exp\left({i\mathfrak{n}S_{n}\tau_{F}(g_{\mathbf{a}_{j-1}}(x))}\right)\quad,\quad\forall
x\in U_{b},\ b\neq b(\mathbf{a}),\
\mathfrak{h}_{\mathbf{A},j}^{\mathfrak{n}}(x):=0.$

With this notation, we may rewrite the sum on $\mathbf{b}$ as follows:

$\underset{\mathbf{a}_{j-1}\rightsquigarrow\mathbf{b}\rightsquigarrow\mathbf{a}_{j}}{\sum_{\mathbf{b}\in\mathcal{W}_{n+1}}}w_{\mathbf{b}}(x_{\mathbf{a}_{j}})\exp\left({i\mathfrak{n}S_{2n}{\tau}_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))}\right)$
$=\underset{\mathbf{a}_{j-1}\rightsquigarrow\mathbf{b}\rightsquigarrow\mathbf{a}_{j}}{\sum_{\mathbf{b}\in\mathcal{W}_{n+1}}}w_{\mathbf{b}}(x_{\mathbf{a}_{j}})e^{i\mathfrak{n}S_{n}{\tau}_{F}(g_{\mathbf{b}}(x_{\mathbf{a}_{j}}))}\exp\left({i\mathfrak{n}S_{n}{\tau}_{F}(g_{\mathbf{a}_{j-1}}g_{\mathbf{b}}(x_{\mathbf{a}_{j}}))}\right)$
$=\underset{\mathbf{b}\rightsquigarrow\mathbf{a}_{j}}{\sum_{\mathbf{b}\in\mathcal{W}_{n+1}}}w_{\mathbf{b}}(x_{\mathbf{a}_{j}})e^{i\mathfrak{n}S_{n}{\tau}_{F}(g_{\mathbf{b}}(x_{\mathbf{a}_{j}}))}\mathfrak{h}_{\mathbf{A},j}^{\mathfrak{n}}\left(g_{\mathbf{b}}(x_{\mathbf{a}_{j}})\right)$
$=\mathcal{L}_{i\mathfrak{n}}^{n}\left(\mathfrak{h}_{\mathbf{A},j}^{\mathfrak{n}}\right)(x_{\mathbf{a}_{j}}).$

A direct computation allows us to estimate the $C^{\alpha}$ norm of
$\mathfrak{h}_{\mathbf{A},j}^{\mathfrak{n}}$. We get, uniformly in $n$:

$\|\mathfrak{h}_{\mathbf{A},j}^{\mathfrak{n}}\|_{C^{\alpha}(\mathcal{U},\mathbb{R})}\lesssim(1+\mathfrak{n}).$

We can now break the estimate into two pieces: high frequencies are controlled
by the contraction property of this transfer operator, and the low frequencies
are controlled by the Gibbs property of $\mu$. We also use the estimates on
the Fourier coefficients on $h$.

$\\#\left\\{\mathbf{b}\in\mathcal{Z}_{j},\
S_{2n}{\tau}_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))\in[a-\sigma,a+\sigma]\right\\}$
$\leq
RN\sum_{\mathfrak{n}\in\mathbb{Z}}c_{\mathfrak{n}}(h)\underset{\mathbf{a}_{j-1}\rightsquigarrow\mathbf{b}\rightsquigarrow\mathbf{a}_{j}}{\sum_{\mathbf{b}\in\mathcal{W}_{n+1}}}w_{\mathbf{b}}(x_{\mathbf{a}_{j}})e^{i\mathfrak{n}S_{2n}\tau_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))}$
$\leq
RN\left(\sum_{|\mathfrak{n}|\leq\rho}|c_{\mathfrak{n}}(h)|\sum_{\mathbf{b}\in\mathcal{W}_{n+1}}w_{\mathbf{b}}(x_{\mathbf{a}_{j}})+\sum_{|\mathfrak{n}|>\rho}|c_{\mathfrak{n}}(h)||\mathcal{L}_{i\mathfrak{n}}^{n}(\mathfrak{h}_{\mathbf{A},j})(x_{\mathbf{a}_{j}})|\right)$
$\lesssim
RN\left(\sigma\sum_{\mathbf{b}\in\mathcal{W}_{n+1}}\mu(P_{\mathbf{b}})+\sum_{|\mathfrak{n}|>\rho}|c_{\mathfrak{n}}(h)|\|\mathcal{L}_{i\mathfrak{n}}^{n}(\mathfrak{h}_{\mathbf{A},j})\|_{\infty,\mathcal{U}}\right)$
$\lesssim
RN\sigma+RN\sum_{|\mathfrak{n}|>\rho}|c_{\mathfrak{n}}(h)|\mathfrak{n}^{\rho}e^{-n/\rho}\|\mathfrak{h}_{\mathbf{A},j}^{\mathfrak{n}}\|_{C^{\alpha}(\mathfrak{U},\mathbb{C})}$

$\lesssim
RN\sigma+RNe^{-n/\rho}\sum_{|\mathfrak{n}|>\rho}|c_{\mathfrak{n}}(h)||\mathfrak{n}|^{\rho+3}|\mathfrak{n}|^{-2}$
$\leq CRN(\sigma+e^{-n/\rho}\sigma^{-(\rho+2)}),$

for some constant $C>0$. We are nearly done. Since
$\sigma\in[e^{-5\varepsilon_{0}n},e^{-\varepsilon_{1}\varepsilon_{0}n/4}]$, we
know that $\sigma^{-(\rho+2)}\leq e^{5(\rho+2)\varepsilon_{0}n}$. Now is the
time where we fix $\varepsilon_{0}$: choose

$\varepsilon_{0}:=\min\left(\frac{1}{5\rho(3+\rho)},\frac{\alpha\lambda}{8}\right).$

Then $e^{-n/\rho}\sigma^{-(\rho+2)}\leq
e^{(-1/\rho+5(\rho+2)\varepsilon_{0})n}\leq e^{-5\varepsilon_{0}n}\leq\sigma$
for $n$ large enough. Hence, we get

$\\#\left\\{\mathbf{b}\in\mathcal{Z}_{j},\
S_{2n}\tau_{F}(g_{\mathbf{a}_{j-1}^{\prime}\mathbf{b}}(x_{\mathbf{a}_{j}}))\in[a-\sigma,a+\sigma]\right\\}\leq
2CRN\sigma.$

Finally, since $\sigma^{1/2}$ is quickly decaying compared to $R$, we have

$2CRN\sigma\leq N\sigma^{1/2}$

provided $n$ is large enough. The proof is done. ∎

##  Appendix A The nonlinearity condition is generic

######  Theorem A.1.

The condition
$\text{dim}_{\mathbb{Q}}\text{Vect}_{\mathbb{Q}}\widehat{\lambda}\left(\Omega_{\text{per}}\right)=\infty$
is generic in the following sense: for any given Axiom A diffeomorphism
$f:M\rightarrow M$ and a fixed basic set $\Omega$ that is not an isolated
periodic orbit, a generic $C^{1}$ perturbation $\tilde{f}$ of $f$ have an
hyperbolic set $\tilde{\Omega}$ on which the dynamic is conjugated with
$(f,\Omega)$, and
$\text{dim}_{\mathbb{Q}}\text{Span}_{\mathbb{Q}}\widehat{\lambda}\left(\tilde{\Omega}_{\text{per}}\right)=\infty$.

###### Proof.

Let $f:M\longrightarrow M$ be a $C^{2}$ Axiom A diffeomorphism, and fix
$\Omega$ a basic set for $f$ that is not an isolated periodic orbit. Recall
that this implies that $\Omega$ is infinite (and even perfect). Since $f$ is
Axiom A, $\Omega_{\text{per}}$ is then infinite. Let $U\supset\Omega$ be a
small open neighborhood in $M$. Consider a small enough open neighborhood
around $f$ in the space of $C^{1}$ maps, $\mathfrak{U}\subset C^{1}(M,M)$.
Then, there exists a map

$\Phi:\mathfrak{U}\rightarrow 2^{M}\times\mathcal{C}^{0}(\Omega,M)$

such that for any $\tilde{f}\in\mathfrak{U}$,
$\Phi(\tilde{f})=(\tilde{\Omega},h)$ satisfies the following properties:

  * •

$\tilde{\Omega}\subset U$ is a hyperbolic set for $\tilde{f}:M\rightarrow M$,

  * •

$\tilde{\Omega}_{\text{per}}$ is dense in $\tilde{\Omega}$,

  * •

$h:\Omega\rightarrow\tilde{\Omega}$ is an homeomorphism and conjugates
$(\Omega,f)$ with $(\tilde{\Omega},\tilde{f})$.

  * •

The map $\tilde{f}\in\mathfrak{U}\mapsto h\in C^{0}(\Omega,M)$ is continuous.

This is Theorem 5.5.3 in [BS02]. See also [KH95], page 571: stability of
hyperbolic sets.
So let $\tilde{f}\in\mathfrak{U}$. Since $h$ conjugates $(f,\Omega)$ with
$(\tilde{f},\tilde{\Omega})$, it is also a bijection between
$\Omega_{\text{per}}$ and $\tilde{\Omega}_{\text{per}}$. For
$x\in\Omega_{\text{per}}$, we write $\tilde{x}\in\tilde{\Omega}$ for $h(x)$.

We are ready to prove cleanly that the condition

$\dim_{\mathbb{Q}}\text{Vect}_{\mathbb{Q}}\widehat{\lambda}\left(\tilde{\Omega}_{\text{per}}\right)=\infty$

is generic in $\tilde{f}\in\mathfrak{U}$. The condition may be rewritten in
the following way:

$\left\\{\tilde{f}\in\mathfrak{U}\ |\ \forall N\geq
1,\exists\tilde{x_{1}},\dots,\tilde{x_{N}}\in\tilde{\Omega}_{\text{per}}\ ,\
\left(\widehat{\lambda}(\tilde{x}_{1}),\dots,\widehat{\lambda}(\tilde{x}_{N})\right)\
\text{is linearly independant over}\ \mathbb{Q}\right\\}.$

ie

$\bigcap_{N\geq 1}\bigcup_{\Omega_{\text{per}}^{N}}\
\left\\{\tilde{f}\in\mathfrak{U}\ |\
\left(\widehat{\lambda}(\tilde{x}_{1}),\dots,\widehat{\lambda}(\tilde{x}_{N})\right)\
\text{is}\ \mathbb{Q}-\text{independant}\right\\}$

Fix once and for all a sequence of distinct periodic orbits $(y_{N})_{N\geq
1}\in\Omega_{\text{per}}$. Then:

$\left\\{f\in\mathfrak{U}\ |\
\dim_{\mathbb{Q}}\text{Vect}_{\mathbb{Q}}\widehat{\lambda}\left(\tilde{\Omega}_{\text{per}}\right)=\infty\right\\}\supset\bigcap_{N\geq
1}\left\\{\tilde{f}\in\mathfrak{U}\ |\
\left(\widehat{\lambda}(\tilde{y}_{1}),\dots,\widehat{\lambda}(\tilde{y}_{N})\right)\
\text{is}\ \mathbb{Q}-\text{independant}\right\\}$ $=\bigcap_{N\geq
1}\bigcap_{(m_{1},\dots,m_{N})\in\mathbb{Z}^{N}\setminus\\{0\\}}\left\\{\tilde{f}\in\mathfrak{U}\
|\ \sum_{i=1}^{N}m_{i}\widehat{\lambda}(\tilde{y}_{i})\neq 0\right\\}$

And this is a countable intersection of dense open sets. Indeed, fix some
$N\geq 1$ and some integers
$(m_{1},\dots,m_{N})\in\mathbb{Z}^{N}\setminus\\{0\\}$, and denote
$\mathfrak{U}_{m_{1},\dots,m_{N}}:=\left\\{\tilde{f}\in\mathfrak{U}\ |\
\sum_{i=1}^{N}m_{i}\widehat{\lambda}(\tilde{y}_{i})\neq 0\right\\}$. Without
loss of generality, we may suppose that $m_{1}\neq 0$.

First of all, $\mathfrak{U}_{m_{1},\dots,m_{N}}$ is open. This is because the
function

$\begin{array}[t]{lrcl}&\mathfrak{U}&\longrightarrow&\mathbb{R}^{N}\\\
&\tilde{f}&\longmapsto&(\widehat{\lambda}(\tilde{y}_{1}),\dots,\widehat{\lambda}(\tilde{y}_{N}))\end{array}$

is continuous. Indeed, recall that

$\widehat{\lambda}(\tilde{y}_{i})=\frac{1}{n_{i}}\sum_{k=0}^{n_{i}}\log|\partial_{u}\tilde{f}(\tilde{f}^{k}(\tilde{y}_{i}))|,$

where $n_{i}$ is the period of $y_{i}\in\Omega_{\text{per}}$ so is constant in
$\tilde{f}$. Moreover, $\tilde{f}$ varies smoothly in $C^{1}$ norm,
$\tilde{y}_{i}$ varies continuously, and finally the unstable direction of
$\tilde{f}$ also varies continuously in $\tilde{f}$, see Corollary 2.9 in
[CP15].

Now, we check that $\mathfrak{U}_{m_{1},\dots,m_{N}}$ is dense in
$\mathfrak{U}$. Without loss of generality, it suffices to prove that
$f\in\overline{\mathfrak{U}_{m_{1},\dots,m_{N}}}$. If
$f\in\mathfrak{U}_{m_{1},\dots,m_{N}}$ then we have nothing to prove. So
suppose $f\notin\mathfrak{U}_{m_{1},\dots,m_{N}}$. We are going to construct a
perturbation $\tilde{f}$ in $\mathfrak{U}_{m_{1},\dots,m_{N}}$.

Notice that the set

$\Lambda:=\left\\{f^{k}(y_{i})\ |\ k\in\mathbb{Z},i\in\llbracket
1,N\rrbracket\right\\}\subset{\Omega}_{\text{per}}$

is discrete and finite. So there exists a small open neighborhood $U$ of
$y_{1}$ such that $U\cap\Lambda=\\{y_{1}\\}$. Then, choose
$\varphi:M\rightarrow M$ a diffeomorphism of $M$ that is a small perturbation
of the identity on $M$ such that $\varphi=Id$ on $M\setminus U$,
$\varphi(y_{1})=y_{1}$, $(d\varphi)_{y_{1}}(E^{u}(y_{1}))=E^{u}(y_{1})$ and
such that $\partial_{u}\varphi(y_{1})=1+\varepsilon$ for some small
$\varepsilon>0$.

Define $\tilde{f}:=f\circ\varphi$. Then $\tilde{f}$ is a small perturbation of
$f$. Moreover, the $y_{i}$ are periodic orbits for $\tilde{f}$. Since $h$
sends periodic orbits to periodic orbits of the same period, and since $h$ is
a small perturbation of the identity, it follows necessarily that
$\tilde{y_{i}}=h(y_{i})=y_{i}$. Hence, the Lyapunov exponents of interests
are:

$\widehat{\lambda}(\tilde{y}_{1})=\frac{1}{n_{1}}\log(|\partial_{u}f(y_{1})(1+\varepsilon)|)+\frac{1}{n_{1}}\sum_{k=1}^{n_{1}-1}\log|\partial_{u}\tilde{f}(f^{k}(y_{1}))|=\frac{1}{n_{1}}\log(1+\varepsilon)+\widehat{\lambda}(y_{1}).$
$\widehat{\lambda}(\tilde{y}_{i})=\widehat{\lambda}(y_{i}),\forall
i\in\llbracket 2,N\rrbracket.$

Hence

$\sum_{i=1}^{n}m_{i}\widehat{\lambda}(\tilde{y}_{i})=\frac{1}{n_{1}}\log(1+\varepsilon)+\sum_{i=1}^{N}m_{i}\widehat{\lambda}(y_{i})=\frac{m_{1}}{n_{1}}\log(1+\varepsilon)\neq
0,$

since $f$ is supposed to be
$\mathfrak{U}\setminus\mathfrak{U}_{n_{1},\dots,n_{N}}$ and $m_{1}\neq 0$. The
proof is done. ∎

##  Appendix B An explicit solenoid satisfying Theorem 1.2

###  B.1 A nonlinear perturbation of the doubling map

Our goal is to construct a nonlinear perturbation of the doubling map on the
circle on which we will be able to compute some periodic orbits and Lyapunov
exponents. We want them to be linearly independant over $\mathbb{Q}$.

######  Lemma B.1.

Let $\Lambda:=\\{\frac{1}{2^{K}-1}\ |\ K\geq 2\\}$ and

$\tilde{\Lambda}:=\bigsqcup_{K\geq 2}\left[\frac{1}{2^{K}-1}-\frac{1}{8^{K}}\
,\ \frac{1}{2^{K}-1}+\frac{1}{8^{K}}\right]$

Let $N\geq 2$ and $k\in\llbracket 1,N-1\rrbracket$. Then

$\frac{2^{k}}{2^{N}-1}\notin\tilde{\Lambda}.$

###### Proof.

First of all, the union is indeed disjoint. Indeed, for $K\geq 2$, we see that

$\frac{1}{2^{K+1}-1}+\frac{1}{8^{K+1}}<\frac{1}{2^{K}-1}-\frac{1}{8^{K}}$

Since

$\frac{1}{2^{K}-1}-\frac{1}{2^{K+1}-1}>\frac{1}{2^{K}}-\frac{1}{\frac{3}{2}2^{K}}=\frac{1}{3}\frac{1}{2^{K}}>\frac{9}{8}\frac{1}{8^{K}}.$

Now, fix $N\geq 2$. We first do the case where $k=N-1$. In this case,

$\frac{2^{k}}{2^{N}-1}=\frac{1}{2}\frac{1}{1-2^{-N}}>1/2,$

which proves that $\frac{2^{k}}{2^{N}-1}\notin\tilde{\Lambda}$ since
$\tilde{\Lambda}\subset[0,0.4]$.

We still have to do the case where $k\in\llbracket 1,N-2\rrbracket$, and
$N\geq 3$. First of all, we check that

$\frac{2^{k}}{2^{N}-1}<\frac{1}{2^{N-k}-1}-\frac{1}{8^{N-k}}.$

Indeed:

$\frac{1}{2^{N-k}-1}-\frac{2^{k}}{2^{N}-1}=\frac{1}{2^{N-k}-1}-\frac{1}{2^{N-k}-2^{-k}}$
$\geq\frac{1}{2^{N-k}-1}-\frac{1}{2^{N-k}-2^{-1}}$
$=\frac{1}{2(2^{N-k}-1)(2^{N-k}-2^{-1})}$ $>\frac{1}{2\cdot
4^{N-k}}>\frac{1}{8^{N-k}}.$

Second, we check that

$\frac{1}{2^{N-k+1}-1}+\frac{1}{8^{N-k+1}}<\frac{2^{k}}{2^{N}-1}.$

Indeed,

$\frac{2^{k}}{2^{N}-1}-\frac{1}{2^{N-k+1}-1}=\frac{1}{2^{N-k}-2^{-k}}-\frac{1}{2^{N-k+1}-1}$
$>\frac{1}{2^{N-k}}-\frac{1}{\frac{3}{2}\cdot 2^{N-k}}$
$=\frac{1}{3}\frac{1}{2^{N-k}}>\frac{1}{8^{N-k+1}}$

And this conclude the proof.

∎

######  Lemma B.2.

Fix $(\alpha_{N})_{N\geq 2}$ a family of real numbers so that $\sum_{K\geq
2}|\alpha_{K}|8^{Kr}<\infty$ for any $r\geq 1$. There exists a
$\mathcal{C}^{\infty}$ function $g:\mathbb{R}\rightarrow\mathbb{R}$ such that:

  * •

$\text{supp g}\subset\tilde{\Lambda}$,

  * •

$\|g\|_{C^{1}}<1$.

  * •

$\forall N\geq 2,\ \forall k\in\llbracket 1,N-1\rrbracket,\
g\left(\frac{2^{k}}{2^{N}-1}\right)=0$

  * •

$\forall N\geq 2,\ g^{\prime}\left(\frac{1}{2^{N}-1}\right)=\alpha_{N}$.

###### Proof.

Let $\theta:\mathbb{R}\rightarrow[0,1]$ be a smooth bump function such that
$\text{supp}(\theta)\subset[-1/2,1/2]$ and such that $\theta=1$ on
$[-1/4,1/4]$. Let $\chi(x):=x\theta(x)$.

The map $\chi$ is supported in $[-1/2,1/2]$ and satisfy $\chi(0)=0$,
$\chi^{\prime}(0)=1$. Define, for $N\geq 2$,

$\chi_{N}(x):=\chi\left(8^{N}\left(x-\frac{1}{2^{N}-1}\right)\right).$

Then $\chi_{N}$ is supported in $\left[\frac{1}{2^{N}-1}-\frac{1}{8^{N}}\ ,\
\frac{1}{2^{N}-1}+\frac{1}{8^{N}}\right]$, and so the function

$g(x):=\sum_{K\geq 2}\alpha_{K}{8^{-K}}{\chi_{K}(x)}$

is well defined and supported in $\tilde{\Lambda}$. Since, for all $r\geq 1$,
$\sum_{K\geq 2}|\alpha_{K}|8^{Kr}<\infty$, $g$ is a $\mathcal{C}^{\infty}$
function, with derivatives

$g^{(r)}(x)=\sum_{K\geq
2}\alpha_{K}8^{(r-1)K}\chi^{(r)}\left(8^{N}\left(x-\frac{1}{2^{N}-1}\right)\right).$

In particular,

$\|g\|_{C^{r}}\leq\sum_{K\geq 2}|\alpha_{K}|8^{(r-1)K}$

And so we see that replacing the sequence $(\alpha_{K})_{K\geq 2}$ by
$(\alpha_{K+K_{0}})_{K\geq 2}$ allows us to choose $\|g\|_{C^{r}}$ as small as
we want if so desired. The fact that $g$ vanishes on $\Lambda$ and the
computation of its derivative on this set follows from the previous formulae.
∎

######  Lemma B.3.

The map $f(x):=2x+g(x)$ can be seen as a smooth map
$\mathbb{S}^{1}\longrightarrow\mathbb{S}^{1}$.

###### Proof.

We will identify $[0,1]/\\{0\sim 1\\}$ to the circle. The only thing to check
is if $g$ stays smooth after the quotient. It stays smooth indeed, since
$g^{(r)}(0)=g^{(r)}(1)=0$ for all $r\geq 0$. ∎

######  Lemma B.4.

For $g$ small enough, $(f,\mathbb{S}^{1})$ is a nonlinear perturbation of the
usual doubling map. A familly of periodic points for this dynamical system is
the set $\Lambda$, seen as a subset of the circle. The associated lyapunov
exponents are $\ln(2)+\frac{1}{N}\ln\left(1+\alpha_{N}/2\right)$.

###### Proof.

Let $N\geq 2$ and let $x_{N}:=\frac{1}{2^{N}-1}$. We check that $x_{N}$ is
periodic with period $N$. Indeed, for all $k\in\llbracket 1,N-1\rrbracket$, we
see that $f^{k}(x_{N})=2^{k}x_{N}$ by construction of $g$, and then

$f^{N}(x_{N})=\frac{2^{N}}{2^{N}-1}=\frac{1}{2^{N}-1}\ \text{mod}\ 1.$

Hence $x_{N}$ is $f$-periodic with period $N$. Its lyapunov exponent is then

$\widehat{\lambda}(x_{N}):=\frac{1}{N}\sum_{k=0}^{N-1}\ln|f^{\prime}(f^{k}(x_{N}))|=\frac{1}{N}\sum_{k=0}^{N-1}\ln|2+g^{\prime}(2^{k}x_{N}))|=\ln(2)+\frac{1}{N}\ln\left(1+\alpha_{N}/2\right).$

∎

######  Lemma B.5.

We can choose the sequence $(\alpha_{N})$ so that the Lyapunov exponents
$\widehat{\lambda}(x_{N})$ is a family a real numbers that are linearly
independent over $\mathbb{Q}$.

###### Proof.

By the Lindemann–Weierstrass theorem [Ba90], we just have to choose the
$\alpha_{N}$ so that

$\widehat{\lambda}(x_{N})=e^{\beta_{N}}$

where $\beta_{N}$ are distinct algebraic numbers (while still ensuring that
the sums of Lemma B.2 converges).

In other words, it suffice to choose $\alpha_{N}$ of the form