Je Souhaite

Hans Henrik Rugh

Masato Tsujii

Nalini Anantharaman

Frédéric Naud

Stéphane Nonnenmacher

Maciej Zworski

Thibault

Mihajlo, Benjamin, Victor, Nir Hugo ) Yann

Keywords: Hyperbolic dynamics, Ruelle spectrum, trace formula, dynamical determinant, ultradierentiability

a laissée que je voudrais la remercier.

sur les familles à paramètre de projecteurs. Leur retour sempiternel a quelque chose de rassurant dans un monde au futur incertain.

Il me faut également remercier Nicolas Bergeron : j'ai beaucoup apprécié d'enseigner le sujet fascinant qu'est l'analyse complexe avec lui.

Mes pensées vont ensuite à toutes les personnes avec qui j'ai pu me lier d'amitié à l'IMJ-PRG puis au LPSM, et qui ont contribué à une ambiance de travail agréable. Je pense bien sûr en premier lieu à Vincent et Jean-Michel, et aux samedis après-midi passés à regarder Survivor. Il me semble important de remercier Hugo pour son accueil bienveillant dans le couloir des doctorants de l'IMJ. Enn, je voudrais exprimer ma gratitude envers tous les doctorants avec qui j'ai pu partager un bureau. Je tiens en particulier à préciser à Xavier que je ne lui tiens pas rigueur d'avoir voulu faire croire à ma mort.

Il y a quelques personnes qui ne sont pas directement lié au cadre professionel qu'il me semble raisonnable de remercier. Tout d'abord, les Arcueillais.e.s (Gwenaël, Pierre-Louis, Trace formulae and dynamical determinants for hyperbolic maps . . . . . .

Introduction

The interest in hyperbolic dynamics goes back at least to the work of Hadamard [START_REF] Hadamard | Les surfaces à courbures opposées et leurs lignes géodésiques[END_REF] on the geodesic ow on negatively curved surfaces. An important milestone in the history of the study of such systems is the denition by Anosov [START_REF] Anosov | Geodesic ows on closed Riemannian manifolds with negative curvature[END_REF] of the class of ows that now bear his name. The introduction of this notion by Anosov was motivated by the study of the dynamical properties of the geodesic ow on the unit tangent bundle of a Riemannian manifold with negative, a priori non-constant, sectional curvature. The seminal work of Anosov was the starting point of a systematic study of Anosov ows, and of their discrete-time counter parts, Anosov dieomorphisms. This line of research has been very active since then, and, while many questions remain unanswered, the understanding of hyperbolic dynamics has greatly improved since the work of Anosov, due to the development of various tools to study those systems: Markov partitions, specication property, coupling arguments, Young towers, etc.

In this thesis, we focus on a particular way to deal with hyperbolic dynamical systems:

the so-called functional approach. The basic idea behind this approach is simple: in order to study a dynamical system T , one may study the associated composition operator (also called Koopman operator ): v → v • T . In particular, we expect that the spectral theory of the Koopman operator contains relevant information concerning the statistical properties of the system T . Koopman operator technique goes back to the work of Koopman and Von Neumann in the early 30's. However, in the particular case of uniformly hyperbolic systems, the work of Ruelle [START_REF] Ruelle | Statistical mechanics of a one-dimensional lattice gas[END_REF][START_REF] Ruelle | Thermodynamic Formalism[END_REF] plays a founding role that can hardly be overestimated.

We will see that the composition operator associated to a uniformly hyperbolic dynamical system may be used to dene a relevant notion of spectrum that enables to describe the statistical properties of this system: the Ruelle or RuellePollicott spectrum. The main focus of this thesis is a ne property of this spectrum: a trace formula (TFF) for C ∞ Anosov ows conjectured by Dyatlov and Zworski in [START_REF] Dyatlov | Dynamical zeta functions for Anosov ows via microlocal analysis[END_REF]. We will develop the idea that the relevant notion to study this trace formula and related questions is the notion of ultradierentiability, that is we will study hyperbolic dynamical systems that belong to classes of regularity that are intermediate between C ∞ and analytic. This approach originated in the suggestion by Sébastien Gouëzel that something could be said about trace formulae for Gevrey hyperbolic systems, and we give indeed in the last chapter of this the-

INTRODUCTION

sis a detailed study of the spectral theory for Gevrey Anosov ows. However, we consider in this thesis much more general classes of regularity than Gevrey, using the language of DenjoyCarleman classes. We give an analogue in this setting of the methods initiated by Ruelle [START_REF] Ruelle | Zeta-functions for expanding maps and Anosov ows[END_REF], Rugh [START_REF] Henrik | The correlation spectrum for hyperbolic analytic maps[END_REF][START_REF] Henrik | Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems[END_REF] and Fried [START_REF] Fried | Meromorphic zeta functions for analytic ows[END_REF] to study analytic hyperbolic dynamics. Their works was based on the use of strong functional analytic tools available in the real-analytic category, relying in particular on the theory of nuclear operators developed by Grothendieck [START_REF] Grothendieck | Produits Tensoriels Topologiques et Espaces Nucléaires[END_REF]. We will see that, going to larger classes of regularity, weaker versions of these tools may still be constructed. In particular, the trace formula (TFF) will be deduced for Anosov ow that satisfy a certain ultradierentiability condition, while we do not expect (TFF) to hold for all C ∞ Anosov ows.

In the remainder of this introduction, we will recall needed facts about the functional approach to statistical properties of hyperbolic dynamical systems. After that, we will describe our main results and propose further lines of work. Most of the results from this thesis may be found in the articles [START_REF] Jézéquel | Local and global trace formulae for smooth hyperbolic dieomorphisms[END_REF] (published in Journal of Spectral Theory), [START_REF] Jézéquel | Global trace formula for ultra-dierentiable Anosov ows[END_REF] (submitted for publication), [START_REF] Jézéquel | Transfer operator for ultradierentiable expanding maps of the circle[END_REF] (published in Ergodic Theory and Dynamical Systems) and [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF] (submitted for publication).

Statistical properties of hyperbolic systems

Let us recall that a dynamical systems is said to be hyperbolic if it is expanding in a direction (the unstable direction) and contracting in a supplementary direction (the stable direction). Hyperbolic systems are notoriously chaotic: the long-time asymptotic of an orbit is highly sensitive to initial conditions. While the systems that we study in this thesis are deterministic, they tend to behave like random systems on large scales of times.

The dynamics of a contracting map is well-known and far from chaotic. Consequently, the source of chaos in hyperbolic systems should be looked for in the unstable direction, and it seems natural to try to understand rst statistical properties for expanding maps.

Statistical properties for expanding maps of the circle

Expanding maps of the circle are the simplest examples of hyperbolic systems that we consider in this thesis. They are indeed hyperbolic systems, but with trivial stable direction.

Denition 1 (Expanding map of the circle). We say that a C 1 map T from S 1 = R/Z to itself is expanding if there is λ > 1 such that, for every x ∈ S 1 , we have |T (x)| ≥ λ. We say that λ is a dilation constant for T .

We are interested in these systems precisely because of their simplicity. Indeed, we will use expanding maps of the circle as a toy model in 2.2 to illustrate our methods to study transfer operators associated to ultradierentiable maps. Considering expanding maps, we do not need to worry about the geometry of the stable and unstable directions, and we can consequently focus on how the regularity of the map and the dilatation contribute to produce rich statistical properties (and associated spectral theory). In order to understand how a chaotic behavior can arise from a deterministic system, it is interesting to consider the most basic example of expanding map.

Example 1. The doubling map x → 2x mod 1 is an expanding map of the circle.

If x and y are two points on S 1 at distance ∼ 2 -n , then after n iterations of the doubling map, the images of the points x and y are at distance ∼ 1. If we were considering the doubling map on R, the images of x and y would keep diverging under further iterations.

However, since we are looking at the doubling map on the quotient S 1 = R/Z, it is not possible for their images to keep diverging after n iterations since their distance has become similar to the diameter of the circle. Actually, after these rst n iterations, the relative position of orbits of x and y could be almost anything: loosely speaking, they became independent.

These considerations suggest to apply methods from probability theory to study expanding maps and more generally hyperbolic systems. The spectral theory of the transition matrix plays a key role in the study of Markov chains. When studying an expanding map T , the analogue of the transition matrix is the already mentioned Koopman operator v → v • T acting on some space of functions or distributions on the circle (the choice of the space on which the Koopman operator acts is in fact of the utmost importance, as we will see later). In the particular case of an expanding map of the circle T , it is more convenient to consider its adjoint, the transfer operator or RuellePerronFrobenius operator:

L = L T : v → T y=x 1 |T (y)|
v(y).

(1)

An advantage of the functional approach based on the study of the transfer operator is that we replaced a potentially complicated non-linear dynamics by a linear one: the iterations of the linear operator L. It is natural to consider that the long-time asymptotics of such a linear system should be ruled by the spectral theory of the transfer operator. The price to pay for this simplication is to replace the nite-dimensional compact manifold S 1 by an innite-dimensional vector space, but we hope that the spectral properties of the transfer operator are reminiscent in some sense of the nite dimension. The best situation possible would be for the operator L to be compact. However, the operator L acting on L 2 S 1 or C 0 S 1 is not compact.

This situation may be improved by changing the space on which L acts. The most comfortable situation is when the expanding map T is real-analytic. Indeed, it has been noticed by Ruelle [Rue76] that, in that case, the operator L acting on a space of holomorphic functions on a complex neighbourhood of S 1 is not only compact but even nuclear of order 0. This very powerful property allowed him to use the theory developed by Grothendieck in [START_REF] Grothendieck | Produits Tensoriels Topologiques et Espaces Nucléaires[END_REF] to study the spectral theory for L. To put it more concretely, due to Cauchy's Formula, the operator L acting on a space of holomorphic functions is morally an operator with smooth kernel (and hence a compact operator). Analytic expanding maps of the circle have been widely studied using renements of this method recently [SBJ17, SBJ13, BJ08, Nau12, BJS17, BN19].

When T is only C 1+k for some k > 0, the example of real-analytic maps suggest to make L acts on a space of functions as smooth as possible. However, even acting on the space C k (S 1 ) of C k functions from S 1 to C, the operator L is not compact (see for instance [START_REF] Chicone | Evolution Semigroups in Dynamical Systems and Dierential Equations[END_REF]Theorem 8.53] or [START_REF] Volker | A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces[END_REF]). In order to bypass this diculty, one may introduce the notions of essential spectral radius and quasi-compactness.

Denition 2. Let B be a Banach space and L be a bounded linear operator from B to itself. The essential spectral radius r ess (L) of L is the inmum of the r's such that the intersection of the spectrum σ (L) with {z ∈ C : |z| > r} is made of a nite number of eigenvalues with nite algebraic multiplicities. We say that L is quasi-compact if the essential spectral radius of L is strictly smaller than its spectral radius r (L) see Figure 1.

Using these notions, one can prove:

Theorem 1 ( [START_REF] Ruelle | The thermodynamic formalism for expanding maps[END_REF]). Let k > 0 and T be a C 1+k expanding map of the circle. Then, the transfer operator (1) induces a bounded operator L on the space C k S1 . Moreover, the spectral radius of L is 1 and its essential spectral radius is less than λ -k where λ is a dilation constant for T . In particular, L is quasi-compact.

One of the modern proofs of this theorem is based on a so-called LasotaYorke inequality 1 combined with an argument due to Hennion [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux lipchitziens[END_REF] Corollaire 1] based on Nussbaum's formula for the essential spectral radius [START_REF] Roger | The radius of the essential spectrum[END_REF]. The intuition behind this proof is very simple: the transfer operator L acts like a contraction of factor λ -k on the derivative of order k of a C k function, while the action of the transfer operator on the other derivatives may be considered as a compact perturbation that may consequently impact the spectral radius of L but not its essential spectral radius. This proof is in fact very general and can be adapted to prove that L is quasi-compact on any reasonable space of regular functions2 .

Here, the word regular can be understood in a very loose sense, including even some kind of discontinuities (for instance, the transfer operator L may be proved to be quasi-compact when acting on the space of functions with bounded variations on S 1 ). See for instance [START_REF] Baladi | Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps[END_REF] Part I] and references therein for a comprehensive treatment of this topic.

From the topological mixing of T , we get further information on the spectrum of the operator L acting on C k S 1 . We nd indeed that 1 is the only eigenvalue of L on the unit circle and that its algebraic multiplicity is 1. Moreover, L has an eigenvector ρ associated to the eigenvalue 1 which is everywhere positive. If we normalize ρ so that its mean (for the Lebesgue measure) is 1, then the probability measure µ = ρdx on S 1 is invariant by the action of T that is we have µ T -1 A = µ (A) for every Borel subset A of S 1 . Moreover, it follows from the spectral decomposition for L that T is exponentially mixing for µ: if f, g : S 1 → C are Hölder functions then

S 1 f • T n .gdµ → n→+∞ S 1
f.dµ

S 1 g.dµ (2)
at an exponential rate that only depends on f and g through their Hölder exponent see [START_REF] Ruelle | The thermodynamic formalism for expanding maps[END_REF] for a proof of this fact. Going slightly further into the study of the transfer operator L, we may establish even stronger statistical properties for the expanding map T such as the central limit theorem and the almost sure invariance principle (see for instance [START_REF] Hofbauer | Ergodic properties of invariant measures for piecewise monotonic transformations[END_REF]).

Hence, considerations about the peripheral spectrum of the operator L led to a very ne understanding of statistical properties for the map T . One can go further by considering the deeper part of the spectrum. In particular, it is possible to give a ner asymptotics for the correlations than (2) in terms of spectral data for the operator L. Indeed, if the INTRODUCTION expanding map T is C k+1 with dilation constant λ, and f, g are C k observables, we can write for any r > λ -k the asymptotic expansion [Rue89, Proposition 5.3]

S 1 f • T n .gdµ = n→+∞ z∈σ C k (L) |z|>r m(z)-1 j=0 c j,z (f, g)n j z n + O (r n ) . (3)
Here, σ C k (L) denotes the spectrum of L acting on C k , and if z is an eigenvalue for L, its multiplicity is denoted by m(z). The c j,z 's are continuous bilinear forms on C k (S 1 )

(the value of c 0,1 is given by ( 2)). Notice that, when T is C ∞ , we get an asymptotic at any geometric rate r for the correlations of C ∞ functions. We see here the importance of the discrete spectrum of the operator L, also called Ruelle spectrum or PollicottRuelle spectrum.

In this thesis, we are mostly interested in innitely dierentiable hyperbolic dynamics.

If T is a C ∞ expanding map of the circle, then Theorem 1 may be applied for any k > 0, making the essential spectral radius of L arbitrarily small. It happens that in that case the discrete spectrum of L acting on dierent spaces are coherent in the following sense: Consequently, letting k go to innity, we dene a discrete spectrum for L that we will call Ruelle spectrum.

Denition 3 (Ruelle spectrum for expanding map). Let T be a C ∞ expanding map3 of the circle. We say that z ∈ C is a Ruelle resonance of multiplicity m ∈ N * for T (or L) if, for every k large enough, z is an eigenvalue of multiplicity m of the transfer operator (1) acting on C k S 1 . The associated generalized eigenvectors are called the resonant states for T or L. The set of Ruelle resonances for L is called its Ruelle spectrum and is denoted by σ R (L).

In view of the importance of the Ruelle spectrum when investigating the statistical properties of T , it is legitimate to wonder if there is any ecient way to compute it.

Following Ruelle [START_REF] Ruelle | Zeta-functions for expanding maps and Anosov ows[END_REF], one may tackle this question by introducing the dynamical determinant dened for z ∈ C small enough4 by

d(z) = d T (z) := exp - +∞ n=1 1 n tr (L n ) z n , (4)
where the at trace of the operator L n is dened for n ∈ N * by tr (L n ) :=

T n x=x 1 (T n ) (x) -1 .

(5)

Here, we take (5) as the denition of the at trace of L n . This expression is motivated by a formal integration of the Schwartz kernel of L n over the diagonal of S 1 × S 1 . However, it is possible to make this computation rigorous and then see (5) as a consequence of a general denition of the at trace using distributional considerations 5 . The denition (4)

for the dynamical determinant is motivated by the following computation, valid for any square matrix A and z ∈ C small enough:

det (I -zA) = det (exp (ln (I -zA))) = exp (tr (ln (I -zA)))

= exp - +∞ n=1 1 n tr (A n ) z n .
Hence, formally we have d(z) = det (I -zL).

We already mentioned that when T is real-analytic then the operator L acting on a space of holomorphic functions is nuclear of order 0. It implies in particular that L has a well-dened Fredholm determinant det (I -zL) and it happens that this determinant is given for z small enough by the expression (4). Consequently, the dynamical determinant d(z) has a holomorphic extension to C whose zeros are the inverses of the Ruelle resonances for 6 L. This fact can be generalized to C ∞ expanding map:

Theorem 2 ( [START_REF] Ruelle | An extension of the theory of Fredholm determinants[END_REF]). Let T be a C ∞ expanding map of the circle. Then the dynamical determinant d(z) has a holomorphic extension to C whose zeros are exactly the inverses of the Ruelle resonances for T (counted with multiplicities).

The proof of Theorem 2 is based on a precise analysis of the action of the transfer operator L on the spaces C k 's for k > 0. When T is real-analytic, the structure of nuclear operator for L implies much stronger results than Theorem 2 in that case. For instance, Ruelle [START_REF] Ruelle | Zeta-functions for expanding maps and Anosov ows[END_REF] proves the following bound on the growth of the dynamical determinant for analytic expanding maps of the circle: there is C > 0 such that, for every z ∈ C, we have

|d(z)| ≤ C exp C (log (1 + |z|)) 2 . ( 6
)
The estimate (6) and Jensen's formula [Boa54, 1.2.1 p.2] implies an upper bound on the number of Ruelle resonances, that has been proven generically sharp by Bandtlow and

INTRODUCTION

Naud [START_REF] Bandtlow | Lower bounds for the Ruelle spectrum of analytic expanding circle maps[END_REF]. Another features of nuclear operator of order 0 is that they have a welldened trace that coincides with the sum of their non-zero eigenvalues according to the work of Grothendieck [Gro55, Corollaire 4 p.18 chap.II] (working on a Hilbert space, one can just use Lidskii's Trace Theorem). It happens that, when T is real-analytic, for n ∈ N * , the trace of L n coincides with its at trace, so that we have the following trace formula:

tr (L n ) = λ∈σ R (L)
λ n .

(TFM)

This formula is the natural analogue for expanding maps of the trace formula (TFF)

proposed by Dyatlov and Zworski for C ∞ Anosov ows. While the holomorphic extension to C of the dynamical determinant (4) remains valid for any C ∞ expanding map of the circle, it is not clear whether the same is true or not for the trace formula (TFM) or the bound (6). Notice for instance that there is no reason a priori for the right-hand side of (TFM) to converge, since no general bound on the number of Ruelle resonances for a C ∞ expanding map is known (and we no not expect that there is any). We will study these questions in Chapters 1 and 2.

Statistical properties for Anosov dieomorphisms

Let us now explain how the picture from the last section is modied when we replace the expanding map T by a hyperbolic dieomorphism, still denoted by T . For the sake of simplicity, we will assume in this introduction that T is a transitive Anosov dieomorphism.

We will need at some point to consider more general hyperbolic dieomorphisms (including in particular Smale's horseshoe [Sma67, I.5]), but let us ignore this technical subtleties for now. We start by recalling the denition of an Anosov dieomorphism.

Denition 4. Let M be a compact manifold and T : M → M be a C 1 dieomorphism.

We say that T is an Anosov dieomorphism if, for every x ∈ M there is a splitting of the tangent space

T x M = E u x ⊕ E s x ,
and there are constants C > 0, λ > 1 and a smooth Riemannian metric on M such that (i) for every x ∈ M and σ ∈ {s, u}, we have DT (x) (E σ x ) = E σ T x ;

(ii) for every x ∈ M, v ∈ E u x and n ∈ N we have |DT -n (x)v| ≤ Cλ -n |v|;

(iii) for every x ∈ M, v ∈ E s x and n ∈ N we have |DT n (x)v| ≤ Cλ -n |v|.

Example 2. The most classical examples of Anosov dieomorphism are the so-called cat maps. These are the maps on the torus T 2 = R 2 /Z 2 induced by matrices A ∈ SL (2, Z)

with no eigenvalue of modulus 1.

Remark 1. We did not make any regularity assumption on the stable and unstable directions (E u and E s ) in Denition 4. It is a well-known fact though that the stable and unstable directions of a smooth uniformly hyperbolic dynamical systems are automatically Hölder-continuous, and that they integrate into Hölder foliations (called the stable and unstable foliations). However, these directions and the associated foliations do not satisfy a priori any better regularity hypothesis. This is an important feature of hyperbolic systems since the regularity of the foliations limit the applications of certain tools to study hyperbolic systems. One of the main advantages of the functional approach is that it bypasses this limitation (see Remark 2 for more details). The same remark applies to Anosov ows dened below (see Denition 6).

If T is a transitive Anosov dieomorphism, it is convenient to introduce a smooth weight g : M → C and then dene the associated weighted Koopman operator 7 by

L = L T,g : v → g.v • T. (7)
In the invertible case, the Koopman operator and its adjoint, the transfer operator, are very much alike, so that there is no particular reason to consider one or the other.

As in the case of expanding maps, one wants to nd a Banach space on which L denes a quasi-compact operator and L 2 (M ) or C 0 (M ) do not fulll that requirement. However, the situation is slightly more complicated here due to the coexistence of the stable and unstable directions. In the expanding case, we saw that composition by a contracting map had good properties (such as quasi-compactness) as an operator acting on a space of regular functions. By duality considerations, this fact suggests that the operator of composition by an expanding map should have a good behavior on a space of distributions of negative order. Hence, to achieve quasi-compactness for the operator (7), we are looking for a space of distributions that are smooth in the stable direction and dual of smooth in the unstable direction. Such a space is now called a space of anisotropic distributions (see [START_REF] Baladi | The quest for the ultimate anisotropic Banach space[END_REF][START_REF] Mark | A gentle introduction to anisotropic Banach spaces[END_REF] for surveys on this topic).

The rst appearance of spaces of anisotropic distributions in the dynamic literature is often considered to be in [START_REF] Blank | Ruelle-Perron-Frobenius spectrum for Anosov maps[END_REF]. However, this notion was already implicitly present in previous work by Rugh [START_REF] Henrik | The correlation spectrum for hyperbolic analytic maps[END_REF][START_REF] Henrik | Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems[END_REF], Fried [START_REF] Fried | Meromorphic zeta functions for analytic ows[END_REF] and Kitaev [START_REF] Yu | Fredholm determinants for hyperbolic dieomorphisms of nite smoothness[END_REF][START_REF] Yu | Corrigendum: Fredholm determinants for hyperbolic dieomorphisms of nite smoothness[END_REF]. In [START_REF] Blank | Ruelle-Perron-Frobenius spectrum for Anosov maps[END_REF], a space on which the transfer operator (the adjoint of ( 7)) associated to a C 3 Anosov map T is quasi-compact is constructed no weight is considered though, which amounts to take g = 1 in (7). However, this construction is restricted by the regularity of the stable and unstable directions of T (see Remark 1), so that, even if T is C ∞ , it does not provide a scale of spaces on which the essential spectral radius of the transfer operator is arbitrarily small as it was the case for expanding maps in Theorem 1. 7 We will also denote this operator by L. The convention throughout this thesis is that the operator of most interest at some point of the text will always be called L.

INTRODUCTION

After the rst construction in [START_REF] Blank | Ruelle-Perron-Frobenius spectrum for Anosov maps[END_REF], plenty of dierent spaces of anisotropic distributions have been introduced. In [GL06], Gouëzel and Liverani improved the geometric construction from [START_REF] Blank | Ruelle-Perron-Frobenius spectrum for Anosov maps[END_REF], lifting in particular the restriction due to the regularity of the stable and unstable directions. As a consequence, if T is a C ∞ Anosov map, the essential spectral radius of the transfer operator (without weight) can be made arbitrarily small on the spaces from [START_REF] Blank | Ruelle-Perron-Frobenius spectrum for Anosov maps[END_REF]. This fact is reproved by Baladi in [START_REF] Baladi | Anisotropic Sobolev spaces and dynamical transfer operators: C ∞ foliations[END_REF] using tools from microlocal analysis under the (very strong) assumption that the stable or unstable foliation of

T is C ∞ . The case of general hyperbolic basic sets (rather than Anosov dieomorphisms, see 1.2.1) and of general weights for the Koopman operator ( 7) is dealt with by Baladi and Tsujii in [START_REF] Baladi | Anisotropic Hölder and Sobolev spaces for hyperbolic dieomorphisms[END_REF]. As in [START_REF] Baladi | Anisotropic Sobolev spaces and dynamical transfer operators: C ∞ foliations[END_REF], the construction from [START_REF] Baladi | Anisotropic Hölder and Sobolev spaces for hyperbolic dieomorphisms[END_REF] is based on tools from micro-local analysis. However, the analysis from [START_REF] Baladi | Anisotropic Sobolev spaces and dynamical transfer operators: C ∞ foliations[END_REF] relies on pseudo-dierential calculus while [START_REF] Baladi | Anisotropic Hölder and Sobolev spaces for hyperbolic dieomorphisms[END_REF] rather uses PaleyLittlewood decomposition a more convenient tool if one wants to study map of nite dierentiability. The case of weighted Koopman operators associated to hyperbolic sets can also be tackled by geometric constructions of spaces of anisotropic distributions, see [START_REF] Gouëzel | Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties[END_REF]. These results have also been translated in the language of semi-classical analysis by Faure, Roy and Sjöstrand [START_REF] Faure | Semi-classical approach for Anosov dieomorphisms and Ruelle resonances[END_REF]. From all these constructions, we have in particular the following result the analogue of Theorem 1 for Anosov dieomorphisms.

Theorem 3 [START_REF] Baladi | Anisotropic Hölder and Sobolev spaces for hyperbolic dieomorphisms[END_REF][START_REF] Gouëzel | Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties[END_REF]). Let T : M → M be a C ∞ transitive Anosov dieomorphism and g : M → C be a C ∞ function. For every > 0 there is a Banach space B with the following properties:

(i) B is continuously contained in the space D (M ) of distributions on M ;

(ii) the space C ∞ (M ) of C ∞ functions on M is contained in B, and the inclusion is continuous with dense image;

(iii) the Koopman operator (7) induces a bounded operator on B with essential spectral radius less than .

Hence, the essential spectral radius of the operator L dened by (7) can be made arbitrarily small. As in the case of expanding maps of the circle, one may see, using for instance Lemma B.1, that the intersection of the spectrum of L with {z ∈ C : |z| > } does not depend on the choice of the space B in Theorem 3. In this context, the notion of Ruelle resonance may be dened as follow.

Denition 5 (Ruelle spectrum for Anosov dieomorphism). Let T : M → M be a C ∞ transitive Anosov dieomorphism and g : M → C be a C ∞ function. Let z ∈ C and m ∈ N * . We say that z is a Ruelle resonance of multiplicity m for (T, g) (or just for the operator L), if there is 8 a Banach space B satisfying the points (i)(iii) from Theorem 3 8 Here, we could replace there is by for any without changing the denition.

for some ∈ ]0, |z|[ such that z is an eigenvalue of algebraic multiplicity m for the operator L dened by (7) acting on B. The associated generalized eigenvectors are called resonant states for L. The set of Ruelle resonances is called the Ruelle spectrum and denoted by σ R (L).

As in the case of expanding maps, the understanding of the spectral theory for the Koopman (or transfer) operator gives a lot of information on the statistical properties of T , see for instance [START_REF] Blank | Ruelle-Perron-Frobenius spectrum for Anosov maps[END_REF][START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF][START_REF] Gouëzel | Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties[END_REF]. In particular, we get a precise asymptotic for the correlations associated to Gibbs states for T , similar to (3) (see [GL08, Theorem 1.2]).

Remark 2. A lot was already known on statistical properties of expanding and hyperbolic maps before the functional approach was developed in the form that we exposed here (based on Theorems 1 and 3). For instance, the existence of an invariant measure absolutely continuous with respect to Lebesgue for smooth expanding maps was proved by Krzy»ewski and Szlenk [START_REF] Krzy | On invariant measures for expanding dierentiable mappings[END_REF]. Concerning hyperbolic maps, many results were obtained rst by means of Markov partitions and symbolic dynamics 9 (see for instance [START_REF] Bowen | Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms[END_REF][START_REF] Ruelle | Thermodynamic Formalism[END_REF]). The main idea of this approach is to code a hyperbolic dynamical systems in order to replace it by a one-dimensional lattice that can then be studied using classical notions from statistical mechanics [START_REF] Ruelle | Statistical mechanics of a one-dimensional lattice gas[END_REF].

However, the approach based on Markov partitions is restricted by the regularity of the coding, which is a priori only Hölder (this is due to the low regularity of the stable and unstable foliations, see Remark 1). Hence, it is very unlikely that an asymptotic expansion at any geometric order for correlations of smooth observables such as (3) may be obtained by the method of Markov partitions. Indeed, the size of the remainder in (3) is deeply related with the regularity of both the dynamics and the observables.

We also have a notion of dynamical determinant for the operator L dened by (7): the associated dynamical determinant d(z) = d T,g (z) may still be dened by (4) for z ∈ C small enough10 , but the at traces of the powers of L are now dened for n ∈ N * by tr (L n ) :=

T n x=x n-1 k=0 g T k x |det (I -DT n (x))| . ( 8
)
The analogue of Theorem 2 in this context reads:

Theorem 4 ( [START_REF] Liverani | Zeta functions and dynamical systems[END_REF][START_REF] Baladi | Dynamical determinants and spectrum for hyperbolic diemorphisms. Geometric and probabilistic structures in dynamics[END_REF]). Let T : M → M be a C ∞ transitive Anosov dieomorphism and g : M → C be a C ∞ function. Then the dynamical determinant d(z) has a holomorphic extension to C whose zeros are exactly the inverses of the Ruelle resonances for the weighted Koopman operator L dened by (7) (counted with multiplicities).

INTRODUCTION

If T is a small real-analytic perturbation of a cat map, then there is a space [START_REF] Faure | Ruelle-Pollicott resonances for real analytic hyperbolic maps[END_REF] on which the operator L is nuclear of order 0. As in the case of expanding maps, the dynamical determinant d(z) coincides then with the Fredholm determinant of L, and a bound on the growth of the dynamical determinant follows. For the same reasons as in the case of expanding map, the trace formula (TFM) holds consequently for small real-analytic perturbations of cat maps. This fact has been used by Adam to prove the genericity of the existence of non-trivial resonances among analytic perturbations of a cat map [START_REF] Adam | Generic non-trivial resonances for Anosov dieomorphisms[END_REF].

It follows from results of Rugh [START_REF] Henrik | The correlation spectrum for hyperbolic analytic maps[END_REF][START_REF] Henrik | Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems[END_REF] that the trace formula (TFM) also holds for real-analytic Anosov dieomorphisms in dimension 2. The question of the validity of (TFM) for more general Anosov dieomorphisms will be discussed in Chapters 1 and 2.

However, notice that, as in the case of expanding maps, there is no reason a priori for the right-hand side of (TFM) to converge when T is a general C ∞ Anosov dieomorphism.

Statistical properties for Anosov ows

We turn now to the continuous-time analogues of the hyperbolic dieomorphisms from the previous section: Anosov ows. As usual, let us start with a denition and an example.

Denition 6 (Anosov ows). Let M be a C ∞ compact manifold and (φ t ) t∈R be a C 1 ow on M with zero-free generator X. We say that (φ t ) t∈R is Anosov if, for every x ∈ M , there is a splitting of the tangent space

T x M = E 0 x ⊕ E s x ⊕ E u x ,
and there are constants C > 0, λ > 1 and a smooth Riemannian metric on M such that:

(i) for every x ∈ M, t ∈ R and σ ∈ {0, s, u}, we have Dφ t (x) (E σ x ) = E σ φt(x) ;

(ii) for every x ∈ M , the space E 0 x is the span of X(x);

(iii) for every x ∈ M, v ∈ E u x and t ∈ R + we have |Dφ -t (x)v| ≤ Cλ -t |v|;

(iv) for every x ∈ M, v ∈ E s x and t ∈ R + we have |Dφ t (x)v| ≤ Cλ -t |v|.

Example 3. As mentioned above, the example that motivated the denition of Anosov ows is the geodesic ow on the unit tangent bundle of a compact Riemannian manifold with negative sectional curvature. Other specimens of Anosov ows are obtained by considering suspensions of Anosov dieomorphism. Notice also that any small C 1 perturbation of an Anosov ow is also Anosov (the same is true for expanding maps and Anosov dieomorphisms).

Let us consider a C ∞ Anosov ow (φ t ) t∈R on a compact manifold. We denote its generator by X and choose a C ∞ weight V : M → C. In this context, the Koopman operator denes a semi-group of operators:

L t = L X,V t : v → exp t 0 V • φ τ dτ v • φ t (9)
for t ≥ 0. Notice that the generator of this semi-group is formally the dierential operator P := X + V : whenever it makes sense we have

d dt (L t v) = P (L t v) . ( 10
) For instance, if v ∈ C ∞ (M ) then (10) holds in C ∞ (M ).
As in the case of expanding maps or Anosov dieomorphisms, we replaced the nite-dimensional non-linear dynamics (φ t ) t∈R by an innite-dimensional linear dynamics. However, since (φ t ) t∈R is a continuous-time dynamical system, we are now studying the linear ODE (10) rather than the iterations of a single operator. Consequently, we want to understand the spectral theory of the dierential operator P , hoping that it would allow us to describe the long-time asymptotic of the solutions of the linear ODE (10).

Once again, the (unbounded) operator P does not have discrete spectrum on L 2 (M )

or C 0 (M ), and we need consequently to nd new spaces on which the spectral theory of P is better-behaved the analogues of the spaces from Theorem 3. Adapting ideas from Theorem 5 ([BL07, [START_REF] Butterley | Robustly invariant sets in ber contracting bundle ows[END_REF][START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov ows[END_REF]). Let (φ t ) t∈R be a C ∞ Anosov ow on a compact manifold M with generator X and V : M → C be a C ∞ function11 . Then, for every A > 0, there is a Banach space B such that

(i) C ∞ (M ) ⊆ B ⊆ D (M )
, both inclusions are continuous and the rst one has dense image;

(ii) (L t ) t≥0 , dened by (9), induces a strongly continuous semi-group of operator on B whose generator is P = X + V with its natural domain The statement of Theorem 5 deserves a little explanation. The point (i) is just a nontriviality condition: it ensures that the space B is made out of objects that live on M .

D (P ) =
This rst condition can be mitigated in order to allow wilder objects in B than distributions this fact will be crucial in Chapters 3 and 4. The point (ii) may seem technical but it is very important. The fact that (L t ) t≥0 is a strongly continuous semi-group makes it possible to link the spectral theory of P with the long-time asymptotic of (L t ) t≥0 . It implies in particular that, if the real part of z ∈ C is positive and large enough, then the resolvent (z -P ) -1 of P at z coincides with the Laplace transform of the Koopman operator:

R(z) := +∞ 0 e -zt L t dt (11)
where the integral converges in operator norm on B. Finally, the point (iii) implies that the spectrum of P on B looks as depicted in Figure 2. As in the discrete-time case, the isolated eigenvalues of nite multiplicity on the right part of the complex plane in Figure 2 are intrinsically dened by

R(z) from C ∞ (M ) to D (M ).
We saw that R(z) coincides with the resolvent of P acting on the space B from Theorem 5. Consequently, it follows from Theorem 6 that R(z) has a meromorphic continuation to C, as an operator from C ∞ (M ) to D (M ), with residues of nite rank. The Ruelle resonances of P are then the poles of R(z), and the multiplicity of a resonance is given by the rank of the residue of the corresponding poles of R(z) (the image of this residue is the space of resonant states associated to the resonance).

When considering Anosov ows, the structure of the Ruelle spectrum is slightly more complicated than in the discrete-time case (it is not a bounded subset of C for instance).

As a consequence, the study of the Ruelle spectrum, and hence of the statistical properties for the ow (φ t ) t∈R , is slightly more involved. For instance, it is much trickier to prove exponential decay of correlations in this context. Consider for example the case V = 0

(that corresponds to the SRB measure), in that case 0 is a Ruelle resonance. Assuming that (φ t ) t∈R is transitive, then 0 is a simple eigenvalue and there is no other Ruelle resonance with non-negative real part. However, in order to prove exponential decay of correlation we need to establish a spectral gap for P : the existence of A > 0 such that the only Ruelle resonance for P with real part greater than -A is 0 (some additional technical estimates are required, see [START_REF] Butterley | A note on operator semigroups associated to chaotic ows[END_REF]). This is tricky because Ruelle resonances with large imaginary parts could accumulate on the vertical line iR without touching it.

Due to the complexity and the importance of the problem of spectral gaps for Anosov ows, there are a lot of works on that topic. Most proofs of spectral gaps for Anosov ows 12 As in Denition 5, we could replace here there is by for any.

INTRODUCTION

are based on some version of Dolgopyat's argument [START_REF] Dolgopyat | On Decay of Correlations in Anosov Flows[END_REF]. Liverani gave a functional analytic version of this argument [START_REF] Liverani | On contact Anosov ows[END_REF] that allowed him to prove exponential decay of correlations for C 4 contact Anosov ows (for the measure induced by the contact form).

This result has been reproved in the C ∞ case by Tsujii [START_REF] Tsujii | Quasi-compactness of transfer operators for contact Anosov ows[END_REF][START_REF] Tsujii | Contact Anosov ows and the FourierBrosIagolnitzer transform[END_REF]. Exponential decay of correlations for C ∞ contact Anosov ows can also be seen as a consequence of a general result by Nonnenmacher and Zworski [START_REF] Nonnenmacher | Decay of correlations for normally hyperbolic trapping[END_REF] that gives a spectral gap of explicit size.

Faure and Tsujii [START_REF] Faure | Band structure of the Ruelle spectrum of contact Anosov ows[END_REF] went further than the spectral gap and stated a band structure for C ∞ contact Anosov ows.

d(z) = d X,V (z) := exp - γ T # γ T γ e γ V |det (I -P γ )| e -zTγ . (12)
The sum in the right-hand side of (12) is over periodic orbits γ for the ow (φ t ) t∈R . We write T γ for the length of γ and T # γ for its primitive length (that is the shortest length of a periodic orbit for (φ t ) t∈R with the same image than γ). The integral of V along γ is dened by γ V := Tγ 0 V (φ t (x)) dt for any x in the image of γ. Finally, P γ denotes the linearized Poincaré map associated to γ, dened by P γ := Dφ Tγ (x) E u x ⊕E s x . The denition of P γ depends on the choice of a point x in the image of γ. However, the conjugacy class of P γ does not, and hence the determinant in the right-hand side of (12) is well-dened.

The intuition behind the denition is that, formally, d(z) = det (z -P ). This heuristic understanding of d(z) will be made rigorous in 4.2. The equivalent of Theorems 2 and 4 in this context is: Theorem 6 ( [START_REF] Giulietti | Anosov ows and dynamical zeta functions[END_REF][START_REF] Dyatlov | Dynamical zeta functions for Anosov ows via microlocal analysis[END_REF]). Let (φ t ) t∈R be a C ∞ Anosov ow, with generator X, on a compact manifold M . Let V : M → C be a C ∞ function 14 . Then the dynamical determinant 14 Actually, [START_REF] Giulietti | Anosov ows and dynamical zeta functions[END_REF][START_REF] Dyatlov | Dynamical zeta functions for Anosov ows via microlocal analysis[END_REF] do not consider the case of a general weight V . However, it is probably clear to specialists that their methods at least those from [START_REF] Dyatlov | Dynamical zeta functions for Anosov ows via microlocal analysis[END_REF] apply to the case of general weights and even to the case of vector bundles. Theorem 6 is proven in a much more general setting in [START_REF] Dyatlov | PollicottRuelle resonances for open systems[END_REF].

(φ t ) t∈R is given for Re z 1 by ζ (z) = γ primitive 1 -e -zTγ , (13)
where the product is over primitive periodic orbits of the ow (φ t ) t∈R . An argument due to

Ruelle [START_REF] Ruelle | Zeta-functions for expanding maps and Anosov ows[END_REF] allows to write the zeta function (13) as an alternated product of dynamical determinants, so that Theorem 6 implies that ζ has a meromorphic extension 15 to C.

As in the discrete-time case, when studying dynamical determinants and zeta functions for Anosov ows, there are functional-analytic tools that are only available in the realanalytic category. These tools have been developed by Ruelle [START_REF] Ruelle | Zeta-functions for expanding maps and Anosov ows[END_REF] (under the very strong assumption that the stable or unstable foliation of the ow is real-analytic, see also [START_REF] Fried | The zeta functions of Selberg and Ruelle[END_REF]), Rugh [START_REF] Henrik | The correlation spectrum for hyperbolic analytic maps[END_REF][START_REF] Henrik | Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems[END_REF] (in dimension 3 but without any condition on the stable or unstable foliation) and Fried [START_REF] Fried | Meromorphic zeta functions for analytic ows[END_REF] (in any dimension). They do not work directly with the Koopman operator L t or its generator P , but rather with a family of auxiliary nuclear operators [START_REF] Grothendieck | Produits Tensoriels Topologiques et Espaces Nucléaires[END_REF] dened using symbolic dynamics. One of the consequences of their results is that if (φ t ) t∈R and V are real-analytic, then the dynamical determinant d(z) has nite order (see Denition 1.2). It is then folklore (see Proposition 1.27) that the nite order for d(z) implies a trace formula for the ow (φ t ) t∈R that reads, in the sense of

distributions on R * + , λ∈σ R (P ) e λt = γ T # γ e γ V |det (I -P γ )| δ Tγ . (TFF)
This means that for every

C ∞ function ϕ ∈ C ∞ c R * + compactly supported in R * + we have λ∈σ R (P ) Lap (ϕ) (-λ) = γ T # γ e γ V |det (I -P γ )| ϕ(T γ ), (14)
where Lap (ϕ) denotes the Laplace transform of ϕ:

Lap(ϕ) : z → +∞ 0 e -zt ϕ(t)dt.
The notations on the right-hand side of (TFF) and ( 14) are the same as in (12) and we 15 Actually, Ruelle arguments require an orientability condition on the stable and unstable directions of (φt) t∈R . A strategy to overcome this condition may be found in [START_REF] Bornsweil | Dynamical Zeta functions in the nonorientable case[END_REF], see [START_REF] Baladi | Dynamical determinants and spectrum for hyperbolic diemorphisms. Geometric and probabilistic structures in dynamics[END_REF] for the discrete-time analogue.

INTRODUCTION

recall that σ R (P ) is our notation for the Ruelle spectrum of P . The trace formula (TFF) is a continuous-time analogue of (TFM). Indeed, according to Guillemin's trace formula, the right-hand side of (TFF) is a generalized trace for the Koopman operator (9) see also [START_REF] Dyatlov | Dynamical zeta functions for Anosov ows via microlocal analysis[END_REF]Appendix B]. The formula (TFF) gives consequently a spectral interpretation for Guillemin's trace formula in the case of real-analytic Anosov ows.

In [START_REF] Dyatlov | Dynamical zeta functions for Anosov ows via microlocal analysis[END_REF], Dyatlov and Zworski gave a new proof of Theorem 6 (the rst proof being in [START_REF] Giulietti | Anosov ows and dynamical zeta functions[END_REF]). One of their motivations was to investigate the validity of the trace formula The question asked by Dyatlov and Zworski of the validity of the trace formula (TFF)

(and of its discrete-time analogue (TFM)) beyond real-analytic systems is the main thread of this thesis. Let us describe now our approach to this problem.

Main results

Maps: Trace formulae and counter-examples (Chapter 1)

Our rst line of approach to the problem of the trace formula (TFF) for C ∞ Anosov ow was to consider the discrete-time analogue of this question (TFM), where L is either the transfer operator (1) associated to a C ∞ expanding map of the circle or a weighted Koopman operator (7) associated to an Anosov dieomorphism. The advantage of considering a map rather than a ow is that the relationship between trace formulae and dynamical determinants is much simpler in that case:

Proposition 1 (See Theorem 1.3). The trace formula (TFM) holds for n large enough with absolute convergence of the right-hand side if and only if the dynamical determinant (4) has nite order.

We will prove a slightly more precise version of this result (Theorem 1.3) in 1.1. As mentioned above, it remains true in the continuous-time setting that nite order for the associated dynamical determinant (12) implies that the trace formula (TFF) holds, and we shall prove an analogue of Proposition 1 for Anosov ows (see Proposition 1.27). However,

16
Their argument is written in the case V = 0 but it is not hard to see that it still applies as soon as V is real-valued.

the link between these two notions is much more explicit in the discrete-time case. For instance, in order to produce a counter-example to (TFM), one only needs to nd a system for which the dynamical determinant d has innite order.

In order to construct counter-examples to (TFM), we need to consider slightly more general systems than expanding maps or Anosov ows, namely hyperbolic basic sets. In- Proposition 3 (Proof in 1.2.3). Let N 0 : R * + → R + be a locally bounded function. Then there are a dieomorphism T : S 4 → S 4 with hyperbolic basic set K and a smooth function

g : S 4 → R + , positive 17 on K, such that, if we dene for r ∈ R * + N (r) = # {z ∈ C : |z| ≥ r and z is a Ruelle resonance associated to (T, g)} , then N 0 (r) = r→0 o (N (r)).

DenjoyCarleman classes and compact transfer operator (Chapter 2)

There is a fundamental dierence between the trace formulae (TFM) and (TFF) and, for instance, Theorems 2, 4 and 6: the formulae (TFF) and (TFM) involve the entire Ruelle spectrum at once, while Theorems 2, 4 and 6 are usually proved by analyzing the Ruelle spectrum little by little.

INTRODUCTION

This fact is essential because the scales of spaces from Theorems 1, 3 and 5 only unveil the Ruelle spectrum gradually. Consequently, they are suited to prove statements like Theorems 2, 4 and 6, but they are not the most convenient tools in order to establish a global formula like (TFF) or (TFM). Hence, we want to construct a single space that unveils the entire Ruelle spectrum. As already mentioned, this had already been done in certain cases, but always for real-analytic dynamics. We will see that this phenomenon is not specic to the real-analytic case, and present tools that we developed to study C ∞ hyperbolic dynamics.

We will start with the simplest example and explain in Chapter 2 how one can construct a space tted to study a given C ∞ expanding map of the circle T and its associated transfer operator (1). In 2.2, we will prove the following theorem, which is in some sense the C ∞ generalization of what Ruelle [START_REF] Ruelle | Zeta-functions for expanding maps and Anosov ows[END_REF] did for real-analytic expanding maps.

Theorem 7 (See Theorem 2.9). Let T be a C ∞ expanding map of the circle and denote by L the associated transfer operator (1). Then, there is a Hilbert space H such that:

(i) H is continuously contained in the space of C ∞ functions from S 1 to C;
(ii) H contains the trigonometric polynomials on S 1 as a dense subspace;

(iii) L induces a compact operator on H whose non-zero spectrum is the Ruelle spectrum of L.

Theorem 7 is not sucient to prove the trace formula (TFM). Indeed, in order to prove a trace formula, it seems natural to try to make L trace class rather than compact. An important dierence between these two notions is that being trace class is a quantitative statement (on the decay of the singular values of L). Consequently, we need to make the fact that T is C ∞ quantitative. To do so, we will use the language of DenjoyCarleman classes or classes of ultradierentiable functions.

A short introduction to the topic of DenjoyCarleman classes is given in 2.1 the interested reader may refer to the more complete survey [START_REF] Kriegl | The convenient setting for non-quasianalytic Denjoy-Carleman dierentiable mappings[END_REF] and references therein.

Let us just say for now that this theory provides a general procedure to produce regularity classes intermediate between C ∞ and real-analytic. The most famous such classes are maybe the classes of Gevrey functions that have been introduced by Gevrey in his seminal paper [START_REF] Gevrey | Sur la nature analytique des solutions des équations aux dérivées partielles[END_REF] to study the regularity of the solutions of certain PDE. Using the language of DenjoyCarleman classes, we will give in 2.2 a quantitative version, Theorem 2.9, of Theorem 7 we will bound the singular values of L acting on H.

When L acting on H is trace class, we will see that the trace formula (TFM) holds. In particular, in that case the dynamical determinant d(z) Theorem 8 (See Proposition 2.18). Under the assumptions of Theorem 7, we may write

L = L b + L c
, where L b is a quasi-nilpotent operator and L c is a nuclear operator of order 0 on H. The dynamical determinant (4) may then be rewritten as

d(z) = det I -z (I -zL b ) -1 L c .
This statement can also be made quantitative using the language of DenjoyCarleman classes yielding in particular a bound on the growth of the dynamical determinant d(z) dynamics. We will also discuss the implication of Theorem 7 in terms of regularity of the resonant states for L and of the conjugacy problem (see Corollaries 2.11 and 2.12).

In 2.3, we recall and discuss Theorem 2.27, a result on Koopman operators associated to Gevrey hyperbolic dieomorphisms that we obtained by similar methods in [START_REF] Jézéquel | Local and global trace formulae for smooth hyperbolic dieomorphisms[END_REF] (before Theorems 7 and 8). In order to keep the page count reasonable, we will not give the proof of this result here.

Trace formulae for ultradierentiable Anosov ows (Chapter 3)

We return to Anosov ows and the question of the validity of the trace formula (TFF) in Chapter 3. Our rst idea when trying to prove trace formula (TFF) was to prove that the dynamical determinant (12) has nite order. Heuristic considerations (involving in particular Theorem 2.27, see Remark 2.28) let us think that Gevrey classes of regularity are a natural setting when it comes to nite order of the dynamical determinant. However, when applying the methods based on PaleyLittlewood decompositions exposed in Chapter 2, we were not able to get a good enough control of the Koopman operator (9) for small t > 0. Hence, we missed the nite order for the dynamical determinant and needed to nd another way to prove the trace formula (TFF). The solution that we found was to consider operators of the form

+∞ 0 ϕ(t)L t dt (15) for a test function ϕ ∈ C ∞ c R *
INTRODUCTION from Lidskii's Trace Theorem. Indeed, we expect the trace of (15) to be equal to the right-hand side of (14) because of Guillemin's trace formula [START_REF] Victor Guillemin | Lectures on spectral theory of elliptic operators[END_REF].

The advantage of considering (15) is that, since ϕ is supported away from 0, we do not need to control the Koopman operator L t for t near 0 (compare for instance with the expression (11)). It turned out that this strategy allows not only to retrieve the trace formula (TFF) using PaleyLittlewood like decompositions similar to those of Chapter 2, but also to work with ultradierentiable classes that are much larger than Gevrey classes.

Thus, we obtain in Chapter 3:

Theorem 9 (See Theorem 3.1 and Corollary 3.2). There is a class of regularity, larger than all Gevrey classes, such that the trace formula (TFF) holds for Anosov ow (φ t ) t∈R and weight V that belong to this class of regularity..

The classes of Anosov ows and weights that are allowed in Theorem 9 will be discussed in 3.1, using the language of DenjoyCarleman classes. We obtain in Proposition 3.3 an estimate on the number of Ruelle resonances (for the allowed ultradierentiable ows) that implies that the left hand side of (TFF) denes a distribution on R * + .

Finite order of dynamical determinants for Gevrey Anosov ows (Chapter 4)

After we established Theorem 9, the question of the nite order of dynamical determinants associated to Gevrey Anosov ows remained open. With Yannick Guedes Bonthonneau, we applied methods from analytic and Gevrey micro-local analysis to study this problem.

The tool that turned out to be the most convenient for us is an analytic FBI transform in the spirit of Heler and Sjöstrand [START_REF] Heler | Résonances en limite semi-classique[END_REF][START_REF] Sjöstrand | Density of resonances for strictly convex analytic obstacles[END_REF]. Similar methods have been used recently by Galkowski and Zworski [GZ19a,[START_REF] Galkowski | Viscosity limits for 0th order pseudodierential operators[END_REF][START_REF] Galkowski | Analytic hypoellipticity of keldysh operators[END_REF] to deal with certain PDEs problems in the real-analytic category. Our methods however are not restricted to the real-analytic regularity, but also apply in the Gevrey category. In particular, we nally prove that the order of the dynamical determinant associated to a Gevrey Anosov ow is nite.

Theorem 10 (Proof in 4.2). Let s ≥ 1. Let (φ t ) t∈R be an s-Gevrey Anosov ow on an s-Gevrey manifold. Let V : M → C be an s-Gevrey function. Denote by d(z) the associated dynamical determinant (12). Then there is a constant C > 0 such that, for every z ∈ C, we have

|d(z)| ≤ C exp (C |z| ns ) ,
where n is the dimension of M . In particular, d(z) has order less than ns. In Appendix D, we discuss the optimality of the regularity hypothesis in Theorem 9. In Appendix E, we detail a heuristic computation that suggests that trace formulae such as (TFM) or (TFF) are rather exceptional phenomena in the C ∞ category.

Perspectives

This thesis is dedicated to the investigation of the trace formulae (TFF) and (TFM), and of the related question of the order of the dynamical determinants (4) and (12). Using the notion of ultradierentiability, we design tools tted to the study of innitely dierentiable uniformly hyperbolic dynamics. These tools allow us to prove positive results concerning trace formulae and order of dynamical determinants, such as Theorems 9 and 10. The main idea behind the proof of these results is that stronger assumptions of ultradierentiability lead to better estimates on the singular values of the relevant operator acting on well-suited spaces. These estimates lead then naturally to bounds on the growth of the dynamical determinant, the number of Ruelle resonances, etc.

However, while we tend to think that phenomena such as the trace formula (TFF) are rather exceptional in the Conjecture 2. Let M be a compact real-analytic manifold of dimension n. Then the dynamical determinant associated to a generic real-analytic Anosov ow on M has order exactly n.

Let us explain why Conjecture 2 could be easier to approach than Conjecture 1. First of all, a similar statement 20 has been proven by Bandtlow and Naud [START_REF] Bandtlow | Lower bounds for the Ruelle spectrum of analytic expanding circle maps[END_REF] in the case of real- analytic expanding map of the circle, and their methods could be a source of inspiration.

18 We state here a conjecture about the order of the dynamical determinant rather than the trace formula since it is easier to state. Indeed, considering the trace formula, we always have the issue that the spectral size could be ill-dened, which makes harder to write down a precise conjecture.

19 Considering the probabilistic approach, the work of Gossart [START_REF] Gossart | Flat trace statistics of the transfer operator of a random partially expanding map[END_REF] could maybe be a source of inspiration, since it involves at trace computations for random hyperbolic dynamical systems. However, the random systems from [START_REF] Gossart | Flat trace statistics of the transfer operator of a random partially expanding map[END_REF] Moreover, the real-analytic case is the one for which we have the strongest functional analytic tools: we construct in Chapter 4 a scale of spaces on which the generator of an Anosov ow behaves like an elliptic operator of order 1. One could then hope that our tools could be combined with recent methods that have been developed in order to establish Weyl's law for random perturbations of non self-adjoint elliptic operators (see [START_REF] Sjöstrand | Non-Self-Adjoint Dierential Operators, Spectral Asymptotics and Random Perturbations[END_REF] and references therein) in order to approach Conjecture 2.

We also hope that the tools that we developed with Bonthonneau in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF] Chapter 1

Trace formulae, dynamical determinants and counter-examples

In this chapter, we explain the complex analytic links between dynamical determinants and trace formulae for expanding and hyperbolic maps in 1.1 and for Anosov ows in 1.3 , proving in particular Proposition 1 (and its continuous-time analogue Proposition 1.27). We use this dictionary to give quite striking counter-examples to the trace formula (TFM) in 1.2, proving in particular Propositions 2 and 3.

The results from 1.1 and 1.2 may be found in [START_REF] Jézéquel | Local and global trace formulae for smooth hyperbolic dieomorphisms[END_REF]3 and 4]. The proofs from 1.3 were already exposed in [Jéz19a, Appendix E], up to some additional details that we give here.

Trace formulae and dynamical determinants for hyperbolic maps

We explain now the link between the trace formula (TFM) and the order of the dynamical determinant (4) where L is either the transfer operator (1) associated to an expanding map of the circle or the weighted Koopman operator (7) associated to an Anosov dieomorphism, and the at trace tr (L n ) is dened by (5) or (8) accordingly. It is convenient to introduce the following denition.

Denition 1.1. If f is an entire function, we say that (z m ) m 0 is an ordering of the zeros of f if z 0 , z 1 , . . . , z m , . . . are the zeros of f counted with multiplicities and the sequence (|z m |) m 0 is non-decreasing.

In this section and in 1.2, we will always order the zeros of an entire function in this way. Recall the following denitions.

1 1 + |z m | p+1 < +∞ (1.1)
where (z m ) m 0 is an ordering of the zeros of f (the integer p is well-dened thanks to Jensen's formula). By Hadamard's Factorization Theorem [START_REF] Boas | Entire Functions[END_REF]2.7.1], if m 0 denotes the order of 0 as a zero of f , then there is a polynomial Q such that, for all z ∈ C, we have

f (z) = z m 0 e Q(z) m m 0 E z z m , p (1.2)
where the function E is the Weierstrass primary factor dened by

E (u, p) = (1 -u) exp p k=1 1 k u k = exp   - +∞ k=p+1 1 k u k   , (1.3)
where the last expression is only valid when |u| < 1. The genus of f is then dened as max (deg Q, p). We will say that the genus of an entire function of innite order is innite.

As explained in Remark 1.4 below, the following theorem is an abstract way to express the link between trace formula and order of the dynamical determinant.

Theorem 1.3. Let f be an entire function such that f (0) = 1. Let G be a holomorphic function dened on a neighbourhood of 0 such that G (0) = 0 and f (z) = e G(z) for z in a neighbourhood of zero. Write

G (z) = - +∞ n=1 1 n a n z n (1.4)
and denote by (z m ) m 0 an ordering of the zeros of f . Then for all r > 0 such that f has no zero of modulus r, we have

a n = n→+∞ |zm|<r 1 z n m + O 1 r n .
(1.5) Furthermore, the following properties are equivalent :

(i) the order of f is nite;

(ii) there is a natural integer n 0 such that for all integers n n 0 + 1 the series

m 0 1 z n m (1.6)
converges absolutely and its sum is a n .

If (i) or (ii) holds then the minimal value of n 0 so that (ii) holds is the genus of f .

Remark 1.4. Taking f = d the dynamical determinant (4), we have a n = tr (L n ), the at trace of L dened by (5) or (8) according to the context. Recalling Theorems 2 and 4, it appears then that the trace formula (TFM) holds with absolute convergence of the right-hand side if and only if n n 0 + 1, where n 0 denotes the genus of the dynamical determinant d. In particular, Proposition 1 follows immediately from Theorems 2, 4 and 1.3.

In 1.2, we will construct dynamical determinants with arbitrary (nite or innite) genus, so that all the behaviours described in Theorem 1.3 may be realized by dynamical determinants. Notice also that, as in the continuous-time case, we have a local version of the trace formula: for every r > 0 such that there is no resonance of modulus r we have

tr (L n ) = r→0 λ Ruelle resonance |λ|>r λ n + O(r n ).
(1.7)

From Theorem 1.3, the local trace formula (1.7) is in fact just a reformulation of Theorem 2 or 4 according to the context.

Remark 1.5. As we will see in Proposition 1.6 below, the absoluteness of the convergence in (ii) is essential to get an equivalence. This is quite unfortunate especially as we will realize the counter-examples from Proposition 1.6 below as dynamical determinants in 1.2. On the other hand, it is very easy to construct an example for which the series (1.6) converges to a sum dierent from a n (for any chosen values of n): just multiply f by the exponential of an entire function.

Proof of Theorem 1.3.

• To prove (1.5), one only needs to notice that the holomorphic

function z → f (z) i∈N |z i |<r 1 -z z i = exp   - +∞ n=1 1 n   a n - |zm|<r 1 z n m   z n  
does not vanish on a disc of center 0 and radius a little bigger than r, and so admits a holomorphic logarithm there.

• Suppose (i). Recall p from Denition 1.2 and notice that the series (1.6) converges absolutely for n p + 1. Let r be a positive real number such that r |z m | for all CHAPTER 1 m. Then dene for |z| r 2 and m 0

f m (z) = - +∞ k=p+1 1 k z z m k , (1.8)
and notice that |f m (z)| r p+1 2 p 1 |zm| p+1 . Then, recalling (1.1), the series m 0 f m converges on the disc of center 0 and radius r 2 to a holomorphic function F and, recalling (1.2), we have for z close enough to 0

e G(z) = f (z) = e Q(z)+F (z) .
Thus we may identify the coecients of order greater than deg Q in the expansions in power series of F and G, which ends the proof of (ii) recalling (1.4) and (1.8).

• Suppose (ii). Using the hypothesis for n = n 0 + 1, the innite product

P (z) = m 0 E z z m , n 0 ,
converges on C to a holomorphic function of nite order smaller than n 0 + 1 and genus n 0 (see [Boa54, Theorem 2.6.5]). But since a n = m 0 1 z n m for n n 0 + 1, we have, recalling the denition (1.3) of E, for z close enough to 0,

P (z) = exp - +∞ n=n 0 +1 1 n a n z n and consequently f (z) = exp - n 0 n=1 1 n a n z n P (z) .
Thus, f has nite order smaller than n 0 + 1 (and genus smaller than n 0 ).

We now give two counter-examples that highlight the necessity to ask for absolute convergence in (ii) in Theorem 1.3.

Proposition 1.6. (a) There exists an entire function f with f (0) = 1 such that if (z m ) m 0 is an ordering of the zeros of f (as dened in Denition 1.1) then for all n 1 the series m 0 1 z n m converges with sum a n (dened in (1.4)) but the convergence is not absolute.

(b) There exists an entire function f with f (0) = 1, an ordering (z m ) m 0 of the zeros of f and a permutation σ of N such that z σ(n) n∈N is an ordering of the zeros of f and, for all n 1, the series m 0

1 z n m converges with sum a n , but m 0 1 z n σ(m) does not converge.
Remark 1.7. Theorem 1.3 implies that the functions constructed by Proposition 1.6 have innite order, while the associated trace formula (1.6) holds in some weak sense.

To prove Proposition 1.6, we will need the following lemma, whose proof is straightforward using an Abel transform.

Lemma 1.8. Let (b m ) m 0 be a sequence of complex numbers such that there is a constant M such that, for all ∈ N, we have m=0 b m M . Let (c m ) m 0 be a decreasing sequence of positive real numbers that converges to 0. Then the series Σ m 0 b m c m converges, and we have the estimates

+∞ m=0 b m c m 2M c 0 .
Proof of Proposition 1.6. We start by proving (a). Choose an irrational real number θ for which there is a constant c > 0 such, that for all n ∈ N * , we have 1 -e 2iπnθ c n 2 (almost any real number may be chosen thanks to BorelCantelli's lemma). For every integer n,

set a n = +∞ m=2 e 2iπmθ ln (m) n ,
which is well-dened thanks to Lemma 1.8, but the convergence is clearly not absolute.

Furthermore, for all integers m 0 2, we have . Since there is only one way to order the zeros of f with increasing moduli, point (a) is proven.

a n - m 0 -1 m=2 e 2iπmθ ln (m) n 4 c n 2 ln (m 0 ) n (1.9) (take b m = e 2iπn(m+m 0 )θ and c m = (ln (m + m 0 )) -n in Lemma 1.8). Now (1.9) with exp - +∞ n=1 1 n m 0 -1 m=2 e 2iπmθ ln (m) n z n = m 0 -1 m=2 1 - e 2iπmθ ln (m) z implies that the function f dened by f (z) = exp -+∞ n=1 1 n a n z n ,
We turn now to the proof of (b). Choose θ as above and denote by (n k ) k 0 the sequence of integers dened by n 0 = 0 and n

k = k! for k 1. Dene I 0 = {0} and I k = [[n k +1, n k+1 ]]
for k 1. for m ∈ N.

We will see that there is another way to order the zeros of f , which breaks the convergence of the series (1.6) for all n 1, but preserves the monotonicity of the sequence of moduli. Choose 0 < < 1 such that for all x ∈ [0, ] we have Re e 2iπx

k =0 N (n) n k+1 → k→+∞ , but k-1 =0 N (n) n k + 1 = k→+∞ o (n k+1
), and thus

N (n) k ln (k + 2) → k→+∞ +∞. Now choose φ : N → N * such that for all n 1 the reciprocal image φ -1 ({n}) is innite (for instance 1, 1, 2, 1, 2, 3, . . . ) and set σ = +∞ k=0 σ (φ(k)) k . If n 1 the series m 0 1 z n σ(m)
does not converge. Indeed, for all k such that φ (k) = n, we have

Re Sn

k-1 +N (n) k Re S n k-1 + 1 2 N (n) k ln (k + 2) , (1.10)
where (S m ) m 0 is the sequence of partial sums of the series m 0

1 z n m
, and ( Sm ) m 0 is the sequence of partial sums of the series

m 0 1 z n σ(m)
. We let k tend to +∞ with φ (k) = n, which is possible thanks to our choice of φ. We saw that the rst term in the right-hand side of (1.10) converges but the second one tends to +∞, and thus the left-hand side of (1.10) does not converge.

In order to realize the counter-examples from Proposition 1.6 as dynamical determinants in 1.2, we will need the two following, merely technical, lemmas. Lemma 1.9. For every > 0 and ρ > 0, the counter-examples from Proposition 1.6 may be realized as entire functions f of the form

f : z → 1 -2z -z (1 -z) h (z)
, where h is an entire function such that, for all z ∈ C, we have h (z) = +∞ =0 α z , where α ∈ρ , ρ for all integers .

Proof. For all k 2 and n 1 set either

a (k) n = +∞ m=k e 2iπmθ ln (m) n or a (k) n = +∞ m=k-2 e 2iπmθ ln (k (m) + 2) n ,
depending on whether you want to get a counter-example of type (a) or (b). Then set for

all k 1 fk (z) = exp - +∞ n=1 1 n a (k) n + ā(k) n z n .
Estimate (1.9) (and its analogue for the case (b) of Proposition 1.6) implies that fk converges to 1 uniformly on all compact subsets of C as k goes to +∞. Then set

f k : z → 1 -z λ k fk (z), where λ k = fk (1) 1+ fk (1) . Thus we have f k (0) = 1, f k (1) = -1,
and it is easy to check that f k is a counter-example of type (a) or (b), according to the way the a

n have been dened. We will see that that for large enough k the function f k satises the conditions of Lemma 1.9. Let h k be the entire function dened by h k

(z) = -f k (z)-1+2z z(1-z) .
We will also need the auxiliary function H k (z) = fk (z)-1 z -( fk (1) -1) which vanishes at z = 1 and tends to 0 uniformly on all compact subsets of C when k tend to +∞. Then notice that

-h k (z) = H k (z) 1 -z 1 - z λ k + fk (1) 2 -1 fk (1)
and write

H k (z) 1 -z = +∞ =0 β z then we have α 0 = -β 0 + fk (1) 2 -1 fk (1) and α +1 = -β +1 + β λ k if 0.
But the sequence (λ k ) k 0 converges to 1 2 and (we may suppose ρ 2)

|β | = 1 2π D(0,ρ) H k (z) (1 -z) z +1 dz 2 ρ sup |z| r fk (z) -1 ,
which ends the proof, recalling that fk converges to 1 uniformly on all compact subsets of C as k goes to +∞.

Lemma 1.10. Let f (z) be an entire function such that f (0) = 1, and (c k ) k 0 be a sequence CHAPTER 1 of positive real numbers such that k 0 c k < +∞. Then the innite product

k 0 f (c k z) (1.11)
converges uniformly on all compact subsets of C to an entire function d that has same genus 1 than f . Furthermore, if f is one of the counter-example of type (a) or (b) constructed in Lemma 1.9, then d also satises point (a) or (b) respectively of Proposition 1.6.

Proof. If K is a compact subset of C then, since f (0) = 1, there is a constant C > 0 such that for all z ∈ K we have |f (z) -1| C |z|. Thus for all z ∈ K and k 0, we have |f (c k z) -1| C |c k | |z|.
out that if f (z) = exp -+∞ n=1 1 n a n z n then d (z) = exp -+∞ n=1 1 n a n +∞ k=0 c n k z n .
Suppose now that f is the counter-example of type (a) constructed in Lemma 1.9 and denote by (z m ) m 0 an ordering of its zeros. Let (w m ) m 0 be an ordering of the zeros of d , then there is a bijection (φ, ψ)

: N → N 2 such that for all m ∈ N we have w m = z φ(m) c ψ(m)
, and for every k ∈ N the sequence z φ(m) m∈ψ -1 ({k}) is an ordering of the zeros of f . Let n 1.

It is clear from our construction that f has no more than two zeros of a given modulus 2 , and so there is a constant M such that for all k ∈ N and m 0 ∈ N we have We suppose that f is a counter-example of type (b). There are two natural partitions of the zeros of d : the partition Z 0 , Z 1 , . . . , Z k , . . . by modulus (Z 0 contains the element of 1 If f has non integral order δ and c δ k < +∞, then one may show using [Boa54, 2.9.1] that d has also same order than f . 2 That's where we use that f is precisely the counter-example constructed above. We will not need it for the case (b) minimal modulus, the following are in Z 1 , etc) and the partition Z 1 , Z 2 , . . . dened by

m m 0 ψ(m)=k 1 z n φ(m) M. (1.12) Now, for k ∈ N, let u k be the sequence m m 0 ψ(m)=k 1 w n m m 0 0 whose limit is c n k a n by construction of f . From (1.12), the sup norm of u k is smaller than c n k M ,
Z k = z c k : z is a zero of f .
Both partitions are endowed with the natural notion of multiplicity. Now, we get an ordering for which (1.6) holds in the following way : we put rst the elements of Z 0 ∩ Z 0 in the order which gave (1.6) for f , then we put the elements of Z 0 ∩ Z 1 (according to the same order), then Z 0 ∩ Z 2 , etc, when we are done with Z 0 (which happens in a nite number of steps), we do the same with Z 1 , then Z 2 , etc. The proof that (1.6) holds in this case is as in case (a) (in fact a bit easier). To get an ordering for which there is divergence of the inverse of the zeros of d at any power, we do exactly the same, except that at each step we put the elements of Z 0 in the order which gave the divergence for f .

We end this section with the two following lemmas, that will be used to prove Propositions 2 and 3 in 1.2.

Lemma 1.11. Let E be a subset of N * . Then there is an entire function Q such that

Q (0) = Q (1) = 0 and if Q : z → +∞ n=1 β n z n then β n = 0 if and only if n ∈ E, and β n ∈ R for all n ∈ N * . Moreover, for every > 0 and ρ > 0, if α > 0 is suciently small, then there is an entire function h : z → +∞ n=0 α n z n such that (1 -2z) e αQ(z) = 1-2z -z (1 -z) h (z), for all z ∈ C, and α n ∈ -ρ n , ρ n for all n ∈ N. Proof. We will construct Q of the form Q : z → z (1 -z) +∞ n=0 b n z n . Then we have β 1 = b 0 and β n+1 = b n -b n-1 , for all n 1. If E contains a nal segment of N * then it is easy
to see that there is a polynomial Q with real coecients that satises the rst part of Lemma 1.11. If E does not contain a nal segment of N * then the sequence (b n ) n∈N may be recursively dened by

b 0 = 1 if 1 / ∈ E, 0 otherwise; b n = b n-1 if n 1 and n + 1 ∈ E; b n = 1 min { n, + 2 / ∈ E}! if n 1 and n + 1 / ∈ E.
The second part of Lemma 1.11 may be proven in a similar way than Lemma 1.9.

Lemma 1.12. Let N 0 : R * + → R be a locally bounded function. Then for every > 0 and ρ > 0 there is an entire function

h : z → +∞ k=0 α k z k such that for all k ∈ N we have α k ∈ -ρ k , ρ k , and if f : z → 1 -2z -z (1 -z) h (z) and (z m ) m∈N is an ordering of the zeros of f then N 0 (r) = r→0 o # m ∈ N : |z m | < 1 16r
.

(1.13) CHAPTER 1

Proof. Choose a sequence (z m ) m∈N of non-zero positive real numbers such that z m → m→+∞ +∞ and

N 0 (r) = r→0 o # m ∈ N : z m 1 16r
.

(

f m 0 (z) = (1 -λ m 0 z) m≥m 0 E z z m , p m = 1 -2z -z(1 -z)h m 0 (z)
where

λ m 0 = 1 + 1 m≥m 0 E 1 zm , p m .
Using Cauchy's formula, it is easy to see that h m 0 converges to 1 uniformly on all compact subsets of C when m 0 → +∞. Thus h = h m 0 satisfy the rst condition when m 0 is large enough. Moreover, we have for all r > 0

# m ∈ N : z m 1 16r = # m m 0 : z m 1 16r + m 0 , (1.15)
and thus

N 0 (r) = r→0 o # m m 0 : z m 1 16r
with (1.14), and since the right-hand side of (1.15) tends to +∞ when r tends to 0. This ends the proof because the z m 's are zeros of f .

Counter-examples to trace formulae

In this section, we realize a wide class of entire functions as dynamical determinants. In particular, all the possibilities considered in Theorem 1.3 as well as the counter-examples from Proposition 1.6 will materialize. We will also construct dynamical determinants, associated with nitely dierentiable weights, which cannot be holomorphically continued to the whole complex plane. Our construction will be based on a well-known example of zeta functions for hyperbolic ows which cannot be continued meromorphically to the whole complex plane (see [START_REF] Baladi | Optimality of Ruelle's bound for the domain of meromorphy of generalized zeta functions[END_REF] and [PP90, Example 1 p.165]). The strategy is the following: we rst construct a subshift of nite type and a weight for which the associated zeta function (1.19) is explicit, then we use Whitney's extension theorem [START_REF] Whitney | Analytic extensions of dierentiable functions dened in closed sets[END_REF] as in [START_REF] Bowen | One-dimensional hyperbolic sets for ows[END_REF] to get a hyperbolic dynamics on a manifold with the same dynamical zeta function, and nally we show that in this particular case the dynamical determinant may be obtained from the dynamical zeta function.

However, this construction produces an open system rather than an Anosov dieomorphism, so that we explain rst how the picture described in the introduction adapts to hyperbolic basic sets.

Hyperbolic basic sets

Let us recall the denition of a hyperbolic basic set. Denition 1.13 (Hyperbolic basic set). Let M be a C ∞ manifold and K be a compact subset of M . Let T be a C 1 dieomorphism from M to itself. We say that K is hyperbolic for T if for every x ∈ K there is a splitting of the tangent space

T x M = E u x ⊕ E s x ,
and constants C > 0, λ > 1 and a smooth Riemannian metric on M such that:

(i) for every x ∈ K and σ ∈ {u, s}, we have DT (x) (E σ x ) = E σ T x ;

(ii) for every x ∈ K, v ∈ E u x and n ∈ N, we have |DT -n (x)v| ≤ Cλ -n |v|;

(iii) for every x ∈ K, v ∈ E s x and n ∈ N, we have |DT n (x)v| ≤ Cλ -n |v|.

We say that K is a hyperbolic basic set for T if in addition:

• K is T -invariant, that is T -1 (K) = K; • K is isolated, i.e. there is an open neighbourhood U of K such that K = n∈Z T n (U )
(we say that U is an isolating neighbourhood for K);

• T |K is transitive, that is T has a dense orbit in K.
Example 1.14. If T is a transitive Anosov dieomorphism on a compact manifold M , then M is a hyperbolic basic set for T . Another classical example of hyperbolic basic set is Smale's horseshoe [Sma67, I.5]. In this section, we are mainly interested in this last example because the associated combinatorics is simple (it is exactly conjugated to a full shift), and it is thin (it has small Hausdor dimension).

CHAPTER 1

The functional approach of statistical properties for hyperbolic basic sets is very similar to the case of Anosov dieomorphisms. Let T : M → M be a C ∞ dieomorphism with hyperbolic basic set K. Let g : M → C be a C ∞ weight. Here, we must assume in addition that g is supported in a small enough isolating neighbourhood U for K. As in the case of Anosov dieomorphism, the associated Koopman operator is dened by

L : v → g.v • T.
(1.16)

Then, as in Theorem 3, there are Banach spaces on which the essential spectral radius of L is arbitrarily small [BT07, GL08]. However, these spaces are not intermediate be-

tween C ∞ (M ) and D (M ), but between C ∞ c (
d(z) = exp   - n≥1 1 n tr (L n ) z n   , (1.17)
where the at traces of the powers of L are now dened by tr (L n ) :=

T n x=x x∈K n-1 k=0 g T k x |det (I -DT n (x))| (1.18)
for n ∈ N * . In particular, the Ruelle resonances of (T, g) only depend on g through its values on K. Beware that it is not the case of the associated resonant states. Notice that Proposition 1 also applies in that case, in view of Theorem 1.3.

Symbolic dynamics with explicit weighted zeta functions

Denote by (Σ, σ) the full (two-sided) shift on two symbols, that is

Σ = {0, 1} Z and σ : (x i ) i∈Z → (x i+1 ) i∈Z . For θ ∈ ]0, 1[, dene a distance on Σ by d θ (x, y) = θ k , where k is the integer inf {i ∈ N : x i = y i or x -i = y -i } (with the convention θ ∞ = 0). Recall that if G : Σ → C
is a function, the weighted zeta function associated to (σ, G) is the formal power series dened by

ζ σ,G (z) = exp +∞ n=1 1 n σ n x=x n-1 k=0 G σ k x z n . (1.19)
Notice that ζ σ,1 is the well-known ArtinMazur zeta function, and that the radius of convergence of ζ σ,G is non-zero as soon as G is bounded. We are going to construct weights G for which ζ σ,G is given by (1.20), adapting a construction from [START_REF] Baladi | Optimality of Ruelle's bound for the domain of meromorphy of generalized zeta functions[END_REF] and

[PP90, Example 1 p.165].
Proposition 1.15. Let h be a holomorphic function dened on a neighbourhood of 0 and whose expansions in power series at zero is h (z) = +∞ k=0 α k z k . Denote by ρ its convergence radius, and assume that for all k ∈ N we have

α k = -1. Then there is a function G : Σ → C such that ζ σ,G (z) -1 = 1 -2z -z (1 -z) h (z) . (1.20) Moreover, for every θ ∈] 1 ρ , 1[, the function G is Lipschitz for the distance d θ and, if α k ∈ ]-1, +∞[ for all k ∈ N, then G is strictly positive. Proof. Set β m = 1+αm 1+α m-1 if m 1 and β 0 = 1 + α 0 and dene G : Σ → C by G (x) = β m if x 0 = • • • = x m-1 = 0 and x m = 1 1 if x 0 = • • • = x i = • • • = 0, where x = (x i ) i∈Z . An easy computation shows that G is Lipschitz for the distance d θ provided that θ ∈] 1 ρ , 1[. For N > 0 dene a (N + 1) × (N + 1) matrix P N by                (P N ) 0,i = β i if 0 i N -1 (P N ) 0,N = 1 (P N ) i+1,i = β i if 0 i N -1 (P N ) N,N = 1
the other entries are zero, that is,

P N =            β 0 β 1 β 2 . . . β N -1 1 β 0 0 . . . 0 0 β 1 0 . . . . . . . . . 0 β 2 . . . . . . 0 . . . 0 β N -1 1           
.

Then an elementary graph-theoretic argument provides that, for all integers k 1 and all N > k, we have

x∈Σ σ k x=x k-1 i=0 G σ i x = tr P k N .
(1.21)

Using an argument of dominated convergence (it is easy to show that tr

P k N 2 k G k
∞ by reducing to the positive case), one may then show that, for positive small CHAPTER 1

enough z, ζ σ,G (z) -1 = lim N →+∞ det (I -zP N ) .
A computation provides

det (I -zP N ) = (1 -z) 1 - N -1 k=0 k i=0 β i z k+1 -z N +1 N -1 i=0 β i = (1 -z) 1 - N -1 k=0 (1 + α k ) z k+1 -z N +1 (1 + α N -1 )
and thus

ζ σ,G (z) -1 = (1 -z) 1 - +∞ k=0 (1 + α k ) z k+1 = 1 -2z -z (1 -z) h (z) .
Remark 1.16. We could get a more general expression for (1.20), for instance by allowing more than two symbols. However, we will not need this here.

Smooth hyperbolic dynamics with explicit dynamical determinants

We want now to conjugate our symbolic examples to smooth ones. To do so, we use a method due to Bowen [START_REF] Bowen | One-dimensional hyperbolic sets for ows[END_REF] to conjugate a subshift of nite type to a piecewise ane horseshoe.

Proposition 1.17. There are a C ∞ dieomorphism T of the sphere S 4 and a hyperbolic basic set K for T such that, if h is as in Proposition 1.15 with in addition that ρ > 1, then there is a function g : S 4 → C such that the dynamical determinant d(z) dened by (1.17) and (1.18) is given by 3

d (z) = +∞ k=0 ζ σ,G z 4 k+2 -1 (k+1)(k+2)(k+3) 6 , (1.22)
where ζ σ,G is from Proposition 1.15. Moreover, g is C r for all integers r strictly smaller than ln ρ ln 4 , and, if α k ∈ ]-1, +∞[ for all integers k, then g is strictly positive on K.

Proof. Let G : Σ → C be the function given by Proposition 1.15. We next recall a construction due to Bowen [START_REF] Bowen | One-dimensional hyperbolic sets for ows[END_REF], in order to check that it has some extra properties that suit us. Let (e i ) 0 i 3 be the standard basis in R 4 . Set R (k) = 0 if k 0 and R (k) = 1

3 The innite product converges for the same reason as in Lemma 1.10.

COUNTER-EXAMPLES TO TRACE FORMULAE

57 if k < 0. Then, for x = (x i ) i∈Z ∈ Σ, dene I (x) = k∈Z 4 -|k| e 2x k +R(k) .
Then one easily checks that for x, y ∈ Σ we have

5 6 d 1 4 (x, y) d (I (x) , I (y)) 8 3 d 1 4 (x, y) , (1.23)
where d is the euclidean distance on R 4 . Thus I induces a homeomorphism on its image K, which is a compact subset of R 4 . Dene then

V i = (x 0 , x 1 , x 2 , x 3 ) ∈ R 4 : 1 x 2i 3 2 , 0 x k 1 2 for k = 2i and F i = I ({x ∈ Σ : x 0 = i}) for i = 0, 1. It is easy to check that F i is contained in V i . Dene L =       4 0 0 0 0 1 4 0 0 0 0 4 0 0 0 0 1 4       . For x ∈ V i , set F i x = Lx -4e 2i + 1 4 e 2i+1 (for i = 0, 1). Then, dene F on V 0 ∪ V 1 by F | V i = F i . One easily checks that F • I = I • σ.
Viewing R 4 as embedded in S 4 , one may extend F to a dieomorphism T of S 4 , that coincides with F i on a neighbourhood U i of V i (see for instance [START_REF] Palais | Local triviality of the restriction map for embeddings[END_REF]). Setting U = U 1 ∪ U 2 one has k∈Z T k (U ) = K. Thus K is a hyperbolic basic set for T with isolating neighbourhood U . Now, dene g on K by g = G • I -1 . Let r be an integer strictly smaller than ln ρ ln 4 .

Choose θ ∈] 1 ρ , 4 -r [. Next, recalling (1.23) and that G is Lipschitz for the distance d θ , there exists a constant C such that for all x, y ∈ K we have

|g (x) -g (y)| d (x, y) r C d θ I -1 (x) , I -1 (y) d 1 4 (I -1 (x) , I -1 (y)) r = C (4 r θ) m(x,y)
where m (x, y) is the smallest integer such that I -1 (x) and I -1 (y) do not coincide at the position m (x, y) or -m (x, y). .

Consequently, Whitney extension's theorem [START_REF] Whitney | Analytic extensions of dierentiable functions dened in closed sets[END_REF] ensures that g may be extended to a CHAPTER 1 C r function g on S 4 . If ρ = +∞, then g may be chosen C ∞ . Moreover, up to multiplying g by a bump function, one may assume that g is supported in U and, if G is positive, that g is positive on K.

Since I conjugates (σ, G) and ( T | K , g), one has for z small enough

1 -2z -z (1 -z) h (z) = ζ σ,G (z) -1 = exp   - +∞ n=1 1 n    x∈K T n x=x n-1 k=0 g T k x    z n    .
Notice that for all n ∈ N * we have

1 |det (I -L n )| = 1 16 n +∞ k=0 (-1) k -4 k 1 4 nk ,
and recall that (-1)

k -4 k = (k+1)(k+2)(k+3)
6 is an integer. Fubini's theorem gives

d (z) = exp   - +∞ n=1 1 n x∈K T n x=x n-1 k=0 g T k x |det (I -D x T n )| z n    = exp   - +∞ n=1 1 n 1 det (I -L n )    x∈K T n x=x n-1 k=0 g T k x    z n    = exp   - +∞ n=1 +∞ k=0 (-1) k -4 k n    x∈K T n x=x n-1 k=0 g T k x    z 4 k+2 n    = +∞ k=0 ζ σ,G z 4 k+2 -1 (k+1)(k+2)(k+3) 6
.

Remark 1.18. In Proposition 1.17, the dynamical determinant d(z) is written as an innite product involving the weighted zeta function ζ σ,G (z) associated to the symbolic dynamical systems from the last section. As we mentioned it in Remark 5, zeta functions are usually written as alternated product of dynamical determinants. The fact that we can go here in the other way and deduce the dynamical determinant from the zeta function is quite exceptional and due to the ane structure of the horseshoe.

We use now the preparatory lemmas from 1.1 to deduce various corollaries from Proposition 1.17. From Proposition 1.17 and Lemma 1.11, we deduce the following corollary, which is a slightly more precise statement than Proposition 2.

Corollary 1.19. Let E be a subset of N * . Then there are a C ∞ dieomorphism T of S4 , a hyperbolic basic set K for T , and a C ∞ function g : S 4 → R, strictly positive on K, such that, for any ordering (λ m ) m 0 of the resonances of (T, g), and for all n ∈ N * , the series m 0

λ n m converges absolutely, and its sum is tr L n (dened by (1.18)) if and only if n ∈ E.

Proposition 3 is an immediate consequence of Proposition 1.17 and Lemma 1.12. From Lemmas 1.9 and 1.10, Proposition 1.17 also gives:

Corollary 1.20. The counter-examples from Proposition 1.6 may be produced as dynamical determinants. Namely : a) There are a C ∞ dieomorphism T of S 4 , a hyperbolic basic set K for T , and a C ∞ function g : S 4 → R, strictly positive on K, such that for any ordering (λ m ) m 0 of the resonances of (T, g) (see Denition 1.1) we have, for every n 1, the trace formula

tr (L n ) = T n x=x x∈K n-1 k=0 g T k x |det (I -D x T n )| = m 0 λ n m (1.24)
but the convergence of the right-hand side is never absolute.

b) There are a C ∞ dieomorphism T of S 4 , a hyperbolic basic set K for T , a C ∞ function g : S 4 → R strictly positive on K, an ordering (λ m ) m 0 of the resonances of (T, g), and a permutation σ of N such that λ σ(m) m 0 is an ordering of the resonances of (T, g) and, for every n 1, the trace formula (1.24) holds but the series m 0 λ n σ(m)

does not converge.

From Lemma 1.10 and Proposition 1.17, it also follows that:

Corollary 1.21. The dynamical determinant (1.17) for a C ∞ dieomorphism with C ∞ weight on a hyperbolic basic set may be of any (nite or innite) genus 4 .

Recall that Theorem 1.3 gives a characterization of the genus of the dynamical determinant in terms of the trace formula (TFM). Moreover from Corollary 1.21 and [Gro55, Corollary 1 p.17, second part of the book], we deduce that there are dynamical determinants that are not Fredholm determinants of any nuclear operators (and so there is no good Banach space on which the associated Koopman operators are nuclear). Finally, we notice that Proposition 1.17 can also be used to construct systems without any resonances.

Corollary 1.22. There are a C ∞ dieomorphism T of S 4 , a hyperbolic basic set K for T , and a C ∞ function g : S 4 → C, such that the system (T, g) has no Ruelle resonances.

CHAPTER 1

In Corollary 1.22, it is fundamental that g takes value in C: if g was positive then e Ptop(T,log g-Ju) would be a resonance, where P top (T, log g -J u ) is the topological pressure of log g -J u (where J u is the unstable Jacobian) with respect to the dynamics T (see Remark 1.25).

Proof of Corollary 1.22. The function h : z → e iπz -1+2z z(z-1) continues holomorphically to C and may be written as h

(z) = +∞ k=0 α k z k with α k = -k+1 l=1 (iπ) l
l! -2. Thus α k = -1 for every n ∈ N (this is a consequence of the fact that π is transcendental). Thus applying Proposition 1.17, we nd (T, g) such that the dynamical determinant

d (z) = +∞ k=0 e iπz 4 k+2 (k+1)(k+2)(k+3) 6
does not vanish. Hence, (T, g) has no resonances.

Remark 1.23. The weight g produced by Corollary 1.19, 1.20 or 1.21 being strictly positive on K, it is associated to some physically meaningful Gibbs measure µ g (see [Bal18,

Chapter 7] for details). For example if g = 1

or g = |det ( DT | E u )| ( = 16 in our case ), µ g
is respectively the physical measure or the measure of maximal entropy for T | K (for the T we constructed these measures coincide). It may be noticed that the weights produced by Corollaries 1.19, 1.20 and 1.21 may be chosen arbitrary close to 1 in the C ∞ topology on a neighbourhood of K. The proof of this relies on the fact that, according to Lemmas 1.9 and 1.11, the function h may be taken arbitrarily close to 0 in the topology of the uniform convergence on all compact subsets of C (but, to actually prove it, an investigation of a proof of Whitney's extension theorem is needed).

Remark 1.24. Proposition 1.17 realizes a lot of entire functions as dynamical determinants, thus we could have stated many variations on Corollaries 1.19, 1.20 and 1.21. For instance, one may construct a weight g for which the trace formula (1.24) always holds but the convergence is absolute only when n is bigger than some xed integer (replace

1 (ln m) n
in the expression of a n in the proof of Proposition 1.6 by 1 m αn for some α > 0 and then state analogues of Lemma 1.9 and Lemma 1.10).

Remark 1.25. If in Proposition 1.17 we take h (z) = h a,ρ (z) = a ln 1 + z ρ , where ρ > 1 and a > 0 is small, then we get weights g = g a,ρ , strictly positive on K. From formulae (1.20) and (1.22), we know that the radius of convergence of the associated dynamical

determinant d(z) = d a,ρ (z) is exactly 5 ρ ef f = 16ρ. Let r
2 be an integer, and choose ρ such that r < ln ρ ln 4 , then [BT08, 1.5] predicted a radius of convergence greater than ρ pred = exp (-P top (log g a,ρ -log 16)) 4 r-1 for d a,ρ . However since g is strictly positive, [Bal18, Theorem 6.2] and [Bal18, Theorem 7.5] imply that exp (-P top (log g a,ρ -log 16))

5 The dynamical determinant da,ρ(z) cannot even be continued meromorphically outside the disc of center 0 and radius 16ρ.

TRACE FORMULAE AND DYNAMICAL DETERMINANTS FOR ANOSOV FLOWS61

is the smallest zero of d a,ρ (z), which can be made arbitrary close to 1 32 by taking a close enough to 0. On the other hand, we may chose ρ arbitrary close to 4 r . Thus, for all > 0, there is a choice of a and ρ such that

ρ ef f ρ pred 2048 + .
This means that [BT08, Theorem 1.5] described accurately the way the radius of convergence of the dynamical determinant grows when the regularity of the weight grows (up to a bounded multiplicative constant that could probably be made smaller than 2048 by adding extra parameters to the construction from Proposition 1.17).

Remark 1.26. The Ruelle resonances of the systems constructed in Proposition 1.17 comes as innite families. In particular, Proposition 1.17 does not allow to construct a system with a nite non-zero number of resonances. As far as we know, the only known examples of systems with nitely many resonances have either one or zero resonance.

Trace formulae and dynamical determinants for Anosov ows

We go back to the continuous-time setting and prove the following analogue of Proposition 1. The implication (i) ⇒ (ii) in Proposition 1.27 is folklore, while the implication (ii) ⇒ (i) seems to be new. Notice that the condition (1.25) implies that the left-hand side of (TFF) converges to a distribution on R * + .

Proposition 1.27. Let (φ t ) t∈R be a C ∞ Anosov ow, with generator X, on a compact manifold M . Let V : M → C be a C ∞ function and set P = X + V . Then the following statements are equivalent:

(i) the dynamical determinant d dened by (12) has nite order;

(ii) the trace formula (TFF) holds and there is ρ > 0 such that λ∈σ R (P )

1 1 + |λ| ρ < +∞. (1.25)
We recall that σ R (P ) is our notation for the Ruelle spectrum of P . Moreover, when these statements hold, the inmum of the ρ's such that (1.25) hold is the order of d.

When trying to prove trace formula (TFF), one can certainly try to apply the implication (i) ⇒ (ii) from Proposition 1.27. However, if on the contrary one tries to understand an Anosov ow (φ t ) t∈R for which the trace formula (TFF) is known to hold, while we do CHAPTER 1 not know if the associated dynamical determinant (12) has nite order, then the implication (ii) ⇒ (i) in Proposition 1.27 becomes interesting. For instance, using only tools from elementary complex analysis, the following result is obtained.

Corollary 1.28. Let (φ t ) t∈R be a C ∞ Anosov ow, with generator X, on a compact manifold M . Let V : M → C be a C ∞ function and set P = X + V . Then, if the trace formula (TFF) holds and if there is ρ ∈ [0, 1[ such that λ∈σ R (P )

1 1 + |λ| ρ < +∞, (1.26)
then the dynamical determinant d(z) dened by ( 12) is constant equal to one 6 (in particular, P has no resonance).

We chose to present Corollary 1.28 here because its proof is very brief, illustrating the power of the trace formula. However, when the weight V is real-valued, [JZ17, Theorem (1.27)

Then, choose x > 0 large enough so that the series

γ T # γ e -xTγ |det (I -P γ )| exp γ V (1.28)
converges absolutely and x > Re (λ) + for all the resonances λ of P = X + V and some > 0. It follows from standard integration by parts that for every N > 0 there is C N > 0 6 Notice that when V is real-valued, or when φ t t∈R has a periodic orbit γ such that no other periodic orbit has the same length, then d is not constant. (1.29)

Then, if γ is a periodic orbit for (φ t ) t∈R , Laplace Inversion Formula writes ϕ(T γ ) = 1 2iπ {Re z=-x} e zTγ Lap (ϕ) (z)dz = 1 2iπ {Re z=x} e -zTγ Lap (ϕ) (-z)dz.

Thanks to the bound (1.29) and the denition of x, we may apply Fubini's Theorem to

nd that γ T # γ e γ V ϕ(T γ ) |det (I -P γ )| = 1 2iπ {Re z=x} h(z)Lap (ϕ) (-z)dz, (1.30)
where h denotes the logarithmic derivative h(z)

= d (z) d(z) of the dynamical determinant d. Now, for R > 0, dene the contour Γ R by Γ R = [x -iR, x + iR] ∪ C R , where C R denotes the half-circle C R = {z ∈ C : |z -x| = R and Re z ≤ x} .
By residue's formula, we have

1 2iπ Γ R h(z)Lap (ϕ) (-z) = λ Lap (ϕ) (-λ),
where the sum on the right-hand side runs over Ruelle resonances of P that are enclosed by the contour Γ R . By (1.30) and dominated convergence, we see that

1 2iπ [x-iR,x+iR] h(z)Lap (ϕ) (-z)dz → R→+∞ γ T # γ e γ V ϕ(T γ ) |det (I -P γ )|
.

Consequently, we only need to prove that

C Rn h(z)Lap (ϕ) (-z)dz → n→+∞ 0, (1.31)
for a sequence of radii (R n ) n∈N that tends to +∞. Letting p be as in (1.1), we use

Hadamard Factorization Theorem to write d(z) as (1.2). We nd then that

h(z) = Q (z) + m 0 z - m≥m 0 1 λ m z λm p 1 -z λm ,
where (λ m ) m∈N denotes an ordering of the zeros of d (that is of the Ruelle resonances) and m 0 is the multiplicity of 0 as a Ruelle resonance. Then, let N be a large enough integer and write

h(z) = Q(z) + m 0 z - m≥m 0 1 λ N m z N λ m -z where Q is the polynomial Q(z) = Q (z) + N -1 =p (-1) +p   m≥m 0 1 λ +1 m   z .
Thanks to (1.1), we may nd a sequence (R n ) n∈N that tends to +∞ such that for every n ∈ N, z ∈ C Rn and m ≥ m 0 we have |z -

λ m | ≥ |λ m | -p-1 . Then it follows that for z ∈ C Rn we have |h(z)| ≤ Q(z) + m 0 |z| + |z| N m≥m 0 1 |λ m | N -p-
T # γ ϕ n (T γ ) |det (I -P γ )| exp γ V → n→+∞ γ T # γ e -zTγ T k γ |det (I -P γ )| exp γ V .
Now, since the trace formula holds (by assumption), we know that for all n ∈ N we have

γ T # γ ϕ n (T γ ) |det (I -P γ )| exp γ V = λ∈σ R (P ) Lap (ϕ n ) (-λ) .
Using (1.32) and the denition (1.27) of the Laplace transform, we nd that for λ ∈ C such that Re λ < x we have

Lap (ϕ n ) (-λ) → n→+∞ ∞ 0 t k e -(z-λ)t dt = k! (z -λ) k+1 . Now, if λ ∈ C is non-zero, we have Lap (ϕ n ) (-λ) = (-1) k λ k +∞ 0 e λt ϕ (k) n (t) dt.
Thus, (1.33), the fact that x > Re (λ) + for λ resonance, and the assumption (1.25) provide a domination of Lap (ϕ n ) (-λ), so that we have, using the dominated convergence theorem, λ∈σ R (P )

Lap (ϕ n ) (-λ) → n→+∞ k! λ resonances 1 (z -λ) k+1 .
Finally we have (when Re (z) 1)

k! λ∈σ P (P ) 1 (z -λ) k+1 = γ T # γ e -zTγ T k γ |det (I -P γ )| exp γ V = (-1) k+1 (ln d) (k+1) (z),
where d is the dynamical determinant dened by (12). Let F denote the canonical product of genus k -1 whose zeros are the Ruelle resonances of P = X + V (well-dened by [START_REF] Boas | Entire Functions[END_REF] CHAPTER 1

(2.6.4)] thanks to (1.25)). Then we see that, if z is not a Ruelle resonance for P , we have

(ln F ) (k) (z) = (-1) k (k -1)! λ∈σ R (P ) 1 (z -λ) k .
(1.34) It follows that (ln d) (k+1) = (ln F ) (k+1) and consequently there is a complex number a such that for every z ∈ C that is not a Ruelle resonance for X we have

(ln F ) (k) (z) = (ln d) (k) (z) + a.
With assumption (1.25), (1.34) and dominated convergence we see that (ln

F ) (k) (r) → r→+∞ r∈R 0. By direct inspection, we see that (ln d) (k) (r) → r→+∞ r∈R
0, and consequently a = 0. Thus, there is a polynomial Q of degree at most k -1 ≤ ρ such that, for every z ∈ C, we have

d(z) = e Q(z) F (z).
It follows that d has order less than ρ, since F has order less than ρ according to [Boa54, Theorem 2.6.5].

We end this section with the short proof of Corollary 1.28. Chapter 2

DenjoyCarleman classes and transfer operators

In this Chapter, we introduce the notion of DenjoyCarleman classes in 2.1. Then, in 2.2, we explain our methods to study transfer or Koopman operators associated to ultradierentiable hyperbolic dynamics through one of the most basic examples: expanding maps of the circle. In particular, the proofs of Theorems 7 and 8 may be found in 2.2.

As we announced it in the introduction, we will in fact prove quantitative versions of these results (Theorem 2.9 and Propositions 2.16, 2.17 and 2.18) using the language of Denjoy Carleman classes. Finally, in 2.3, we recall and discuss Theorem 2.27, one of the rst result that we obtained by applications of our methods to hyperbolic dieomorphisms.

Most of the contents of 2.1 and 2.2 may be found in [START_REF] Jézéquel | Transfer operator for ultradierentiable expanding maps of the circle[END_REF] (with the noticeable exceptions of Proposition 2.10 and Corollaries 2.11 and 2.12 that are new). Theorem 2.27 from 2.3 is proved in [START_REF] Jézéquel | Local and global trace formulae for smooth hyperbolic dieomorphisms[END_REF].

DenjoyCarleman classes

We give here a very brief introduction to the topic of DenjoyCarleman classes. The interested reader may refer for instance to [START_REF] Kriegl | The convenient setting for non-quasianalytic Denjoy-Carleman dierentiable mappings[END_REF] and references therein. For a more historical point of view on this topic, see [START_REF] Bilodeau | The origin and early development of nonanalytic innitely dierentiable functions[END_REF].

The ideas behind the notion of DenjoyCarleman classes have rst been introduced by Denjoy in [START_REF] Denjoy | Sur les fonctions quasi-analytiques de variable réelle[END_REF] in order to understand the notion of quasi-analyticity that was proposed by Borel in [START_REF] Borel | Les séries de fonctions analytiques et les fonctions quasianalytiques[END_REF]. However, it seems that the notion of DenjoyCarleman classes as we know it today has been introduced by Carleman in [START_REF] Carleman | Sur les fonctions indéniment dérivables[END_REF]. We give now this denition.

Let A = (A k ) k∈N be an increasing and logarithmically convex sequence of positive real

numbers such that A 0 = 1. Recall that the fact that A is logarithmically convex means CHAPTER 2 that ∀k ∈ N * : A 2 k ≤ A k-1 A k+1 .
(2.1)

Let us mention for later use that (2.1) implies that for every k 1 , . . . , k n ∈ N we have

n j=1 A k j ≤ A k , (2.2)
where

k = k 1 + • • • + k n .
This fact may be proved by induction using the fact that the

sequence (A k+1 /A k ) k∈N is increasing. If U is an open subset of R n and f : U → C is a C ∞ function,
we say that f belongs to the DenjoyCarleman class C A if, for every compact subset K of U , there are constants C, R > 0 such that for every x ∈ K and α ∈ N n we have

|∂ α f (x)| ≤ CR |α| α!A |α| .
(2.3)

Replacing the modulus in the left-hand side of (2.3) by a norm, we dene what it means for a function valued in a Banach space to be C A . Parallelizing the circle in the usual way, this denition is immediately adapted to the case of functions on S 1 : we say that a C ∞

function f : S 1 → C is in the DenjoyCarleman class C A if there are constants C, R > 0
such that for all k ∈ N and x ∈ S 1 we have

f (k) (x) ≤ CR k k!A k .
(2.4)

We will say that a map T : S 1 → S 1 belongs to the class C A if its lift from R to R does.

This denition will only be used in Corollary 2.12. We will rather assume when needed that the derivative T : S 1 → R belongs to the class C A . When the class C A is not closed under dierentiation, it does not imply that T belongs to

C A . Example 2.1. Let s ≥ 1. If A = (k! s-1 ) k∈N ,
+∞ k=0 1 (k!A k ) 1 k = +∞.
In Chapters 3 and 4, we will mostly be interested in the non-quasi-analytic case with the exception of the class of real-analytic functions in Chapter 4. However, in the following section 2.2, we do not need to distinguish between quasi-analytic and non-quasi-analytic classes. This is possible thanks to the simple geometry of our context. We think that the following result can be enlightening considering the question of trace formulae.

Proposition 2.4. Let A = (A k ) k∈N be an increasing, logarithmically convex sequence of positive real numbers such that A 0 = 1. Then the DenjoyCarleman class C A is closed under dierentiation if and only if

sup k∈N * A k+1 A k 1 k < +∞.
(2.5)

It is clear that (2.5) is a sucient condition for C A to be closed under dierentiation.

As noticed in [START_REF] Kriegl | The convenient setting for non-quasianalytic Denjoy-Carleman dierentiable mappings[END_REF], it is also necessary as a consequence of [ Let us now discuss some technical facts that are required for the next section. To the class C A , we associate the function w = w A on R *

+ dened by ∀x ∈ R * + : w(x) := inf k∈N x k k!A k .
(2.7)

The function w will play a fundamental role in estimates on singular values and norms of operators appearing in the nuclear power decomposition of the transfer operator. We are not aware of any reference introducing precisely the function w, but it seems common to introduce similar objects adapted to a particular problem (see for instance [START_REF] Fürdös | Almost analytic extensions of ultradierentiable functions with applications to microlocal analysis[END_REF] (1.1)]). The following lemma lists basic properties of the function w.

Lemma 2.6. Let A = (A k ) k∈N be an increasing, logarithmically convex sequence of positive real numbers such that A 0 = 1. Then, the function w = w A is continuous and increasing from R * + to itself. Moreover, w vanishes at all orders in 0, i.e. for all α ∈ R we have

x α w(x) → x→0 0. If µ ∈ ]0, 1[ then w(µx)/w(x) → x→0 0. If in addition γ > 1 is such that there is C > 0 such that for all k ∈ N we have (k + 1)A k+1 ≤ Cγ k A k , (2.8) then, if µ ∈ ]0, 1[, there is a constant C such that for all x > 0 we have w(µx) w(x) ≤ C x δ ,
(2.9)

where δ = -log µ/ log γ.

Remark 2.7.

≤ m then the logarithmic derivative of x → x !A is smaller than that of x → x m m!A m . Consequently, the function x → k(x) is decreasing. Thus if x 0 > 0 then for all x > x 0 since k(x) ≤ k(x 0 ) we have w(x) = x k(x) k(x)!A k(x) = min n=0,...,k(x 0 )
x n n!A n ,

and consequently w is continuous on ]x 0 , +∞[. Since x 0 > 0 is arbitrary, w is continuous on R * + .
Let µ be an element of ]0, 1[. Notice that for all x > 0 we have

w(µx) w(x) = w(µx) x k(x) k(x)!A k(x) ≤ (µx) k(x) k(x)!A k(x) x k(x) k(x)!A k(x) = µ k(x) ,
(2.10)

and since it is clear that k(x) → x→0 +∞, we get that w(µx)/w(x) → x→0 0. Assume now that (2.8) holds. Notice that if 0 < x < C -1 then x γ k(x)+1 (k(x) + 1)!A k(x)+1 ≤ x k(x) k(x)!A k(x) ,
and thus we have

k x γ ≥ k(x) + 1. Now, if 0 < x < C -1 , letting n be the largest integer such that γ n x < C -1 , we nd that k(x) = k γ n x γ n ≥ k (γ n x) + n ≥ n ≥ - log x log γ -a,
where a = log (γC) / log γ. Thus by (2.10) we nd that if 0

< x < C -1 then w(µx) w(x) ≤ C x δ ,
where C = µ -a .

We end this section with a lemma that implies that every C ∞ function on the circle belongs to some DenjoyCarleman class. It allows us to deduce Theorems 7 and 8 from CHAPTER 2 their quantitative versions Theorem 2.9 and Propositions 2.16, 2.17 and 2.18. We omit the elementary proof.

Lemma 2.8. Let (A k ) k∈N be a sequence of non-negative real numbers. Then there are a constant C > 0 and an increasing and logarithmically convex sequence (B k ) k∈N of positive real numbers such that B 0 = 1 and

∀k ∈ N : A k ≤ CB k .

Transfer operator for ultradierentiable expanding maps of the circle

We are now going to use the notion of DenjoyCarleman classes in order to prove Theorems 7 and 8. Let thus T be a C ∞ expanding map of the circle (in the sense of Denition 1).

Denote by λ > 1 a dilation constant for T . Moreover, let A = (A k ) k∈N be an increasing, logarithmically convex sequence of positive real numbers such that A 0 = 1, and assume that T : S 1 → R is of class C A . We recall that the transfer operator L associated to T is dened by

L : u →   x → T y=x 1 |T (y)| u(y)   .
(2.11)

Using the function w = w A dened by (2.7), we can now state a quantitative version of Theorem 7.

Theorem 2.9. For every θ ∈ ]1, λ[ there are constants C, M > 0 and a Hilbert space H continuously contained in C ∞ S 1 and containing trigonometric polynomials as a dense subspace, such that L denes a compact operator from H to itself. Moreover, if (σ k ) k∈N is the sequence of singular values of L acting on H then we have

∀k ∈ N * : σ k ≤ C sup 0<x≤ 1 k w (M x) w (θM x) .
(2.12)

The proof of Theorem 2.9 is carried on in 2.2.1. In order to promote our idea that DenjoyCarleman classes that are closed under dierentiation are much better-behaved than the others, we will also prove the following:

Proposition 2.10.

π n u = u, e 0 L 2 e 0 if n = 0 θ n-1 ≤|k|<θ n u, e k L 2 e k otherwise .
In order to give the denition of the space H from Theorem 2.9, we need to state a technical but fundamental result.

Lemma 2.13. There are constants C, R > 0 such that for all m, n ∈ N and u ∈ L 2 S 1 such that m ≥ n we have

π m Lπ n u L 2 ≤ Cw R θ m θ m+n 2 π n u L 2 .
We can now dene H = H θ,R,A as the space of u ∈ L 2 S 1 such that (R is the constant from Lemma 2.13):

m∈N λ -2m w R θ m-1 -2 π m u 2 L 2 < +∞.
(2.13)

It is easily seen that the square root of the quantity above denes a norm for which H is a Hilbert space. From Lemma 2.6, the quantity λ -m w R θ m-1 -1 tends to innity faster than any geometric sequence when m tends to innity. Consequently, the space H is CHAPTER 2 continuously contained in C ∞ S 1 . One can check easily that trigonometric polynomials form a dense subspace of H.

Before proving Lemma 2.13, we need another technical result.

Lemma 2.14. There are constants C, R > 0 such that for all k, ∈ Z such that

|k| > θ -1 | | we have | Le , e k L 2 | ≤ Cw R |k| .
Proof. Dene the function a k, :

S 1 → C by a k, (x) = 1 2iπ(kT (x)-) and the dierential operator L a k, : u → (a k, u) .
Then for all m ∈ N we have

Le , e k L 2 = S 1 e 2iπ( x-kT (x)) dx = S 1 e 2iπ( x-kT (x)) L m a k, (1)(x)dx, so that | Le , e k L 2 | ≤ L m a k, (1)
∞ .

In order to bound L m a k, (1), we rst investigate the derivatives of a k, . By Faa di Bruno's formula, for all n ∈ N and x ∈ S 1 we have

a (n) k, (x) = 1 2iπ m 1 +2m 2 +•••+nmn=n (-1) m 1 +...mn n!(m 1 + • • • + m n )! m 1 ! . . . m n ! × 1 (kT (x) -) 1+m 1 +•••+mn n j=1 kT (1+j) (x) j! m j . Thus, since T belongs to the class C A , a (n) k, (x) ≤ 1 2π m 1 +2m 2 +•••+nmn=n n!(m 1 + • • • + m n )! m 1 ! . . . m n ! × 1 |kT (x) -| 1+m 1 +•••+mn n j=1 |k| CR j A j m j ,
where C, R > 0 are from the denition of C A . Recall (2.2) and notice that we have

|kT (x) -| ≥ λ |k| -| | > δ |k|
, where δ = λ -θ. Thus, we nd (assuming that C > 1 and δ < 1, which is true without loss of generality)

a (n) k, (x) ≤ A n 2πδ |k| CR δ n m 1 +2m 2 +•••+nmn=n n!(m 1 + • • • + m n )! m 1 ! . . . m n ! . Now, notice that m 1 +2m 2 +•••+nmn=n n!(m 1 + • • • + m n )! m 1 ! . . . m n ! =    1 if n = 0 2 n-1 n! otherwise .
Indeed, as a consequence of Faa di Bruno's formula, the sum in the left hand side is the nth derivative at zero of the function

x → 1 + x 1 -2x = 1 1 -x 1-x .
Notice then that for all m ∈ N there are natural integer coecients that do not depend on a k, such that

L m a k, 1 = n 1 +•••+nm=m c n 1 ,...,nm m j=1 a (n j ) k, .
(2.14) Thus, using (2.2) again,

L m a k, (1)
∞ ≤ n 1 +•••+nm=m c n 1 ,...,nm m j=1 n j ! A n j 2πδ |k| 2CR δ n j ≤ A m CR πδ 2 |k| m n 1 +•••+nm=m c n 1 ,...,nm m j=1 n j !. Now replacing a k, by the function a : x → 1 1-x in (2.14) we have that (notice that L m a (1) : x → (2m)! m!2 m 1 (1-x) 2m
, where L a is the dierential operator dened by L a (u) = (au) )

n 1 +•••+nm=m c n 1 ,...,nm m j=1 n j ! = L m a (1)(0) = (2m)! m!2 m . Thus L m a k, (1) ∞ ≤ m!A m (2m)! m! 2 CR 2πδ 2 |k| m .
We only need to notice that (2m)!/m! 2 grows at most exponentially to end the proof (with dierent values of C and R of course).

Proof of Lemma 2.13. We will only deal with the case n = 0, the case n = 0 is similar.

CHAPTER 2

Let us compute (here L * denotes the L 2 -adjoint of L, that is the Koopman operator):

π m Lπ n u 2 L 2 = θ m-1 ≤|k|<θ m | Lπ n u, e k L 2 | 2 = θ m-1 ≤|k|<θ m | π n u, L * e k L 2 | 2 ≤ π n u 2 L 2 θ m-1 ≤|k|<θ m π n L * e k 2 L 2 ≤ π n u 2 L 2 θ m-1 ≤|k|<θ m θ n-1 ≤| |<θ n | Le , e k L 2 | 2 . Now, if θ m-1 ≤ |k| < θ m and θ n-1 ≤ | | < θ n then we have |k| ≥ θ m-1 ≥ θ n-1 > θ -1 | |
(since m ≥ n) and thus by Lemma 2.14 we have (recall that w is increasing)

| Le , e k L 2 | ≤ Cw R |k| ≤ Cw θR θ m .
Consequently,

π m Lπ n u 2 L 2 ≤ 4C 2 π n u 2 L 2 θ m -θ m-1 + 1 θ n -θ n-1 + 1 w θR θ m 2
and the result follows.

We will need another technical result to prove Theorem 2.9. For N ∈ N, dene the following nite rank operators on H:

L c,N = 0≤n≤m≤N π m Lπ n and L b,N = 0≤m<n≤N π m Lπ n .
(2.15)

We will use these nite rank operators to approximate the transfer operator L, to do so we need the following lemma.

Lemma 2.15. There is a constant C > 0 such that for all M ≥ N ≥ 0 we have

L c,N -L c,M L 2 →H ≤ C sup m>N w R θ m w R θ m-1 and L b,N -L b,M H→H ≤ C sup m≥N w R θ m w R θ m-1 . Proof. If u ∈ H then we have (L c,M -L c,N ) u = 0≤n≤m≤M N <m π m Lπ n u and thus (L c,M -L c,N ) u 2 H = N <m≤M λ -2m w R θ m-1 -2 π m L n≤m π n u 2 L 2 . But if N < m ≤ M we have with Lemma 2.13 π m L n≤m π n u L 2 ≤ n≤m π m Lπ n u L 2 ≤ Cw R θ m θ m 2 n≤m θ n 2 π n u L 2 ≤ Cw R θ m θ m n≤m π n u 2 L 2 ,
and thus (for some new constant C that may change from one line to another)

(L c,M -L c,N ) u 2 H ≤ C sup m>N w R θ m w R θ m-1 2 n≥0 m>N θ 2m λ 2m π n u 2 L 2 ≤ C sup m>N w R θ m w R θ m-1 2 n≥0 π n u 2 L 2 ≤ C sup m>N w R θ m w R θ m-1 2 u 2 L 2 .
Before proving the second estimate, let us show that there is a constant C > 0 such that for every integer n we have

0≤m<n λ -2m w R θ m-1 -2 ≤ Cλ -2n w R θ n-2 -2 . (2.16)
To do so, recall the function k(x) from the proof of Lemma 2.6 and choose m 0 large enough so that

λ 2 θ 2k R θ m 0 -1 < 1.
Then, when n is large enough, we may split the sum in (2.16) between the sum over 0 ≤ m < m 0 and the sum over m 0 ≤ m < n. The rst sum is independent on n, and can consequently be ignored since the right-hand side of (2.16) tends to +∞ when n tends to CHAPTER 2 +∞, according to Lemma 2.6. To bound the second sum, recall (2.10) to see that

λ 2n w R θ n-2 2 m 0 ≤m<n λ -2m w R θ m-1 -2 ≤ m 0 ≤m<n λ 2(n-m) θ m-n+1 2k R θ m-1 ≤ λ 2 ≥0 λ 2 θ 2k R θ m 0 -1 < +∞.
We turn now to the proof of the second estimate and write for u ∈ H

(L b,M -L b,N ) u = 0≤m<n≤M N <n π m Lπ n u
from which we get (we use (2.16) on the fth line and C may change from one line to another)

(L b,M -L b,N ) u 2 H = 0≤m<M λ -2m w R θ m-1 -2 π m L n>max(m,N ) π n u 2 L 2 ≤ C 0≤m<M λ -2m w R θ m-1 -2 n>max(m,N ) π n u 2 L 2 ≤ C 0≤m<M λ -2m w R θ m-1 -2 n>max(m,N ) π n u 2 L 2 ≤ C n>N   0≤m<n λ -2m w R θ m-1 -2   π n u 2 L 2 ≤ C n>N λ -2n w R θ n-2 -2 π n u 2 L 2 ≤ C sup n>N w R θ n-1 w R θ n-2 2 u 2 H .
We are now in position to end the proof of Theorem 2.9.

Proof of Theorem 2.9. Lemma 2.15 implies in particular that the sequence (L c,N ) N ∈N is a Cauchy sequence of bounded operators from L 2 to H and thus converges to a bounded operator L c : L 2 → H. For the same reason, (L b,N ) N ∈N converges to a bounded operator L b : H → H. By checking the identity on trigonometric polynomials, we see that

L = L c + L b .
(2.17)

In particular, L is bounded (and even compact, as a limit of nite rank operators) from H to itself. The only thing that we still need to check is the bound (2.12) on singular values of the operator L acting on H. If N ∈ N, notice that the operator L c,N + L b,N has rank at most 2 θ N + 1. From Lemma 2.15 (letting M tend to innity), we deduce that

L -(L c,N + L b,N ) H→H ≤ 2C sup m≥N w R θ m w R θ m-1
and thus (see [GGK00, Theorem IV.2.5])

σ 2 θ N +2 ≤ 2C sup m≥N w R θ m w R θ m-1
.

The result then follows from the fact that the sequence (σ k ) k∈N is decreasing.

We saw that H is continuously contained in C ∞ . However, when the class C A is closed under dierentiation, we can improve this result and prove Proposition 2.10.

Proof of Proposition 2.10. Let u ∈ H. Since u is C ∞ , we may write it as the sum of its Fourier series u = k∈Z u, e k L 2 e k .

(2.18)

From the denition of H, we see that for k ∈ Z we have

| u, e k | ≤ λ m w R θ m-1 u H , (2.19)
where m is such that θ m-1 ≤ |k| < θ m if k = 0 and 0 otherwise. Since w is increasing,

(2.19) implies that | u, e k | ≤ λ (|k| + 1) log λ log θ w θR max(1, |k|) u H .
(2.20)

Now, choose L > 0 large and notice that if k ∈ Z and n ∈ N then we have for every x ∈ S 1 :

e (n) k (x) = (2π |k|) n = 2π |k| L n (A n n!) -1 L n A n n! ≤ w L 2π max(1, |k|) -1 L n A n n!.
(2.21) CHAPTER 2 From (2.18), (2.20) and (2.21), we nd that for every n ∈ N and x ∈ S 1

u (n) (x) ≤   k∈Z (|k| + 1) log λ log θ w θR max(1,|k|) w L 2π max(1,|k|)   L n A n n! u H .
Consequently, in order to prove that f belongs to C A , we only need to see that for L large enough we have k∈Z

(|k| + 1) log λ log θ w θR max(1,|k|) w L 2π max(1,|k|) < +∞.
Since C A is closed under dierentiation, it follows from Proposition 2.4 that there is γ > 0 such that (2.8) holds for every k ∈ N. Hence, according to Lemma 2.6, there is C > 0 such that for every k ∈ Z we have

w θR max(1,|k|) w L 2π max(1,|k|) ≤ C max(1, |k|) -δ , where δ = -log(2πθR/L) log γ
. Taking L large enough so that log L > log (2πθRγ) + log γ log λ log θ , the lemma is proved.

We end this section with the proof of Corollaries 2.11 and 2.12 of Proposition 2.10. loss of generality, we may assume that 0 is a common xed point for T 1 and T 2 . Then, we let ρ 1 and ρ 2 denote the density of the invariant probability measures absolutely continuous with respect to Lebesgue, respectively for T 1 and T 2 . For j = 1, 2, we dene the map F j from S 1 to itself by

F j (x) = [0,x]
ρ j (y)dy mod 1.

Since ρ 1 and ρ 2 are fully supported, we see that the F j 's are dieomorphisms from S 1 to itself. Moreover [Rue89, Corollary 5.2], ρ 1 and ρ 2 are resonant states associated to the resonance 1 respectively for T 1 and T 2 . Hence, it follows from Corollary 2.11 that ρ 1 and ρ 2 , and hence F 1 and F 2 , are of class C A . Since the Inverse Function Theorem holds in regularity C A , the inverses of the dieomorphisms F 1 and F 2 also belong to the class C A (see [START_REF] Komatsu | The implicit function theorem for ultradierentiable mappings[END_REF] or [BM04, Theorem 4.10]). In particular, the map F -1 2 •(±F 1 ) is of regularity C A . However, it follows from the claim on the bottom of the second page of [START_REF] Arteaga | Dierentiable conjugacy for expanding maps on the circle[END_REF] that

F -1 2 • (±F 1 )
is the conjugacy map between T 1 and T 2 (recall that it is unique once its degree is xed [KH95, Lemma 2.4.10]).

Nuclear power decomposition

We saw in the proof of Theorem 2.9 that the transfer operator L may be written as the sum (2.17) of the operators L b and L c . In this section, we show that this is a nuclear power decomposition in the spirit of [START_REF] Baladi | Dynamical determinants and spectrum for hyperbolic diemorphisms. Geometric and probabilistic structures in dynamics[END_REF], and we investigate the consequences of the existence of such a decomposition, in particular in terms of dynamical determinants (see Propositions 2.18 and 2.23). Thus, we will prove in particular Theorem 8.

We rst investigate the operator L b . To do so, dene the function g : Proposition 2.16. There is a constant C > 0 such that for all n ∈ N * we have

N → R * + by g(N ) = sup m≥N w R θ m w R θ m-1 (2.
L n b H→H ≤ C n n-1 k=0 g(k).
In particular, the spectral radius of L b is zero (i.e. L b is quasi-nilpotent).

Proof. Notice that if k < N then from the denition of L b,N it comes that

(L b,N -L b,k ) L b,N = (L b,N -L b,k ) (L b,N -L b,k+1 ) . Thus if N ≥ n -1 we have L n b,N = n-1 k=0 (L b,N -L b,k ) .
Letting N tends to innity, we get that

L n b = n-1 k=0 (L b -L b,k )
and the result follows from Lemma 2.15.

Then we investigate the operator L c (as an operator from H to itself ).

Proposition 2.17. There are constants C, R > 0 such that, if we dene the function f

for x > 0 by f (x) = x α w R x , (2.23)
where α = log λ/ log θ, and if (s ) ∈N denotes the sequence of singular values of L c acting on H, then, for all ≥ 1, we have

s ≤ Cf ( ).
(2.24)

In particular, L c is nuclear of order 0.

Proof. Since L c is continuous from L 2 to H, we have the following bound on its singular values as a compact operator from H to itself:

∀m ∈ N : s 2 θ m +1 ≤ C sup p≥m λ p w R θ p-1
for some constant C > 0. Since f does not vanish, we only need to prove (2.24) for large. Thus, let be large and let m be the largest integer such that ≥ 2 θ m + 1. Then, we have

s ≤ s 2 θ m +1 ≤ C sup y≥θ m y α w θR y .
(2.25)

Here, we performed the change of variables y = θ p ". Then, notice that θ m ≥ 1 2θ -3 2θ ≥ 1 4θ (provided that is large enough). Hence, we deduce from (2.25) that, for large enough, we have (with the change of variables x = 4θy and taking R = 4θ 2 R in the denition of f )

s ≤ C sup x≥ f (x).
(2.26)

Recall the function k(x) from the proof of Lemma 2.6 and use (2.10) to see that for all

x ≥ we have

f (x) f ( ) ≤ x α-k R , (2.27) but if
is large enough we have k R > α and consequently f (x) ≤ f ( ). Hence, for large enough we have

f ( ) = sup x≥ f (x),
(2.28) and (2.24) follows from (2.26). To see that L c is nuclear of order 0, recall from Lemma 2.6 that w vanishes at all orders in 0. Hence, f decays faster than the inverse of any polynomial and so does the sequence of singular values of L c . Now, following [START_REF] Baladi | Dynamical determinants and spectrum for hyperbolic diemorphisms. Geometric and probabilistic structures in dynamics[END_REF], we want to use the nuclear power decomposition in order to study the dynamical determinant d dened by (4) and (5). This is the point of Proposition 2.18, that completes the proof of Theorem 8.

Proposition 2.18. If z is small enough then we have

d(z) = det I -z (I -zL b ) -1 L c , (2.29)
where the dynamical determinant d is dened by (4) and (5).

Notice in particular that this proposition and Theorem 2 imply that the spectrum of L acting on H coincides with the Ruelle spectrum of L from Denition 3 (we already knew that fact, as this is a consequence of Lemma B.1).

Proof of Proposition 2.18. If N ∈ N then the operators L b,N and L c are trace class (recall that L b,N , dened by (2.15), has nite rank). Moreover, L b,N is nilpotent and thus

det (I -z (L b,N + L c )) = det (I -zL b,N ) det I -z (I -zL b,N ) -1 L c = det I -z (I -zL b,N ) -1 L c → N →+∞ det I -z (I -zL b ) -1 L c , (2.30)
and the convergence holds uniformly on all compact subsets of C. Denote by h(z) the entire function on the right-hand side of (2.29). Since h(0) = 1, there is a sequence (a n ) n≥1 of complex numbers such that for |z| small enough we have

h(z) = exp   - n≥1 a n n z n   .

CHAPTER 2

Then applying Cauchy's formula, we nd for n ≥ 1 (and for small enough)

a n = - 1 2iπ ∂D(0, ) h (z) h(z) dz z n = lim N →+∞ tr ((L b,N + L c ) n ) . Then notice that, since L b,N is nilpotent we have tr ((L b,N + L c ) n ) = tr (L b,N + L c ) n -L n b,N but the operators (L b,N + L c ) n -L n b,N converge in trace class topology to the operator L n -L n b . Thus we have a n = tr (L n -L n b ) = k∈Z (L n -L n b ) e k , e k L 2 .
(2.31) (2.32)

Now, notice that if k ∈ Z and n ∈ N * we have L n b e k , e k L 2 = 0. Indeed, if k ∈ Z * is such that θ m-1 ≤ |k| < θ m ,
Finally, we use a partition of unity in the last integral and locally we perform the change of variable u = x -T n (x). Then, we recognize the Dirichlet kernel and nd a n = tr (L n ),

where the at trace is dened by (5). This proves that h(z) = d(z).

The fact that the zeros of d are exactly the inverses of the non-zero eigenvalues of L counted with multiplicity follows from [Kur61, Theorem 3.1]. However, notice that in our case the situation is simpler than in the general theory of WeinsteinAronszajn determinant, and that the correspondence between the zeros of d and the inverses of the non-zero eigenvalues of L may be deduced from the convergence (2.30).

In some cases, it may happen that L acting on H is trace class, or in some Schatten class. In these cases, we may simplify Proposition 2.18 in the following way.

Proposition 2.19. Assume that there is p > 0 such that L acting on H is in the Schatten class S p . Then, if m denotes the smallest integer larger than p, we have

d(z) = det m (I -zL) exp - m-1 n=1 tr (L n ) n z n , (2.33)
where the tr (L n )'s are dened by (5) and det m denotes the regularized determinant of order m dened in [GGK00, IX] (this is the usual Fredholm determinant when m = 1).

In particular, the order of d is less than p and the trace formula (TFM) holds for every

n ≥ p.
Proof. We know that when |z| is small enough we have

det m (I -zL) = exp   - n≥m tr (L n ) n z n   .
Then, the same computation (2.31)-(2.32) as in the proof of Proposition 2.18 ensures that for n ≥ m we have Remark 2.20. If the sequence A satises (2.8) for some γ > 0, then the estimates (2.9) and (2.12), respectively from Lemma 2.6 and Theorem 2.9 imply that the singular values of L acting on H satisfy

tr (L n ) = k∈Z L n e k , e k L 2 = tr (L n ) , (
σ k = k→+∞ O 1 k δ , (2.35)
where δ = log θ/ log γ. Hence, L acting on H belongs to the Schatten class S p for any p > δ -1 and Proposition 2.19 implies that the order of d is less than log γ/ log θ. Since θ may be chosen arbitrarily close to the expanding constant λ, the following result follows.

Corollary 2.21. If there is γ > 0 such that the sequence A satises (2.8), then the order of the dynamical determinant d is less than log γ/ log λ and the trace formula (TFM) holds for every n > log γ/ log λ.

Remark 2.22. Recall from Remark 2.7 that the validity of (2.8) (for some value of γ > 0)

is equivalent to C A being closed under dierentiation. We advertised in Remark 2.5 that it was a relevant notion when investigating trace formula, we explain now why.

Notice that (2.8) implies that taking a derivative in the class C A results in replacing R by γR in (2.4). Composing by a contraction of factor λ -1 (which is basically what L does) results in replacing R by λ -1 R. Thus L has morally the same regularizing eect in the class C A as taking log γ/ log λ primitives (notice that this number is not necessarily an CHAPTER 2 integer). To put it loosely, L is a smoothing operator from the point of view of the class C A . Consequently, the decay that we obtain on the singular values of L, and ultimately the bound on the order of the dynamical determinant, is natural (considering for instance the case of Sobolev injections) for an operator that makes a function gain log γ/ log λ derivatives. We witness here an interesting phenomenon: the values of n for which Corollary 2.21 ensures that the trace formula (TFM) holds depend a priori on the dilation factor λ for T . Now, if the sequence (A k ) k∈N satises (2.6), we see that (2.8) holds for any γ > 1.

Hence, we see from the discussion above that L is an operator that makes a function gain an innite number of derivatives in the class C A . To put it more rigorously, Proposition 2.19 implies that L acting on H is nuclear of order 0, and following the dynamical determinant d has order zero and trace formula (TFM) always hold.

See the examples from 2.2.3 for further discussions of these phenomena.

Finally, we will use the nuclear power decomposition (2.17) with Propositions 2.16, 2.17 and 2.18 in order to bound the growth of the dynamical determinant d. To do so, dene the entire functions F and G by

F (z) = (1 + z) +∞ m=1 (1 + f (m)z) and G(z) = +∞ n=0 n-1 k=0 g(k) z n ,
|d(z)| = k∈N (1 + λ k ) ≤ 1 + n≥1 k 1 <•••<kn n j=1 λ k j .
Then applying [GGK00, Theorem IV.3.1] we see that Remark 2.24. Assume that the right-hand side of (2.12) in Theorem 2.9 is summable.

|d(z)| ≤ 1 + n≥1 k 1 <•••<kn n j=1 c k j = k≥0 (1 + c k ) . Now if (s k ) k≥0
Then we know that L acting on H is trace class and Proposition 2.19 implies that the order of the dynamical determinant d is less than 1. In particular, d has genus zero, and when the right-hand side of (2.12) decays very fast we may want to get a better bound on d. To do so, we may work as in the proof of Proposition 2.23 to nd that there is a constant C > 0 such that for every z ∈ C we have

|d(z)| ≤ (1 + C |z|) k∈N * 1 + C |z| sup x≤ 1 k w (M x) w (θM x) .
(2.38)

The innite product in the right-hand side of (2.38) may be bounded using [Boa54, Lemma 3.5.1] as we did for F : just replace n(s) by the number of integer k such that

sup x≤ 1 k w(M x) w(θM x) ≥ s -1 in (2.37) or k = 0.
Remark 2.25. Notice that, using Jensen's formula [Boa54, 1.2.1 p.2], a bound on the growth of the dynamical determinant immediately gives an upper bound on the asymptotics of the number of Ruelle resonances outside of D(0, ), when tends to 0.

Examples Gevrey and analytic dynamics

In this section we take A k = k! s-1 for some s ≥ 1, that is we study the Gevrey classes of regularity from Example 2.1. We still denote by T an expanding map of the circle with expanding factor at least λ > 1, and we assume that T is s-Gevrey. In this case, we see CHAPTER 2

that for every γ > 1, we can nd C > 0 such that for all k ∈ N the estimate (2.8) holds.

Thus the dynamical determinant d has order 0. But we can of course get a better bound.

To do so, recall the function k from the proof of Lemma 2.6. Its denition implies that if x > 0 then

x k(x) k(x)! s < x k(x)+1 (k(x) + 1)! s and thus k(x) > x -1 s -1.
Then if θ ∈ ]0, λ[ and M > 0 is the constant from Theorem 2.9, we have when m ≥ 1

σ m ≤ C sup 0<x≤ 1 m w(M x) w(θM x) ≤ sup 0<x≤ 1 m 1 θ k(θM x) ≤ c exp -c -1 m 1 s (2.39)
for some constant c > 0. Thus, by [Jéz20a, Lemma 2.13] (or Remark 2.24), we have that for some constant c > 0 we have

|d(z)| ≤ c exp c log + |z| 1+s .
Notice that we retrieve the optimal result [BN19] when s = 1, which is also the result of Ruelle [START_REF] Ruelle | Zeta-functions for expanding maps and Anosov ows[END_REF].

Remark 2.26. The bound (2.39) on the singular values of L acting on H implies that the transfer operator L belongs to the exponential class of type (c -1 , σ -1 ) dened in [START_REF] Bandtlow | Resolvent estimates for operators belonging to exponential classes[END_REF].

Hence, we may apply the results from [START_REF] Bandtlow | Resolvent estimates for operators belonging to exponential classes[END_REF] to transfer operators associated to Gevrey expanding maps of the circle. For instance, the resolvent estimates [Ban08, Theorem 3.13] may be used to derive a better (super-exponential) remainder in the asymptotics expansion for the correlations of Gevrey observables (see [GL08, Theorem 1.2] for the usual asymptotics of correlation in the case of hyperbolic dieomorphisms). We could probably also use [Ban08, Theorem 4.2] to control globally the Ruelle spectrum of a perturbation of T in the Gevrey category.

The class C κ,υ

We investigate now the classes from Example 2.2 and illustrate the discussion from Remark 2.22. Recall that these classes will be used in Chapter 3 to dene a class of Anosov ow for which the trace formula (TFF). Hence, we choose κ > 0 and υ ≥ 1 and take A k = exp(k υ /κυ). Recall that according to Proposition 2.4 the class C κ,υ is closed under dierentiation if and only if υ ≤ 2. Moreover, the stronger condition (2.6) holds if and only if υ < 2. We assume now that T belongs to the class C κ,υ .

Let us deal rst with the case 1 < υ < 2 (the case in which (2.6) holds), then we see that for 0 < x < 1 and some constants c depending on κ we have (k is still from the proof of Lemma 2.6)

k(x) ≥ c -1 |log x| 1 υ-1 -c, (2.40)
thus if µ ∈ ]0, 1[, we have for some new constant c > 0 and small x > 0

w(µx) w(x) ≤ c exp -c -1 |log x| 1 υ-1 . (2.41)
Then, for some new constant c > 0 the estimates on the singular values of L from Theorem 2.9 becomes (for k ≥ 1)

σ k ≤ c exp -c -1 (log k) 1 υ-1 .
Once again, this gives, with Remark 2.24, that, up to taking larger c,

|d(z)| ≤ k≥1 1 + c |z| exp -c -1 (log k) 1 υ-1 . (2.42)
Using [Boa54, Lemma 3.5.1] to bound the right-hand side of (2.42) (that is using (2.37) with the modication described in Remark 2.24), we get that for some new constant c > 0 and all z ∈ C we have

log + |d(z)| ≤ c exp c log + |z| υ-1 .
(2.43)

In particular, d has order zero, but this could have been seen as a consequence of Corollary 2.21 (see also Remark 2.22). Now, if υ = 2 (the case in which C κ,υ is closed under dierentiation but (2.6) does not hold) then we have

lim k→+∞ (k + 1)A k+1 A k 1 k = e 1 κ .
Thus, by Corollary 2.21, the dynamical determinant d has order less than (κ log λ) -1 .

We have here a very interesting behaviour: the bound on the order of the dynamical determinant depends on the dilation factor (this implies in particular that trace formula (TFM) holds for large n according to Proposition 1). As pointed out in Remark 2.22, it is not surprising that this behaviour occurs for the value of υ that separates classes that are closed under dierentiation and those that are not. As far as we know, it is the rst time that such a behavior is proved and, consequently, it would be particularly interesting CHAPTER 2 to know whether our result is sharp or not in that case.

Finally, we deal with the case υ > 2, when the class C κ,υ is not closed under dierentiation. The estimates (2.40), and thus (2.41), remain true. Thus, for some c > 0, we have for large N (recall that g is dened by (2.22))

g(N ) ≤ c exp -c -1 N 1 υ-1
and thus, changing the value of c,

N -1 k=0 g(N ) ≤ c exp -c -1 N 1+ 1 υ-1 .
Then, in the denition (2.36) of G, we may split the sum between n ≤ 2 log r c υ-1 and n > 2 log r c υ-1 , to nd that for some c > 0 and all r > 0 log + G(r) ≤ c log + r υ + 1

An easy computation shows that for some c > 0 and all m ≥ 1 we have

f (m) ≤ c exp -c -1 (log m) υ υ-1
, where f has been dened by (2.23). Thus reasoning as above in the case υ < 2 (that is using [Boa54, Lemma 3.5.1], which has been stated as (2.37) in this case), we nd that for some c > 0 and r > 0

log + F (r) ≤ c exp c log + r υ-1 υ .
And by Proposition 2.23, we see that there is still a new constant c > 0 such that for all z ∈ C we have

log + |d(z)| ≤ c exp c log + |z| υ-1 .
(2.44)

Notice that this is the same estimate than (2.43) that we established in the case υ < 2, and that it is still true in the case υ = 2 (but we have more precise information in this case).

It is very interesting that the bound (2.44) is true regardless of the value of υ while there is a huge change in the structure of the transfer operator at υ = 2. Hence, it seems that in most cases the nuclear power decomposition contains all the information that we need on the dynamical determinant. This is indeed a very versatile tool that allows also to deal with nitely dierentiable map [START_REF] Baladi | Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps[END_REF], and as we have just seen, it does not seem that we lose much information by using this method in more favorable cases. Notice however

KOOPMAN OPERATORS FOR GEVREY HYPERBOLIC MAPS

91 that in some very favorable cases (such as Gevrey and analytic dynamics), the nuclear decomposition does not seem to give the best bound (this is because in this case, the bounds on the singular values of L and L c are very similar).

Koopman operators for Gevrey hyperbolic maps

In [START_REF] Jézéquel | Local and global trace formulae for smooth hyperbolic dieomorphisms[END_REF], we applied the strategy exposed in 2.2 to more complicated dynamics hyperbolic basic sets for smooth dieomorphisms, see 1.2.1. Hence, we were able to prove the following result. We recall that G s is our abbreviation for s-Gevrey. The spaces G s c (V ) and U s c (V ) and their topologies will be introduced in 4.1.1. The latter is a space of linear functionals on Gevrey functions or ultradistributions.

Theorem 2.27 ([Jéz20a]). Let s > 1. Let M be a G s manifold and T : M → M a G s dieomorphism. Let K be a hyperbolic basic set for T . Then there exist a compact isolating neighbourhood V for K and a separable Hilbert space H such that the following holds (i) the Hilbert space H is contained in U s c (V ) and the inclusion is continuous;

(ii) the Hilbert space H contains G s c (V ) and the inclusion is continuous with dense image;

(iii) if g : M → C is a G s function supported in V , the weighted Koopman operator L dened by (1.16) extends to a trace class operator from H to itself;

(iv) for all n ∈ N * we have tr (L n ) = tr (L n ) where the at trace of L n is dened by

(1.18), in particular the dynamical determinant d, dened by (1.17), is the Fredholm determinant of L, and the trace formula (TFM) holds for every n ∈ N * ;

(v) for all β > 2 + (s + 1) d we have

log + |d (z)| = |z|→+∞ O (log |z|) 1+β ,
in particular d has order zero;

(vi) if N (r) is the number of Ruelle resonances for (T, g) outside of the closed disc of center 0 and radius r (counted with multiplicity) we have for all β > 2 + (s + 1) d

N (r) = r→0 O |log r| 1+β .
Let us discuss a bit Theorem 2.27. First of all, the strategy to construct the space H

is similar to the one exposed in 2.2. However, following the general philosophy exposed in the introduction, the space H from Theorem 2.27 has to be anisotropic. This means that the elements of H are indeed very smooth in the stable direction (as in 2.2) but very irregular in the unstable direction. This is why we end up with a space that is intermediate CHAPTER 2

between Gevrey functions and ultradistributions (rather than between C ∞ function and distributions as in Theorem 3).

There are certain technical dierences between the proofs of Theorems 2.9 and 2.27.

There are indeed some additional geometric complications in the proof of Theorem 2.27 since we need to control the stable and unstable directions for the hyperbolic map T . This is slightly harder than in the C ∞ case, since we need to make the hyperbolicity of T eective after just one iteration of T in order to control the singular values of L. We introduced the notion of generalized cone-hyperbolicity to do so (see [Jéz20a, Denition 7.1]). A similar notion will be used in Chapter 3 to deal with ultradierentiable Anosov ows, see 3.3.

Another technical dierence between the proofs of Theorems 2.9 and 2.27 is the kind of PaleyLittlewood decompositions that we are using. In 2.2, we introduced a Paley Littlewood with blocks of exponential sizes adapted to the hyperbolicity of our map. On the contrary, in [START_REF] Jézéquel | Local and global trace formulae for smooth hyperbolic dieomorphisms[END_REF] we used a PaleyLittlewood decomposition with blocks of polynomial size (depending on the regularity of the map). The construction from 2.2 seems to give slightly better results (see 2.2.3), and it is likely that they can be adapted to the case of hyperbolic maps in order to improve Theorem 2.27 (for instance, one could try to allow smaller values of β in (v) and (vi)). However, one have to keep in mind that the excessively simple geometry of the circle is one of the elements that allowed us to design the spaces from 2.2 more carefully than those of [START_REF] Jézéquel | Local and global trace formulae for smooth hyperbolic dieomorphisms[END_REF].

Remark 2.28. This is one of the reasons why we think that Gevrey regularity is the good setting in order to establish nite order for the dynamical determinant (12) associated to an Anosov ow. Some heuristic computations based on the construction of Hilbert spaces of anisotropic distributions from [START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov ows[END_REF] also support this idea.

Chapter 3

Trace formula for ultradierentiable Anosov ows

This chapter is dedicated to the proof of Theorem 9. The contents of this chapter and of Appendices C and D may also be found in [START_REF] Jézéquel | Global trace formula for ultra-dierentiable Anosov ows[END_REF]. Appendix B is written in a slightly more general way than the corresponding [Jéz19a, Appendix A] in order to cover the dierent types of systems that are considered in this thesis.

We will start by discussing a bit further the classes C κ,υ from Example 2.2 in 3.1.

Using the notations from 3.1, we can get a precise version of Theorem 9.

Theorem 3.1. Let κ > 0 and υ ∈ ]1, 2[. Let M be a C κ,υ compact manifold, (φ t ) t∈R be a C κ,υ Anosov ow on M with generator X and V : M → C a C κ,υ function. Then for every t 0 > 0 there is a separable Hilbert space H such that (i) for every υ > υ suciently close to υ, we have C ∞,υ (M ) ⊆ H ⊆ D υ (M ) , both inclusions are continuous, and the rst one has dense image;

(ii) for every t ∈ R + , the operator L t dened by ( 9) is bounded on H;

(iii) (L t ) t≥0 denes a strongly continuous semi-group of operators on H, whose generator coincides with P = X + V on its domain, which is {u ∈ H : P u ∈ H};

(iv) the spectrum of P acting on H consists of isolated eigenvalues of nite multiplicity which coincide with the Ruelle resonances of P (multiplicity taken into account);

(v) if h : R * + → C is C ∞ and compactly supported in [t 0 , +∞[ then the operator +∞ 0 h (t) L t dt : H → H (3.1)
is trace class and its non-zero spectrum is the intersection of C \ {0} with the image of the spectrum of P by λ → Lap (h) (-λ) (multiplicity taken into account, Lap (h) CHAPTER 3

denotes the Laplace transform of h). Moreover, the trace of the operator (3.1) is given by

tr +∞ 0 h(t)L t dt = γ T # γ h (T γ ) |det (I -P γ )| exp γ V ,
where the sum on the right-hand side runs over periodic orbits γ of the ow (φ t ) t∈R , the notations are as in (12). This is indeed a more precise version of Theorem 9, since, with Lidskii's trace theorem [GGK00, Theorem 6.1 p.63], the last point of Theorem 3.1 implies the following Corollary.

Corollary 3.2 (Trace formula for ultradierentiable Anosov ows). Under the assumptions of Theorem 3.1, the trace formula (TFF) holds. In particular, the right-hand side of (TFF) denes a distribution.

As a by-product of the proof of Theorem 3.1, we get the following bound on the number of Ruelle resonances for ultradierentiable Anosov ow. Notice that it implies that the right-hand side of (TFF) converges indeed to a distribution on R * + .

Proposition 3.3. Under the assumptions of Theorem 3.1, for all > 0, we have λ∈σ R (P )

e Re(λ)

1 + |λ| dim M + < +∞,
where P = X + V .

The bound on the number of resonances given by Proposition 3.3 is not sucient to apply Proposition 1.27 and get a Hadamard factorization [Boa54, Theorem 2.7.1] for the dynamical determinant d dened by (12). However, we will derive in Appendix C a Hadamard-like factorization for d.

Finally, although we need υ < 2 to prove (TFF), most of the statements in Theorem 3.1 remain true when υ ≥ 2. We discuss in Appendix D the relevance and necessity of the condition υ < 2 through the simplest possible example: the doubling map on the circle.

Concerning this condition, we also refer to the discussion from Chapter 2 about Denjoy Carleman classes that are closed under dierentiation (see in particular Remarks 2.5 and 2.22 and the examples from 2.2.3).

Proposition 3.4. If, in Theorem 3.1, we allow υ ≥ 2, then there is still a Hilbert space H satisfying (i),(ii),(iii) and (iv). Moreover, under the hypothesis of (v), the operator (3.1) is compact and its spectrum can be described as in Theorem 3.1 in terms of Ruelle resonances.

This chapter is structured as follow. As already mentioned, 3.1 is dedicated to a discussion of the classes C κ,υ and associated generalized distributions. Then, in 3.2, we construct a local version of the space H from Theorem 3.1. In 3.3, we study the action of a local model for the Koopman operator (9) on these local spaces. In 3.4, we construct a rst version of the space H from Theorem 3.1. However, the action of the Koopman operator (L t ) t∈R from (9) on this rst space will not be controlled for small t's, a aw that will be xed in 3.5, proving Theorem 3.1 as well as Corollary 3.2 and Propositions 3.3 and 3.4.

The classes C κ,υ

In 2.1, Example 2.2, we dened the classes of regularity C κ,υ for κ > 0 and υ > 1. The Fourier transform will be a key tool in this paper, it is thus natural to introduce a suitable class of rapidly decreasing functions and associated spaces of tempered generalized distributions. This is often done in the literature, in particular when dealing with Gevrey classes (see for instance [START_REF] Pilipovi¢ | Tempered ultradistributions[END_REF][START_REF] Carmichael | On the convolution and the Laplace transformation in the space of Beurling-Gevrey tempered ultradistributions[END_REF]). Notice that we will use the following convention for the Fourier transform:

if f ∈ L 1 (R n ) and ξ ∈ R n we set F (f ) (ξ) = f (ξ) = R n e -ixξ f (x) dx.
For all κ > 0, υ >

1 and f ∈ C ∞ (R n ), dene f κ,υ = sup x∈R n α∈N n m∈N (1 + |x|) m |∂ α f (x)| exp - (m + |α|) υ κυ .
Then dene, for υ > 1,

S υ = f ∈ C ∞ (R n ) : ∀κ ∈ R * + , f κ,υ < +∞ , (3.2)
which is a Fréchet space when endowed with the family of semi-norms • κ,υ for κ > 0.

Notice that S υ is contained in the usual space of Schwartz functions and that the elements of S υ are in the DenjoyCarleman class C κ,υ for every κ > 0. One may also check that S υ is closed under dierentiation. We will denote by (S υ ) the space of continuous linear forms on S υ endowed with the weak-star topology. This space will play the role of tempered distributions in our context.

Proposition 3.5. If υ > 1, then the Fourier transform from S υ to itself is a continuous isomorphism.

Proof. We start by proving that the Fourier transform is continuous from S υ to itself. Let 0 < κ < κ. Let f ∈ S υ and recall that for all ξ ∈ R n and α, β ∈ N n we have1

ξ α ∂ β f (ξ) = (-i) |α|+|β| R n e -ixξ ∂ α x β f (x) dx = (-i) |α|+|β| γ 1 +γ 2 =α γ 2 β α!β! γ 1 !γ 2 ! (β -γ 2 )! R d e -ixξ x β-γ 2 ∂ γ 1 f (x)dx,
where γ 2 β means that each coordinate of γ 2 is smaller than the corresponding coordinate of β. Then, notice that there is a constant C > 0 such that, for every γ 1 , γ 2 , β ∈ N d such that γ 2 β, we have

R n e i xξ x β-γ 2 ∂ γ 1 f (x)dx ≤ C f κ,υ exp (|β| -|γ 2 | + |γ 1 | + n + 1) υ κυ .
Moreover, up to making C larger we also have, for every γ 2 ∈ N n ,

γ 2 ! ≤ C exp |γ 2 | υ κυ .
Consequently, we nd that for every ξ ∈ R n and α, β ∈ N n , the quantity

ξ α ∂ β f (ξ) is smaller than C 2 f κ,υ γ 1 +γ 2 =α γ 2 β α! γ 1 !γ 2 ! β! γ 2 ! (β -γ 2 )! exp |γ 2 | υ + (|β| -|γ 2 | + |γ 1 | + n + 1) υ κυ ≤ C 2 f κ,υ 2 |α|+|β| exp (|α| + |β| + n + 1) υ κυ .
Using the fact that for

∈ N |ξ| 2 =   n j=1 |ξ j | 2   = |α|= c (α) ξ 2α ,
where |α|= c (α) = n , we see that, for some new constant C > 0, we have for all m ∈ N, ξ ∈ R n and α, β ∈ N n :

(1 + |ξ|) m ∂ β f (ξ) ≤ C f κ,υ 4 √ n m 2 |β| exp (m + |β| + n + 2) υ κυ .
(3.3) Indeed, we can deal rst with m even and then argue that

(1 + |ξ|) m ≤ (1 + |ξ|) m+1 .
Finally, since κ < κ and (r+n+2) υ κ -r υ κ ∼ r→+∞ -κ-κ κκ r υ , we see that, for some new constant

C > 0, we have f κ ,υ ≤ C f κ,υ ,
and the Fourier transform is indeed continuous from S υ to itself. The same argument gives that the inverse Fourier transform is also continuous from S υ to itself. Moreover, since S υ is included in the space of Schwartz function on R n , the elements of S υ satisfy the Fourier Inversion Formula. Hence, the Fourier transform is indeed a continuous automorphism of S υ .

Proposition 3.5 allows to dene the Fourier transform on (S υ ) by duality in the usual way. Since S υ is closed by multiplication, for ψ ∈ S υ we can dene the Fourier multiplier ψ (D) :

(S υ ) → (S υ ) by ∀u ∈ (S υ ) : ψ (D) u = F -1 (ψ.û) .
It is well-known that the Fourier transform of a C ∞ compactly supported function decays faster than the inverse of any polynomial. For functions in the class C κ,υ this statement is made quantitative in Proposition 3.6 below. This is the key point that will allow us in 3.2 to construct Sobolev-like spaces of anisotropic generalized distributions that are the pieces from which we will construct the space H from Theorem 3.1 in 3.4 and 3.5.

Proposition 3.6. For every R > 0 and υ > 1, there are constants C > 0 and κ > 0 such that, for all f ∈ S υ and ξ ∈ R n , we have

f (ξ) ≤ C f κ,υ exp -R(ln (1 + |ξ|)) υ υ-1 . (3.4) Proof. Choose κ > 0 large enough so that R := κ 1 υ-1 υ -1 υ > R.
Then apply (3.3) from the proof of Proposition 3.5 with β = 0 to get a constant C > 0 such that, for all ξ ∈ R n and m ∈ N, we have

f (ξ) ≤ C f κ,υ 4 √ n 1 + |ξ| m exp (m + n + 2) υ κυ .
When |ξ| is small, we bound f (ξ) by taking m = 0. When |ξ| is large enough so that the following expression makes sense and is non-negative, we take

m = -ln 4 √ n 1 + |ξ| 1 υ-1 κ 1 υ-1 -n -2 .

CHAPTER 3

With this choice of m we have

4 √ n 1 + |ξ| m exp (m + n + 2) υ κυ ≤ exp ln 4 √ n 1 + |ξ| 1 υ-1 κ 1 υ-1 -n -3 ln 4 √ n 1 + |ξ| + κ 1 υ-1 υ ln 4 √ n 1 + |ξ| υ υ-1 ≤ 1 + |ξ| 4 √ n n+3 exp κ 1 υ-1 1 -υ υ ln 1 + |ξ| 4 √ n υ υ-1 ≤ 1 + |ξ| 4 √ n n+3 exp -R ln 1 + |ξ| 4 √ n υ υ-1
, and the result follows then from the fact that (recall that R > R)

1 + r 4 √ n n+3 exp R (ln (1 + r)) υ υ-1 -R ln 1 + r 4 √ n υ υ-1 → r→+∞ 0.
We need to extend the notion of ultradierentiability to more general objects than complex-valued functions in order to make sense of Theorem 3.1. For instance, we will dene what a C κ,υ manifold is. To do it, we follow ideas that may be found in [START_REF] Kriegl | The convenient setting for non-quasianalytic Denjoy-Carleman dierentiable mappings[END_REF], notice however that when υ > 2 the sequence (A m ) m∈N from Example 2.2 is not a DCweight sequence in the sense of [START_REF] Kriegl | The convenient setting for non-quasianalytic Denjoy-Carleman dierentiable mappings[END_REF], so that we cannot apply most of their results.

Hopefully, it will be clear in the remaining of the section that, whereas the general theory of our ultradierentiability classes may not be very satisfactory, this is of no harm in our pedestrian approach to the problem of the trace formula.

We already dened in 2.1 what it means for a map from an open subset of R n to a Banach space to be C κ,υ . We recall that this class of regularity is closed under composition and inversion. A C κ,υ manifold is then dened to be a Hausdor topological space with countable basis endowed with a maximal C κ,υ atlas i.e. a maximal atlas whose change of charts are C κ,υ . As usual, a map f : M → N between two C κ,υ manifolds is said to be C κ,υ if it is C κ,υ in charts.

We dene now topological vector spaces associated to these classes of regularity. If M is a C κ,υ manifold for some κ > 0 and υ > 1 then M has a natural C κ ,υ manifold structure for all κ > 0 and υ > υ, so that we may dene the class C ∞,υ (M ) of functions from M to C that are C κ ,υ for all κ > 0. Notice that all C κ,υ functions from M to C belong to C ∞,υ (M ) if υ > υ.

Notice that if υ > 2 then the class C κ,υ is not closed under dierentiation and in particular in this case the tangent bundle T M has no natural C κ,υ structure. However, derivatives of C κ,υ functions are C κ ,υ for all 0 < κ < κ. Thus the tangent bundle T M may be endowed naturally with a C κ ,υ structure, so that it makes sense to talk about a C κ ,υ vector eld when υ > υ, or υ = υ and κ < κ. Integrating such a vector eld gives rise to a C κ ,υ ow 2 (φ t ) t∈R , see [START_REF] Kriegl | The convenient setting for non-quasianalytic Denjoy-Carleman dierentiable mappings[END_REF][START_REF] Komatsu | Ultradierentiability of solutions of ordinary dierential equations[END_REF]. A consequence of this fact is that if X is a C κ ,υ vector eld on M that does not vanish then X is locally conjugated via C κ ,υ charts to a constant vector eld on R n where n = dim M . This implies in particular that if υ > υ then C ∞,υ is closed under dierentiation with respect to X (this operation is even continuous with respect to the topology that we dene below).

If M is compact, we endow C ∞,υ (M ) with a structure of Fréchet space in the following

way: if U is an open subset of M and V is an open subset of R n , if ψ : U → V is a C ∞,υ chart, ϕ is an element of C ∞,υ supported in U and κ > 0, dene the semi-norm • ψ,ϕ,κ,υ by ∀u ∈ C ∞,υ : u ψ,ϕ,κ,υ = sup α∈N n x∈V ∂ α (ϕu) • ψ -1 (x) exp - |α| υ κυ .
The topology of C ∞,υ (M ) is generated by a countable family of these semi-norms: since M is compact we can cover M by a nite number of domain of charts and take a partition of unity subordinated to this cover, then we only need to let κ runs through the integers.

The completeness of C ∞,υ (M ) is easily veried. One can also check using Leibniz formula

that pointwise multiplication C ∞,υ (M ) × C ∞,υ (M ) → C ∞,υ (M ) is continuous. Notice also that if N is another C κ,υ manifold and ψ : M → N is a C κ,υ local dieomorphism then the map C ∞,υ (N ) u → u • ψ ∈ C ∞,υ (M ) is continuous.
We will also need the space D υ (M ) of C ∞,υ densities on M : this is the space of complex measures of M which are absolutely continuous with respect to Lebesgue and whose density in any C ∞,υ chart is C ∞,υ . We endow D υ (M ) with a Fréchet structure as we did for 3 C ∞,υ (M ). We will denote by D υ (M ) the space of continuous linear functionals on M on D υ (M ), that we endow with the weak-star topology. Notice that if u ∈ C ∞,υ (M ) then u denes an element of D υ (M ) that we also denote by u, by the formula

∀µ ∈ D υ (M ) : u, µ = M udµ.
We dene in this way an injection of C ∞,υ (M ) into D υ (M ) that can be shown to be continuous and to have dense image (by mollifying elements of D υ (M ) , by convolution for instance).

Now, let M be a C κ,υ compact manifold for some κ > 0 and υ > 1 and let n = d + 1

denotes the dimension of M . Let (φ t ) t∈R be a C κ,υ ow on M (that is, the map M × R (x, t) → φ t (x) is C κ,υ ). Then the generator X of the ow (φ t ) t∈R is a C κ ,υ vector eld for 2 That is, the map (x, t) → φt (x) is C κ ,υ . 3 Notice that these two spaces may be identied by the choice of a particular element of D υ (M ).

CHAPTER 3 all κ < κ. Choose V : M → C a C κ,υ potential. Let υ > υ and dene for all t ∈ R the

continuous operator L t on C ∞,υ (M ) by ∀u ∈ C ∞,υ (M ) : ∀x ∈ M : L t u (x) = exp t 0 V • φ τ (x) dτ u • φ t (x) .
Here, let us notice that the prefactor in the denition of L t is a C κ,υ function (since this class of regularity is closed under composition). It is convenient4 to extend L t and P = X + V from D υ (M ) to itself. To do so, we need to compute their adjoints. Choose

d ((φ t+t ) * µ) dµ (x) = d ((φ t ) * µ) dµ (x) d ((φ t ) * µ) dµ (φ -t (x)) , so that we have ∀x ∈ M : ∀t ∈ R : d ((φ t ) * µ) dµ (x) = exp - t 0 div (X) • φ -τ (x) dτ ,
where the divergence of X is dened by

∀x ∈ M : div (X) (x) = - d dt d ((φ t ) * µ) dµ (x) t=0 .
Notice that div (X) is a C κ ,υ function for all κ < κ. Then the formal adjoint of L t may be dened on D υ (M ) by

(L t ) * ν = exp t 0 (V -div (X)) • φ -τ dτ dν dµ • φ -τ dµ
and the formal adjoint of P by

P * ν = (-X -div (X) + V ) dν dµ dµ.
These two operators are continuous on D υ (M ), so that P and L t may be extended as continuous operators on D υ (M ) . Notice that P and L t commute.

We will need Lemmas 3.7 and 3.8 to prove Theorem 3.1.

Lemma 3.7. With the notations above, we have:

(i) If u ∈ C ∞,υ (M ) then the map R t → L t u ∈ C ∞,υ (M ) is C ∞ and its derivative is t → L t P u = P L t u. (ii) If u ∈ D υ (M ) then the map R t → L t u ∈ D υ (M ) is C ∞ and its derivative is t → L t P u = P L t u.
Proof. We only need to prove the rst point: the same argument with C ∞,υ (M ) replaced by D υ (M ), and L t and P replaced by their formal adjoints gives the second point.

We start with the case V = 0. Using the group property of (L t ) t∈R , we only need to prove dierentiability at t = 0. Then we may cover M by ow boxes, and thus we only

need to show that if u ∈ S υ is supported in a compact subset K of R d+1 then u (• + te d+1 ) -u t → t→0 ∂ x d+1 u in S υ, (3.5)
where e d+1 denotes the last vector of the canonical basis of R d+1 . Up to enlarging K we may assume that for all t ∈ [-1, 1] the function u (• + te d+1 ) is supported in K. Then if x ∈ K, α ∈ N d+1 , and t ∈ [-1, 1] we have with Taylor's formula (for any κ > 0):

∂ α u (x + te d+1 ) -∂ α u (x) t -∂ α ∂ x d+1 u (x) = ∂ α u (x + te d+1 ) -∂ α u (x) t -∂ x d+1 ∂ α u (x) ≤ ∂ 2 x d+1 ∂ α u ∞ 2 |t| ≤ |t| 2 u κ ,υ exp (|α| + 2) υ κ υ .
Thus if κ , κ > 0 and for R > 0 depending only on K, we have for all x ∈ R d+1 , α ∈ N d+1 and m ∈ N:

(1 + |x|) m ∂ α u (x + te d+1 ) -∂ α u (x) t -∂ α ∂ x d+1 u (x) exp - (m + |α|) υ κ υ ≤ |t| 2 u κ ,υ R m exp (|α| + 2) υ κ υ - (m + |α|) υ κ υ .
Thus if κ > 0 and κ > κ , then there is a constant C > 0 (that only depends on K, υ, κ , and κ ) such that for all t ∈ [-1, 1] we have

u (• + te d+1 ) -u t -∂ x d+1 u κ ,υ ≤ C |t| u κ ,υ ,
which implies (3.5) and thus ends the proof of the lemma in the case V = 0.

In order to deduce the result in the case of a general V from the case V = 0, we only need to prove that the map

t → exp t 0 V • φ τ dτ (3.6)
exponential size, but the ideas behind are the same (see the discussion in 2.3).

First of all, we need to dene the systems of cones that we will use. As in [START_REF] Jézéquel | Local and global trace formulae for smooth hyperbolic dieomorphisms[END_REF], we need to consider system of potentially a large number of cones, in order to deal with the low hyperbolicity of the ow for small times. The interior and the adherence of a subset Y of a topological space will be denoted respectively by Denition 3.9 (System of cones). Let (E, ., . ) be an Euclidean vector space, e ∈ E and r ≥ 2 be an integer. A system of r + 2 cones with respect to the direction e is a family

Θ = (C 0 , C 1 , . . . , C r , C f ) of non-empty closed cones in E such that (i) • C 0 ∪ • C 1 ∪ • C f = E \ {0};
(ii) C f is one-dimensional and there is c > 0 such that for all ξ ∈ C f we have | ξ, e | ≥ c |ξ|;

(iii) there are integers d u and d s such that

d u + d s + 1 = dim E, C 0 is d s -dimensional and, for i ∈ {1, . . . , r}, the cone C i is d u -dimensional; (iv) if i ∈ {1, . . . , r -1} then C i+1 C i ; (v) C 0 ∩ C 2 = C f ∩ C 2 = {0}.
Let us x d ∈ N. The vector space R d+1 will always be endowed with its canonical Euclidean structure and system of cones in R d+1 will always be with respect to the direction of e d+1 = (0, . . . , 0, 1). We will mainly use Denition 3.9 with E = R d+1 , however, it will be convenient in the proof of Lemma 3.29 to have at our disposal the denition of a system of cones in a general Euclidean space.

If (C 0 , . . . , C r , C f ) is a system of r + 2 cones in R d+1 (with respect to the direction e d+1 ) then we can choose (ϕ 0 , ϕ 1 , . . . , ϕ r-1 , ϕ f ) a Gevrey5 partition of unity on S d such that:

• for i ∈ {0, . . . , r -1, f }, the function ϕ i is supported in the interior of C i ∩ S d ; • if i ∈ {1, . . . , r -2} then ϕ i vanishes on a neighborhood of S d ∩ C i+2 . Indeed, the interiors of C 0 ∩ S d , (C f \ C 2 ) ∩ S d , (C 1 \ C 3 ) ∩ S d , . . . , (C r-2 \ C r ) ∩ S
such that χ (x) = 1 if x ≤ 1 2 and χ (x) = 0 if x ≥ 1. Dene for all n ≥ 1 and ξ ∈ R d+1 , CHAPTER 3 χ n (ξ) = χ (2 -n |ξ|) and χ α,n (ξ) = χ |ξ| -2 n α , set also χ n = χ α,n = 0 if n ≤ 0. Then set for n ∈ N, ψ n (ξ) = χ n+1 (ξ) -χ n (ξ) and ψ α,n (ξ) = χ α,n+1 (ξ) -χ α,n (ξ). Thus we have for n ≥ 1 supp ψ n ⊆ ξ ∈ R d+1 : 2 n-1 ≤ |ξ| ≤ 2 n+1 and supp ψ α,n ⊆ ξ ∈ R d+1 : 2 n α ≤ |ξ| ≤ 2 (n+1) α + 1 .
In addition, supp ψ 0 and supp ψ α,0 are contained in ξ ∈ R d+1 : |ξ| ≤ 5 . Moreover, we have

n≥0 ψ n = n≥0 ψ α,n = 1. Set Γ = N × {0, . . . , r -1, f } . Dene for (n, i) ∈ Γ the function ψ Θ,n,i by ψ Θ,n,i (ξ) =    ψ n (ξ) ϕ i ξ |ξ| if n ≥ 1, ψ 0 (ξ) r-1 if n = 0, if i ∈ {1, . . . , r -2, f }, and by ψ Θ,n,i (ξ) = (1 -ψ 0 (ξ)) ψ α,n (ξ) ϕ i ξ |ξ| if i ∈ {0, r -1}, so that we have (n,i)∈Γ ψ Θ,n,i = 1.
We will give a Sobolev-like denition of the local space H Θ,α (Denition 3.10) by mean of a weight w Θ,α (see (3.9)). If this description is convenient to prove the basic properties of H Θ,α (see Proposition 3.11), we will rather use in the following sections a PaleyLittlewoodlike description of the space H Θ,α (see Proposition 3.13), for any υ ∈ 1, 1 1-α we have:

H Θ,α =    u ∈ (S υ ) :
(n,i)∈Γ

2 nβ i ψ Θ,n,i (D) u 2 2 < +∞    where β 0 = d + 2, β r-1 = -(d + 2) , β f = -(d + 2) (3.7)
and

β i = -(i + 1) (d + 2) for i ∈ {1, . . . , r -2} .
(3.8)

The main idea behind the choice of the β i is that the expected regularity of elements of H Θ,α (measured via integrability of the Fourier transform) must decrease under the action of the linear model of the dynamics (the β i play the role here of an analogue of the escape function from [START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov ows[END_REF]). The particular choice has been made so that computations are as easy as possible. Our parameters have been designed in order to make the Paley Littlewood description as simple as possible, at the cost of a denition of the weight w Θ,α that may seem a bit heavy. It is dened for ξ ∈ R d+1 by

w Θ,α (ξ) = ψ 0 (ξ) + (1 -ψ 0 (ξ))   i∈{0,r-1} ϕ i ξ |ξ| exp β i ln (1 + |ξ|) 1 α (ln 2) 1 α -1 + i∈{1,...,r-2,f } ϕ i ξ |ξ| ξ β i   , (3.9)
where

ξ = 1 + |ξ| 2 for ξ ∈ R d+1 .
Denition 3.10. Dene the space (for any υ ∈ 1,

1 1-α ) H Θ,α = u ∈ (S υ ) : û ∈ L 2 loc and R d+1 |û (ξ)| 2 w Θ,α (ξ) 2 dξ < +∞ endowed with the scalar product u, v Θ,α = R d û (ξ)v (ξ) w Θ,α (ξ) 2 dξ.
Recall (3.2) for the denition of S υ and (3.9) for the denition of w Θ,α .

Proposition 3.11. H Θ,α is a separable Hilbert space that does not depend on the choice of υ. For all 1 < υ < 1 1-α , the space S υ is continuously contained and dense in H Θ,α , and H Θ,α is continuously contained in (S υ ) .

Proof. The map

A :

H Θ,α → L 2 R d+1 u → ûw Θ,α
is clearly an isometry. Choose υ < 1 1-α , thanks to Propositions 3.5 and 3.6 (recall (3.4)), and since

1 α < υ υ-1 , the map u → u.w -1 Θ,α is continuous from S υ to L 2 R d+1 . Thus the map B : u → F -1 uw -1 Θ,α is continuous from L 2 R d+1 to (S υ ) . But if u ∈ L 2 R d+1 then it is clear that Bu ∈ H Θ,
α with Bu Θ,α = u 2 . Now, since A and B are inverses of each other, H Θ,α is isometric to L 2 R d+1 and thus a separable Hilbert space. Proposition 3.6 implies that S υ is continuously contained in H Θ,α and that the inclusion of H Θ,α in (S υ ) is continuous. Let u ∈ H Θ,α be in the orthogonal space to S υ . If ρ is a CHAPTER 3 compactly supported element of S υ , then, for all v ∈ S υ , we have

R d+1 ρ (ξ) û (ξ)w Θ,α (ξ) 2 v (ξ) dξ = u, F -1 (ρ.v) Θ,α = 0.
Thus the function ρ ūw 2 Θ,α ∈ L 1 R d+1 vanishes (take for v a convolution kernel), and so does u. Consequently, S υ is dense in H Θ,α .

To see that H Θ,α does not depend on the choice of υ, just notice that, if we use υ ∈ υ, 1 1-α instead of υ in the denition of H Θ,α , then we obtain another Hilbert space H Θ,α . But then H Θ,α ⊆ H Θ,α , and the inclusion is isometric and has a dense image (because H Θ,α contains S υ ). Since H Θ,α and H Θ,α are both Hilbert spaces, they must coincide.

Remark 3.12. It is clear from the proof that in fact the elements of S υ whose Fourier transform is compactly supported form a dense subset of H Θ,α . Proposition 3.13. Let 1 < υ < 1 1-α and u ∈ (S υ ) . Then u ∈ H Θ,α if and only if

(n,i)∈Γ 2 nβ i ψ Θ,n,i (D) u 2 2 < +∞.
(3.10) Moreover, the square root of this quantity denes an equivalent (Hilbertian) norm on H Θ,α .

Proof. First, notice that there is C > 0 such that, if n ∈ N, i ∈ {1, . . . , r -2, f } and

ξ ∈ supp ψ Θ,n,i , then 1 C 2 nβ i ≤ ξ β i ≤ C2 nβ i . Up to enlarging C, it is also true that if n ∈ N, i ∈ {0, r -1} and ξ ∈ supp ψ Θ,n,i then 1 C 2 nβ i ≤ exp β i ln (1 + |ξ|) 1 α (ln 2) 1 α -1 ≤ C2 nβ i .
Now, using the fact that the intersection number of the support of the ψ Θ,n,i for (n, i) ∈ Γ is nite, we nd another constant C > 0 such that for all ξ ∈ R d+1 we have

1 C w Θ,α (ξ) 2 ≤ (n,i)∈Γ 2 nβ i ψ Θ,n,i (ξ) 2 ≤ Cw Θ,α (ξ) 2 .
(3.11)

From this, we get immediately that if u ∈ H Θ,α then (3.10) holds. Reciprocally, if (3.10) holds, then û is in L 2 loc (the sum (n,i)∈Γ ψ Θ,n,i is locally nite) and from (3.11) we get that u ∈ H Θ,α . The equivalence of norms is an immediate consequence of (3.11). Proposition 3.13 suggests to dene the auxiliary Hilbert space

B =    (u n,i ) (n,i)∈Γ ∈ (n,i)∈Γ L 2 R d+1 : (n,i)∈Γ 2 nβ i u n,i 2 2 < +∞    .
(3.12) Dene the map

Q Θ : H Θ,α → B u → (ψ Θ,n,i (D) u) (n,i)∈Γ .
(3.13) For (n, i) ∈ Γ dene also the natural projection and inclusion

π n,i : B → L 2 R d+1 (u ,j ) ( ,j)∈Γ → u n,i and ι n,i : L 2 R d+1 → B u → uδ (n,i)=( ,j) ( ,j)∈Γ .

Local Koopman operator

We are now going to study a local model for the Koopman operator (9) associated to an Anosov ow (φ t ) t∈R on a (d + 1)-dimensional manifold M . The main result of this section is Proposition 3.17 which is a local version of Theorem 3.1.

As a local model for a ow, we will consider a family (T t ) t∈R of dieomorphisms of R d+1

such that if we dene F : R d → R d+1 by x → T 0 (x, 0) (here we make the identication

R d+1 R d × R) then we have ∀t ∈ R : ∀ (x, y) ∈ R d × R R d+1 : T t (x, y) = F (x) + ye d+1 + te d+1 .
(3.14)

We will say that F is the map associated to the family of dieomorphisms (T t ) t∈R . Reciprocally, if F : R d → R d+1 is an immersion, we dene by (3.14) the associated family of dieomorphisms (T t ) t∈R (provided they actually are dieomorphisms).

Remark 3.14. Let us explain why we use such a family of dieomorphisms as a local model for a ow. We want to study the ow (φ t ) t∈R in the neighbourhood of a xed time t0 . To do it, we take charts κ and κ for M and we study the family of dieomorphisms (T t ) t∈R dened by the formula

T t = κ • φ t0 +t • κ -1 .
Of course, this is not in general a family of dieomorphisms from R d+1 to itself (a priori the domain of T t depends on t). However, it is more convenient to deal with dieomorphisms of the whole R d+1 , and we will consequently provide extensions of the T t to R d+1 when applying Proposition 3.17 in 3.4 (see Lemma 3.29). These extensions are far from canonical, but the use of a cuto function will ensure that none of the objects that we consider in 3.4 depend on the choices we will make in a relevant way.

It is natural to ask for κ and κ to be ow boxes, that is, if X is the generator of the ow (φ t ) t∈R , we require κ * (e d+1 ) = X and κ * (e d+1 ) = X (we identify e d+1 with the constant vector eld with value e d+1 ). This requirement implies (3.14) for small t and y, and, since we are only interested here in the behaviour of (φ t ) t∈R locally in both space and time, we may modify the denition of T t for large t and design our extension to ensure that (3.14) holds (we refer to the proof of Lemma 3.29 for details). Once again, this will be of no harm in the global analysis thanks to the use of cuto functions in both time and space.

In this section, we will study such a family with no reference to a particular Anosov ow. We will need further assumptions to do so. The rst one is that F (or equivalently T 0 or any T t for t ∈ R) is C κ,υ for some κ > 0 and υ > 1. The second one is a condition of hyperbolicity that we will express using cones.

Let r ≥ 2 be an integer and choose two systems of r + 2 cones (with respect to the direction e d+1 as usual) Θ = (C 0 , . . . , C r , C f ) and Θ = C 0 , . . . , C r , C f . We assume that (T t ) t∈R is cone-hyperbolic from Θ to Θ in the following sense:

(i) for all x ∈ R d+1 , i ∈ {1, . . . , r} and t ∈ R we have 6 D x T tr t (C i ) ⊆ C min(i+2,r) ;

(ii) for all x ∈ R d+1 and t ∈ R we have

D x T tr t (C f ) ∩ C 0 = {0} ;
(iii) there is Λ > 1 such that for all x ∈ R d+1 , all ξ ∈ C r-1 , and all t ∈ R we have

D x T tr t (ξ) ≥ Λ |ξ| ;
(iv) for the same Λ > 1, for all x ∈ R d+1 , all ξ ∈ R d , and all t ∈ R such that D x T tr t (ξ) ∈ C 0 we have 7

D x T tr t (ξ) ≤ Λ -1 |ξ| .
Remark 3.15. Notice that the denition of the cone-hyperbolicity of the family (T t ) t∈R only involves the derivatives D x T tr t . However, these derivatives do not depend on t (this is a consequence of (3.14)). Consequently, one only needs to check that (i)-(iv) hold for 6 Here, A tr denotes the transpose of A.

7 Notice that the condition DxT tr t (ξ) ∈ C 0 implies in particular that ξ ∈ C0, as a consequence of (i) and (ii). t = 0. This fact may be surprising since hyperbolicity is usually a phenomenon that can only be observed after a small amount of time, but recall Remark 3.14: in the application, the family (T t ) t∈R will only be used to describe the ow (φ t ) t∈R near some time t0 . Then, provided that t0 > 0, the family (T t ) t∈R will be cone-hyperbolic (see 3.4 for the details).

Remark 3.16.

is cone-hyperbolic from Θ to Θ then T 1 t • T 2 0 t∈R is cone-hyperbolic from Θ to Θ .
We will also consider a C ∞ family (G t ) t∈R of S υ functions from R d+1 to C, such that there is a compact subset K of R d+1 such that, if x ∈ R d+1 \ K and t ∈ R, then G t (x) = 0.

In this section, we study the family (L t ) t∈R of local Koopman operators dened by

L t u = G t (u • T t ) .
(3.15)

This denition makes sense for u ∈ S υ (for any υ > υ) and may be extended by duality to u ∈ S υ . The main result of this section is Proposition 3.17, which can be seen as a local version of Theorem 3.1.

Proposition 3.17. Let α ∈ υ-1 υ , 1 . For every t ∈ R, the operator L t dened by (3.15) is bounded from H Θ,α to H Θ ,α . Moreover, the family (L t ) t∈R is strongly continuous (as a family of operators from H Θ,α to H Θ ,α ), hence it is measurable.

Moreover, if α < 1 2 , if k is a non-negative integer and if h : R → C is a compactly supported kth time dierentiable function whose kth derivative has bounded variation then the operator

R h (t) L t dt : H Θ,α → H Θ ,α (3.16)
is in the Schatten class S q for all q ≥ 1 such that q > d+1 k+1 . Moreover, there is a constant C > 0, which depends on h only through its support, such that

R h (t) L t dt Sq ≤ C h C k-1 + h (k)

BV

, where • Sq denotes the S q Schatten class norm and • BV the bounded variation norm.

If

k + 1 > d + 1 and Θ = Θ we have tr R h (t) L t dt = p•F (x)=x h (T (x)) |det (I -p • D x F )| R G T (x) (x, y) dy,
where p is the orthogonal projection from R d+1 to R d R d × {0} and, for x ∈ R d , the number T (x) is dened by F (x) = p (F (x)) + (0, -T (x)).

CHAPTER 3

Without the hypothesis α < 1 2 , it remains true that the operator (3.16) is compact.

Remark 3.18. Since α > υ-1 υ , we may choose υ > υ such that υ < 1 1-α . Then H Θ,α ⊆ S υ and thus L t u is well-dened as an element of S υ when t ∈ R and u ∈ H Θ,α . Remark 3.19. Notice that the spaces H Θ,α and H Θ ,α depend a priori not only on Θ and Θ (and α) but also on the choice of partitions of unity (ϕ 0 , . . . , ϕ r-1 , ϕ f ) and ϕ 0 , . . . , ϕ r-1 , ϕ f on S d as in 3.2. However, in view of Proposition 3.17, this choice is mostly irrelevant and the dependence on Θ and Θ is the fundamental point.

The remainder of this section is devoted to the proof of Proposition 3.17. For this, we introduce in Lemma 3.20 a family of auxiliary operators (M t ) t∈R acting on the space B dened in (3.12). Then, we prove that the family (M t ) t∈R has the properties that we expect from (L t ) t∈R : boundedness and strong continuity is proven in Lemma 3.20 (with the help of the preparatory Lemmas 3.21 and 3.22, see 3.3.1), that an operator similar to (3.16) is in a Schatten class is proven in Lemma 3.27 (with the help of Lemmas 3.23, 3.24 and 3.25, see 3.3.2) and the formula for the trace is given in Lemma 3.28 (see 3.3.3). Finally, we end the proof of Proposition 3.17 by showing that (L t ) t∈R inherits these properties from (M t ) t∈R .

The auxiliary operators M t .

We will need smooth functions φ0 , . . . , φr-1 , and φf :

S d → [0, 1] such that • if i ∈ {0, . . . , r -1, f } then φi is supported in the interior of C i ∩ S d ; • if i ∈ {1, . . . , r -2} then φi vanishes on a neighborhood of C i+2 ∩ S d ; • if i ∈ {0, . . . , r -1, f }, x ∈ S d ,
and ϕ i (x) = 0 then φi (x) = 1.

Dene then ψn = χ n+2 -χ n-1 and ψα,n = χ α,n+b -χ α,n-b for n ≥ 0, where b is chosen large enough so that for all n ∈ N * we have

2 (n+1) α -2 (n+b) α + 1 ≤ 1 2 and 2 n α -2 (n-b) α ≥ 1. If (n, i) ∈ Γ set ψΘ,n,i (ξ) =    ψn (ξ) φi ξ |ξ| if n ≥ 1, ψ0 (ξ) if n = 0, if i ∈ {1, . . . , r -2, f }, and ψΘ,n,i (ξ) =    ψα,n (ξ) φi ξ |ξ| if n ≥ 1, ψα,0 (ξ) if n = 0, if i ∈ {0, r -1}. Thus ψ Θ,n,i (ξ) = 0 implies ψΘ,n,i (ξ) = 1. Now if (n, i) , ( ,
j) ∈ Γ, and t ∈ R dene an operator S ,j t,n,i :

L 2 R d+1 → L 2 R d+1 by S ,j t,n,i = ψ Θ ,n,i (D) • L t • ψΘ, ,j (D) .
(3.17)

As announced above, we dene in Lemma 3.20 a family of auxiliary operators whose study will take most of the remainder of this section.

Lemma 3.20. For every t ∈ R, the sum

(n,i),( ,j)∈Γ ι n,i • S ,j t,n,i • π ,j (3.18)
converges in the strong operator topology to an operator M t : B → B that depends continuously on t in the strong operator topology.

The proof of Lemma 3.20 is based on Lemmas 3.21 and 3.22 below. In order to prove Lemma 3.20, we rst dene a relation → on Γ that indexes the transitions (in the frequency space) that would occur for a linear dynamics, in the spirit of [START_REF] Adam | Opérateurs de Transfert et Moyennes Horocycliques Sur Les Variétés Fermées[END_REF][START_REF] Adam | Horocycle averages on closed manifolds and transfer operators[END_REF]. Our local space has been designed so that it corresponds either to a transition from high regularity to low regularity (which makes this part of the action smoothing) or to a stationary frequency in the direction of the ow (we will integrate in this direction, so that it also corresponds to a smoothing operator). The other transitions do not happen in the linear case, and we will control this non-linearity using not only the hyperbolicity of the dynamics but also its high regularity. Choose a > 0 such that for all x ∈ K and t ∈ R we have

a < D x T tr t -1 -1
.

Choose also ν such that 0 < ν < log 2 Λ α . We dene now the relation →. For ( , j) , (n, i) ∈ Γ, we say that ( , j) → (n, i) holds if either of the following conditions is satised:

• i = j = 0 and ≥ n + νn 1-α ; • i = j = r -1 and n ≥ + ν 1-α ; • j = 0 and i ∈ {1, . . . , r -1, f }; • j ∈ {1, . . . , r -2, f } , i = r -1 and ≤ n α + 4 -log 2 a; • j = f, i ∈ {1, . . . , r -2} and n ≥ -4 + log 2 a;
• i, j ∈ {1, . . . , r -2} with i ≥ j + 1 and n ≥ -4 + log 2 a;

• i = j = f and | -n| ≤ 10 -log 2 c, where c is such that for all ξ = (ξ 1 , . . . , ξ d+1 ) ∈ C f ∪ C f we have |ξ d+1 | ≥ c |ξ| (such a constant exists by our denition of a system of cones).

D x T tr

t (η) ∈ C 0 , and by cone-hyperbolicity we have

|ξ| -D x T tr t (η) ≥ 2 n α -Λ -1 2 ( +b) α + 1 ≥ 2 n α -Λ -1 2 (n+νn 1-α +b) α + 1 ≥ 2 n α 1 -2 (n+νn 1-α +b) α -log 2 Λ-n α -2 -n α .
We can then conclude if N is large enough, since

n + νn 1-α + b α -log 2 Λ -n α → n→+∞ αν -log 2 Λ < 0 and 2 (n+νn 1-α ) α ≤ C2 n α , (3.20)
for some constant C > 0 that does not depend on n. We used here the asymptotic

expansion n + νn 1-α α = n→+∞ n α + αν + o (1)
.

• If i = j = r -1 and n < + ν 1-α then D x T tr t (η) -|ξ| ≥ Λ2 ( -b) α -2 (n+1) α + 1 ≥ Λ2 ( -b) α -2 ( +ν 1-α +1) α + 1 ≥ 2 α Λ2 ( -b) α -α -2 ( +ν 1-α +1) α -α -2 -α .
We can conclude if N is large enough, since

Λ2 ( -b) α -α -2 ( +ν 1-α +1) α -α -2 -α → →+∞ Λ -2 αν > 0,
and (3.20) still holds when n is replaced by .

• If j ∈ {1, . . . , r -2, f } , i = r -1 and > n α + 4 -log 2 a, then D x T tr t (η) -|ξ| ≥ a2 -2 -2 (n+1) α + 1 ≥ a2 -2 -2 n α +1 -1 ≥ a2 -3 -1. • If i ∈ {1, . . . , r -2} , j = f and n < -4 + log 2 a then D x T tr t (η) -|ξ| ≥ a2 -2 -2 n+1 ≥ a2 -3 .
• The case i, j ∈ {1, . . . , r -2} , i ≥ j + 1 and n < -4 + log 2 a is dealt as the previous CHAPTER 3 one.

• If i = j = f and | -n| > 10 -log 2 c, then just notice that the d + 1th coordinate of • The three last cases are dealt with by cone hyperbolicity using (3.19) (the support of ψ Θ ,n,i and the image of the support of ψΘ, ,j by D x T tr t are contained in transverse cones).

D x T tr t (η) -ξ is η d+1 -ξ d+1 and consequently D x T tr t (η) -ξ ≥ |η d+1 -ξ d+1 | .
We now use Lemma 3.21 to control transitions that do not happen in the linear picture.

Lemma 3.22. There is δ > 1 such that, for every bounded interval I of R, there is C > 0 such that if ( , j) → (n, i) for (n, i) , ( , j) ∈ Γ, then for all t ∈ I we have, recalling (3.17),

S ,j t,n,i L 2 →L 2 ≤ C exp - max (n, ) δ C .
Proof. First of all, notice that L t is bounded from L 2 to L 2 (uniformly when t ∈ I) and, since for all (n, i) , ( , j) ∈ Γ and t ∈ I, we have

S ,j t,n,i L 2 →L 2 ≤ L t L 2 →L 2 ,
the case of max (n, ) ≤ N is dealt with by taking C large enough. We consider consequently (n, i) , ( , j) ∈ Γ and t ∈ I such that ( , j) → (n, i) and max (n, ) > N . If u ∈ L 2 R d+1 then we have, using Plancherel's formula,

(2π) 2(d+1) S ,j t,n,i u 2 2 = R d+1 ψ Θ ,n,i (ξ) 2 (R d+1 ) 2 e -ixξ e iTt(x)η ψΘ, ,j (η) G t (x) û (η) dxdη 2 dξ.
(3.21)

We are going to bound the inner integral. To do so, dene for all x ∈ R d+1 and j ∈ {1, . . . , d + 1} the linear form l j (x) on R d+1 × R d+1 by l j (x) (ξ, η) = i (∂ j T t (x) η -ξ j ).

Dene also for all x ∈ R d+1 the quadratic form Φ (x

) on R d+1 × R d+1 by Φ (x) (ξ, η) = D x T tr t (η) -ξ 2
. Now for all t ∈ I and k ∈ N we dene a kernel K k,t : R d+1 ×R d+1 ×R d+1 → C by induction: we set K 0,t (x, ξ, η) = G t (x) and for all k ∈ N

K k+1,t (x, •, •) = d+1 j=1 ∂ x j l j (x) K k,t (x, •, •) Φ (x) .
Integrating by parts in y we see that the inner integral of (3.21) is equal, for all k ∈ N, t ∈ I and ξ ∈ R d+1 , to (R d+1 ) 2 e -ixξ e iTt(x)η ψΘ, ,j (η) K k,t (x, ξ, η) û (η) dxdη.

(

3.22)

To bound the kernel K k,t , we notice that it is the sum of at most (5

(d + 1)) k k! terms of the form (x, ξ, η) → ± ∂ σ G t (x) (Φ (x) (ξ, η)) k+m ∂ γ 1 l j 1 (x) (ξ, η) . . . ∂ γ k l j k (x)(ξ, η) × ∂ µ 1 Φ (x) (ξ, η) . . . ∂ µm Φ (x) (ξ, η) , (3.23)
where m ≤ k is an integer, j 1 , . . . , j k ∈ {1, . . . , d + 1}, and σ, γ 1 , . . . , γ k , µ 1 , . . . , µ m are

elements of N d+1 such that |σ| + |γ 1 | + • • • + |γ k | + |µ 1 | + • • • + |µ m | = k (all the derivatives
are with respect to the variable x). In order to bound these terms, notice that Lemma 3.21

implies that if x ∈ K, if ξ ∈ supp ψ Θ ,n,i and if η ∈ supp ψΘ, ,j then Φ (x) (ξ, η) ≥ c 2 max 2 n α i , 2 n α j 2 ≥ c 1 max 2 n α i , 2 n α j max (|ξ| , |η|) ≥ c 2 max (|ξ| , |η|) 2 ,
for some positive constants c 1 and c 2 . Consequently, there is a constant C > 0 such that if l is a linear map from R d+1 × R d+1 → C and if q is a quadratic map R d+1 × R d+1 → C then we have, for all x ∈ K, ξ ∈ supp ψ Θ ,n,i and η ∈ supp ψΘ, ,j

l (ξ, η) Φ (x) (ξ, η) ≤ C l max 2 n α i , 2 α j and q (ξ, η) Φ (x) (ξ, η) ≤ C q .
The choice of the norms on the spaces of linear and quadratic maps R d+1 × R d+1 → C is of course irrelevant. Thus for such x, ξ and η any term of the form (3.23) is bounded by

C 2k max 2 n α i , 2 α j -k ∂ σ G t ∞ ∂ γ 1 l j 1 ∞ . . . ∂ γ k l j k ∂ µ 1 Φ ∞ . . . ∂ µm Φ ∞ ,
where • ∞ refers to the supremum of the corresponding norm on K. Now, notice that, since T 0 is C κ,υ then for any κ < κ the maps l 1 , . . . , l d+1 (valued in the space of linear maps from R d+1 × R d+1 to C) and Φ (valued in the space of quadratic maps from R d+1 × R d+1 to C) are C κ ,υ (we can event take κ = κ if υ ≤ 2). Thus there are constants M, R > 0 CHAPTER 3 such that for all µ ∈ N d , we have

∂ µ Φ ∞ ≤ M R |µ| |µ|! exp |µ| υ κ υ ,
for all γ ∈ N d and j ∈ {1, . . . , d + 1}, we have

∂ γ l j ∞ ≤ M R |γ| |γ|! exp |γ| υ κ υ ,
and for all t ∈ I and σ ∈ N d , we have

∂ σ G t ∞ ≤ M R |σ| |σ|! exp |σ| υ κ υ .
Thus each term of the form (3.23) is bounded by

C 2k M 2k+1 R k k k exp k υ κ υ 2 -k max(n, ) α when x ∈ K, ξ ∈ supp ψ Θ ,n,i , η ∈ supp ψΘ, ,
2 -k max(n, ) α (5 (d + 1)) k C 2k M 2k+1 R k k 2k exp k υ κ υ . (3.24)
Now, choose κ > 0 such that 1 κ υ + 2 ≤ 1 κ and pick new values of the constants M and R so that (3.24) is now smaller than

M R 2 max(n, ) α k exp k υ κ .
Now, using this estimate and CauchySchwarz in (3.22), we bound the inner integral in

(3.21) by C u 2 2 (d+1) 2 R 2 max(n, ) α k exp k υ κ , which gives S ,j t,n,i u 2 ≤ C u 2 2 ( +n)(d+1) 2 R 2 max(n, ) α k exp k υ κ . Now take k =        -κ ln R 2 max(n, ) α υ   1 υ-1     
to get (with new constants and δ = αυ υ-1 > 1, see the proof of Proposition 3.6 for a similar computation)

S ,j t,n,i u 2 ≤ C u 2 2 ( +n)(d+1) 2 exp - max (n, ) δ C .
Finally, we get rid of the factor 2 ( +n)(d+1) 2

by taking larger C.

We can now prove Lemma 3.20 about the family (M t ) t∈R of auxiliary operators.

Proof of Lemma 3.20. Let us split the sum (3.18) into (n,i),( ,j)∈Γ ( ,j) →(n,i)

ι n,i • S ,j t,n,i • π ,j and
(n,i),( ,j)∈Γ (n,i) →( ,j)

ι n,i • S ,j t,n,i • π ,j .

(3.25)

Thanks to Lemma 3.22, the rst sum in (3.25) converges absolutely in norm operator topology. To deal with the second sum in (3.25), notice that there is some constant C

depending on I such that, for all t ∈ I and (n, i) , ( , j) ∈ Γ, we have

ι n,i • S ,j t,n,i • π ,j B→B ≤ C2 nβ i 2 -β j .
(3.26)

Then the second sum in (3.25) can be divided into seven sums that correspond to the dierent cases in the denition of →. It is elementary, using (3.26), to see that the rst six converge in norm operator topology. Consequently, we are left with the sum

n, ∈N |n-|≤M ι n,f • S ,f t,n,f • π ,f (3.27)
for some M > 0. For all N 1 ∈ N, dene the operator

P N 1 = 0≤n, ≤N 1 |n-|≤M ι n,f • S ,f t,n,f • π ,f . Pick u = (u m,k ) (m,k)∈Γ ∈ B. Then if N 2 ≥ N 1 ≥ 0, we have (P N 2 -P N 1 ) u 2 B ≤ 2 N 2 n=0 2 -2(d+2)n N 1 < ≤N 2 | -n|≤M S ,f t,n,f u ,f 2 2 + 2 N 2 n=N 1 +1 2 -2(d+2)n 0≤ ≤N 1 | -n|≤M S ,f t,n,f u ,f 2 2 .
(3.28) CHAPTER 3

Next, we have by the triangle inequality,

N 1 < ≤N 2 | -n|≤M S ,f t,n,f u ,f 2 2 ≤     N 1 < ≤N 2 | -n|≤M S ,f t,n,f u ,f 2     2 ≤ C     N 1 < ≤N 2 | -n|≤M u ,f 2     2 ,
for some constant C > 0. Then, from the CauchySchwarz inequality, we get

    N 1 < ≤N 2 | -n|≤M u ,f 2     2 =     N 1 < ≤N 2 | -n|≤M 2 (d+2) 2 -(d+2) u ,f 2     2 ≤     N 1 < ≤N 2 | -n|≤M 2 2 (d+2)         N 1 < ≤N 2 | -n|≤M 2 -2 (d+2) u ,f 2 2     ≤ C 2 2n(d+2) N 1 < ≤N 2 | -n|≤M 2 -2 (d+2) u ,f 2 2
for another constant C > 0. Consequently, we can bound the rst sum in (3.28)

N 2 n=0 2 -2n(d+2) N 1 < ≤N 2 | -n|≤M S ,f t,n,f u ,f 2 2 ≤ CC N 2 n=0 N 1 < ≤N 2 | -n|≤M 2 -2 (d+2) u ,f 2 2 ≤ C >N 1 2 -2 (d+2) u ,f 2 2 ,
where in the last line we noticed that, when is xed, there are at most 2M + 1 values of n for which | -n| ≤ M . Working similarly with the second sum, we see that there is a constant C such that

(P N 2 -P N 1 ) u 2 B ≤ C ≥N 1 -M 2 -2(d+2) u ,f 2 2 ,
and thus the sequence

(P N 1 u) N 1 ≥0 is Cauchy in B. Consequently, the sequence (P N ) N ≥0
converges in strong operator topology, hence, so does the sum (3.27). It remains to prove that M t depends continuously on t in the strong operator topology. To do so, just notice that when u is xed the sum

(n,i),( ,j)∈Γ ι n,i • S ,j t,n,i • π ,j u
converges uniformly (in t ∈ I) to M t u and each of its terms is continuous with respect to t (to see this, notice that if (n, i) , ( , j) ∈ Γ then S ,j t,n,i is locally uniformly bounded as an operator from L 2 to L 2 , and the continuity is easily proven for smooth u).

Schatten class properties

Now let h : R * + → C be a compactly supported function as in Proposition 3.17. If (n, i) , ( , j) ∈ Γ, then write

H ,j n,i = R h (t) S ,j t,n,i dt,
where we recall that S ,j t,n,i is dened by (3.17). Notice that the sum

(n,i),( ,j)∈Γ ι n,i • H ,j n,i • π ,j
converges in strong operator topology to R h (t) M t dt, sincein Lemma 3.20 the convergence is uniform locally in t. To prove Proposition 3.17, we want now to prove that this operator is in a Schatten class (or at least compact), this is the point of Lemma 3.27. To do so we need rst to establish a bunch of lemmas: Lemma 3.23 will be used to deal with the transition of frequencies corresponding to the linear model of the dynamics apart from the direction of the ow, Lemma 3.24 will settle the problem of frequency transitions corresponding to the non-linearity, and Lemmas 3.25 and 3.26 will be used to deal with stationary frequencies in the direction of the ow. Lemma 3.23. There is a constant C > 0 such that, for all (n, i) , ( , j) ∈ Γ, the trace class operator norm of H ,j n,i :

L 2 → L 2 is bounded by C2 (d+1)n α i 2 2 (d+1) α j 2
, where

α i = α if i = 0 or i = r -1 and α i = 1 otherwise. Proof. Notice that if u ∈ L 2 then ψ Θ ,n,i (D) u = F -1 ψ Θ ,n,i * u. Consequently, we have 8 H ,j n,i = K F -1 ψ Θ ,n,i (• -y) ⊗ R h (t) G t (y) δ Tt(y) • ψΘ, ,j (
D) dt dy.

(3.29)

And then the result follows from the fact that

F -1 ψ Θ ,n,i 2 = 1 √ 2π d+1 ψ Θ ,n,i 2 ≤ C2 (d+1)n α i 2 and R h (t) G t (y) δ Tt(y) • ψΘ, ,j (D) dt (L 2 ) * ≤ C ψΘ, ,j 2 ≤ C2 (d+1) α j 2 ,
where • (L 2 ) * denotes the operator norm on the dual of L 2 R d+1 .

8 If E, F are Banach spaces, e ∈ F and l ∈ E , we denote by e ⊗ l the rank 1 operator dened by e ⊗ l(u) = l(u).e for u ∈ E.

Lemma 3.24. There is a constant C > 0 and some δ > 1 such that, if ( , j) → (n, i)

for (n, i) , ( , j) ∈ Γ, then the trace class operator norm of H ,j n,i :

L 2 → L 2 is bounded by C exp -max(n, ) δ C .
Proof. We may assume that max (n, ) > N . Without loss of generality, we may assume that K ⊆ ]-π, π[ d+1 and then, if u ∈ L 2 R d+1 write (the sum converges in L 2 )

H ,j n,i u = k∈Z d+1 c k R h (t) L t ψΘ, ,j (D) udt ψ Θ ,n,i (D) ρ k ,
where ρ is a function supported in ]-π, π[ d+1 that takes value 1 on K, the function ρ k is

dened by ρ k (x) = ρ (x) e ikx and if v is supported in ]-π, π[ d+1 and k ∈ Z d+1 , its kth
Fourier coecient is denoted by c k (v):

c k (v) = 1 (2π) d+1 ]-π,π[ d+1 e -ikx v (x) dx.
By requiring that ρ is s-Gevrey (for some s > 1), we may ensure as in [Jéz20a, Lemma 6.5] that (for some constant C > 0)

ψ Θ ,n,i (D) ρ k 2 ≤ C2 (d+1)n α i 2 exp   - d k, supp ψ Θ ,n,i 1 s C   . Now, if k ∈ Z d+1 and ( , j) ∈ Γ dene δ (k, , j) = sup x∈K d k, D x T tr t supp ψΘ, ,j .
Then integrating by parts as in [Jéz20a, Lemma 6.7] or as in Lemma 3.22 we see that if

δ (k, , j) ≥ 2 α j
(for some arbitrary xed

> 0) then c k • R h (t) L t dt • ψΘ, ,j (D) (L 2 ) * ≤ C2 (d+1) α j exp - ln (1 + δ (k, , j)) υ υ-1 C . But now, if ( , j) → (n, i) and max (n, l) > N , then, for all k ∈ Z d+1 , either the distance d k, supp ψ Θ ,n,i or the distance δ (k, , j) is greater than c 2 max 2 n α i , 2 α j
, thanks to Lemma 3.21. Moreover, if |k| is greater than C2 max(n, ) (for some large C > 0), then we have δ (k, , j) ≥ max(2 n α i , 2 α j ) and d k, supp ψ Θ ,n,i ≥ |k|. Thus, the sum

H ,j n,i = k∈Z d+1 ψ Θ ,n,i (D) ρ k ⊗ c k • R h (t) L t dt • ψΘ, ,j (D)
converges in trace class topology, and from the estimates above we see that the announced result holds with δ = αυ υ-1 .

Lemma 3.25. Assume that h is kth times dierentiable and that is kth derivative has bounded variation. Then there is a constant C > 0 such that for all n, ∈ N we have

H ,f n,f L 2 →L 2 ≤ C2 -(k+1) .
Proof. If u ∈ L 2 R d+1 and x ∈ R d+1 , then we have,

H ,f n,f u (x) = R d+1 V n, (x, η) û(η)dη,
where the kernel V n, is dened by

V n, (x, η) = 1 (2π) 2(d+1) (R d+1 ) 3 ×R e i(x-z)ξ+iT 0 (z)η e itη d+1 ψ Θ ,n,f (ξ) × ψΘ, ,f (η) h (t) G t (z) dzdξdt.
(3.30)

We can assume that is large enough (the H ,f n,f 's are uniformly bounded on L 2 ), which ensures that η d+1 (the last coordinate of η) does not vanish on the support of ψΘ, ,f . Consequently, we can perform k + 1 integrations by parts in t in (3.30) to get

V n, (x, η) = i k+1 (2π) 2(d+1) (R×R d+1 ) 3 e i(x-z)ξ+iT 0 (z)η e itη d+1 ψ Θ ,n,f (ξ) × ψΘ, ,f (η) η k+1 d+1 d k+1 dt k+1 (h (t) G t (z)) dtdzdξ.
Using the Leibniz rule, we see that, if µ denotes the measure of total variation of h (k+1) , the measure

d k+1
dt k+1 (h (t) G t (z)) dt may be written as f (t, z) dµ (t) for all z ∈ R d+1 . Moreover, f has the following properties: it is measurable, f (t, z) = 0 if z ∈ R d+1 \ K, and

R sup z∈R d+1 |f (t, z)| dµ (t) < +∞. Then, dene the function Ψ : R d+1 → R by Ψ (η) = ψΘ, ,f (η) η k+1 d+1 , the operator L t : L 2 R d+1 → L 2 R d+1 by L t u (z) = f (t, z) . (u • T t (z))
, and notice that we have

H ,f n,f = ψ Θ ,n,f (D) • R L t dµ (t) • Ψ (D) .
Finally, notice that Ψ ∞ ≤ C2 -(k+1) to end the proof.

Lemma 3.26. Let s > 0 and > 0. Then there is a constant C > 0 such that for all N > 0 and n ∈ N with n < N there is an operator

F n,N : L 2 (K) → L 2 R d+1 of rank at most 2 (1+ )(d+1)N such that for all u ∈ L 2 (K) we have ψ Θ ,n,f (D) u -F n,N u 2 ≤ C2 -sN .
Proof.

C h C k-1 + h (k)
BV where C depends on h only through its support.

Without the assumption that α < 1 2 , it remains true that the operator dened by (3.31) is compact.

Proof. We know that R h (t) M t dt = (n,i),( ,j)∈Γ ι n,i • H ,j n,i • π ,j (3.32)
where the sum converges in the strong operator topology. From Lemma 3.24, it is clear that the sum

(n,i),( ,j)∈Γ (n,i) →( ,j) ι n,i • H ,j n,i • π ,j
converges in the trace class operator topology. We are left with the sum

(n,i),( ,j)∈Γ (n,i) →( ,j) ι n,i • H ,j n,i • π ,j
that we can divide, as in the proof of Lemma 3.20, into seven sums corresponding to the dierent cases in the denition of →. The rst six sums are dealt with by using Lemma 3.23. We will only detail the computation corresponding to the rst case in the denition of → (i.e. the case i = j = 0, the case i = j = r -1 is dealt with in the same way and the others are easier), in order to highlight where the hypothesis α < 1 d+2) . Thus, in order to deal with the sum corresponding with the case i = j = 0 in the denition of →, we only need to prove that the quantity

2 is used. If n, ∈ N, then the trace class operator norm of ι n,0 • H ,0 n,0 • π ,0 is smaller than C2 (d+1)n α 2 2 (d+1) α 2 2 (d+2)n 2 -(
,n∈N ( ,0) →(n,0) 2 (d+1)n α 2 2 (d+1) α 2 2 (d+2)n 2 -(d+2) (3.33) is nite. Notice that 2 -(d+2) + (d+1) α 2 ∼ →+∞ 1 1 -2 -(d+2) 2 -(d+2) + (d+1) α 2 -2 (d+2)( +1)+ (d+1)( +1) α 2 so that ≥ 0 2 -(d+2) + (d+1) α 2 ∼ 0 →+∞ 2 -(d+2) 0 + (d+1) α 0 2 1 -2 -(d+2) .
In particular, there is a constant C > 0 such that, for all 0 ∈ N. We have

≥ 0 2 -(d+2) + (d+1) α 2 ≤ C2 -(d+2) 0 + (d+1) α 0 2
. Now if n ∈ N, let 0 be the smallest integer such that 0 ≥ n + νn 1-α , we have then (notice that 0 ≤ Bn for some constant B that does not depend on n)

∈N ( ,0) →(n,0) 2 -(d+2) + (d+1) α 2 = ≥ 0 2 -(d+2) + (d+1) α 2 ≤ C2 -(d+2) 0 + (d+1) α 0 2 ≤ C2 -(d+2)n 2 -(d+2)νn 1-α 2 (d+1)B α 2 n α .
Thus, we have

,n∈N ( ,0) →(n,0) 2 (d+1)n α 2 2 (d+1) α 2 2 (d+2)n 2 -(d+2) ≤ C n∈N 2 -(d+2)νn 1-α 2 (d+1)(B α +1) 2 n α , (3.34)
and this sum is nite since α < 1 2 . As explained above, a similar argument allows to deal with the sum corresponding to any of the six rst cases in the denition of →.

Consequently, we are left with the sum

P = n, ∈N |n-|≤M ι n,f • H ,f n,f • π ,f .
Choose s > k + 1 and > 0, and apply Lemma 3.26 to dene for all N > 0 the operator

P N = 0≤n, <N |n-|≤M ι n,f • F n,N • R h (t) L t dt • ψΘ, ,j • π ,f , CHAPTER 3
whose rank is at most N 2 2 (1+ )(d+1)N . Then notice, using Lemma 3.25, that we have

P N -P B→B ≤ C n, <N |n-|≤M F n,N -ψ Θ ,n,f (D) L 2 (K)→L 2 (R d+1 ) × R h (t) L t dt • ψΘ, ,f (D) L 2 →L 2 + C n, ≥N |n-|≤M H ,f n,f L 2 →L 2 ≤ C N 2 2 -sN + 2 -(k+1)N ≤ C 2 -(k+1)N , (3.35)
for some constants C, C and C that do not depend on N . Letting N tend to innity, we see that P is compact. Moreover, if (s m ) m≥0 denotes the sequence of singular values of P , we get from (3.35) and [GGK00, Theorem 2.5 p.51]

s N 2 2 (1+ )(d+1)N +1 ≤ C 2 -(k+1)N .
Thus, the sequence (s m ) m≥0 is in p for all p > (1+ )(d+1) k+1 (the sequence (s m ) m≥0 is decreasing). This ends the proof in the case α < 1 2 since > 0 is arbitrary. Indeed, all the terms in the proof are controlled by the L ∞ norm of h, except the one that we bounded using Lemma 3.25 that is controlled by h

C k-1 + h (k) BV .
In order to deal with the case α ≥ 1 2 , notice that we only used the assumption α < 1 2 to ensure that the series (3.34) converges. However, if we remove the factor 2

(d+1)n α 2 2 (d+1) α 2
from the sum (3.33), this new series converges, just like in the proof of Lemma 3.20. That is, if we consider the operator norm instead of the trace class norm, the sums corresponding to the rst six cases in the denition of → converge, even if α ≥ 1 2 . Consequently, the right-hand side of (3.32) always converges in the operator norm topology, and the left-hand side of (3.32) is always compact. Before proving that (L t ) t∈R inherits of the properties of (M t ) t∈R , thus showing Proposition 3.17, we still need to prove that the operator R h (t) M t dt has the expected trace, when it makes sense. This is the point of the following lemma.

Trace of +∞

Lemma 3.28. Under the hypotheses of Proposition 3.17

, if Θ = Θ ,if α < 1 2 and if k + 1 > d + 1 then tr R h (t) M t dt = p•F (x)=x h (T (x)) |det (I -p • D x F )| R G T (x) (x, y) dy,
where p denotes the orthogonal projection on R d R d × {0} and, for x ∈ R d , the number

T (x) is dened by F (x) = p (F (x)) + (0, -T (x)).
Proof. For all N ∈ N write

A N = (n,i),( ,j)∈Γ 0≤n, ≤N ι n,i • H ,j n,i • π ,
H n,i n,i = R K h (t) G t (w) ψΘ,n,i (D) F -1 (ψ Θ,n,i ) (• -w) (T t (w)) dwdt = R K h (t) G t (w) F -1 (ψ Θ,n,i ) (T t (w) -w) dwdt.
We used in the second line that if ψ Θ,n,i (ξ) = 0 then ψΘ,n,i (ξ) = 1. Now let M be such that

K ⊆ [-M, M ] d+1 and h is supported in [-M, M ]. Dene the map g : R d+1 R d × R → R d+1 by g (x, t) = F (x) -(x, -t).
Notice that for all (x, y) ∈ R d+1 and t ∈ R we have

T t (x, y) -(x, y) = g (x, t) .
Cone-hyperbolicity implies that the Jacobian of g does not vanish. Consequently we can nd a nite family (ρ a ) a∈A of compactly supported C ∞ functions ρ a : R d+1 → [0, 1] such that a∈A ρ a (w) = 1 for all w ∈ [-M, M ] d+1 and for all a ∈ A there is a C ∞ dieomorphism g a : R d+1 → R d+1 that coincides with g on a neighborhood of the support of ρ a . Thus, we nd that tr H n,i n,i is equal to

a∈A [-M,M ] d+2 h (t) ρ a (x, t) G t (x, y) F -1 (ψ Θ,n,i ) (g a (x, t)) dxdtdy = a∈A R d+1 ×[-M,M ] h (t a (z)) ρ a • g -1 a (z) G ta(z) (x a (z) , y) det D g -1 a (z) g a F -1 (ψ Θ,n,i ) (z) dzdy
where w = (x, y) and g -1 a (z) = (x a (z), t a (z)). Since (n,i)∈Γ ψ Θ,n,i = 1 we nd that for all CHAPTER 3

a ∈ A we have lim

N →+∞ (n,i)∈Γ 0≤n≤N R d+1 ×[-M,M ] h (t a (z)) ρ a • g -1 a (z) G ta(z) (x a (z) , y) det D g -1 a (z) g a F -1 (ψ Θ,n,i ) (z) dzdy = [-M,M ] h (t a (0)) ρ a g -1 a (0) G ta(0) (x a (0) , y) det D g -1 a (0) g a dy.
And thus tr

R h (t) M t dt = a∈A [-M,M ] h (t a (0)) ρ a g -1 a (0) G ta(0) (x a (0) , y) det D g -1 a (0) g a dy.
Now, notice that g (x, t) = 0 if and only if p • F (x) = x and t = T (x), thus

[-M,M ] h (t a (0)) ρ a g -1 a (0) G ta(0) (x a (0) , y) det D g -1 a (0) g a dy = p•F (x)=x [-M,M ] ρ a (x, T (x)) h (T (x)) G T (x) (x, y) |det (I -p • D x F )|
dy.

Here we noticed that the Jacobian of g do not depend on the last coordinate. Finally,

summing over a ∈ A we get tr R h (t) M t dt = p•F (x)=x h (T (x)) |det (I -p • D x F )| [-M,M ]
G T (x) (x, y) dy.

We show Proposition 3.17 by proving that (L t ) t∈R also satises the properties established for (M t ) t∈R in Lemmas 3.20, 3.27 and 3.28.

Proof of Proposition 3.17. Recall that Q Θ (dened by (3.13)) induces an isomorphism between H Θ,α and Q Θ (H Θ,α ), which is a closed subspace of B. We denote by Q -1 Θ the inverse isomorphism (and similarly for Q Θ ). Now, if (u n,i ) (n,i)∈Γ is nitely supported (i.e. there are nitely many (n, i) ∈ Γ such that u n,i = 0) and such that for all (n, i) ∈ Γ we have u n,i ∈ S υ (for some υ ∈ υ, 1 1-α )

we write u = (n,i)∈Γ ψΘ,n,i (D) u n,i and notice that L t u ∈ S υ for all t ∈ R, and thus

L t u ∈ H Θ ,α . Consequently, M t (u n,i ) (n,i)∈Γ = Q Θ L t u is in Q Θ H Θ ,α . Since such
elements are easily seen to be dense in B, it appears that M t sends B into Q Θ H Θ ,α .

We can consequently dene the operator

Q -1 Θ • M t • Q Θ . The calculation above also implies that L t and Q -1 Θ • M t • Q Θ coincides on H Θ,α (
since the element of S υ whose Fourier transform is compactly supported are dense in H Θ,α , CHAPTER 3

Proof. Choose a Mather metric |•| x on M (see [START_REF] Mather | Characterization of Anosov dieomorphisms[END_REF]). This metric makes the splitting

T x M = E u x ⊕ E s x ⊕ RX (x) (3.36)
orthogonal and is Hölder-continuous. Moreover, |X (x)| x = 1 for all x ∈ M and, for t ≥ 0,

we have D x φ t | E s x ≤ λ -t and D x φ -t | E u x ≤ λ -t (for the induced norm, λ > 1). Choose γ > 0 such that λ -2t 0 γ 2 + 1 < 1. Then, choose γ 1 ∈]1/γ, λ t 0 2 /γ[ and dene for all i ≥ 2 the number γ i = λ -t 0 (i-1) 2 γ 1 . Now, let r be large enough so that λ 2t 0 1 + γ 2 r-1 > 1.
Since γγ 1 λ -5t 0 8 < 1, we may choose ˜ u > 0 and ˜ s > 0 such that

˜ u > λ -t 0 2 γ 1 ˜ s and ˜ s > γλ -t 0 8 ˜ u ,
and small enough so that

λ -2t 0 ˜ 2 s + γ 2 + 1 < 1 and λ 2t 0 1 + γ 2 r-1 + ˜ 2 u > 1.
Finally, set u = λ -t 0 8 ˜ u and s = λ -t 0 2 ˜ s . Now, for all x ∈ M , if ξ ∈ T x M write ξ = ξ u + ξ s + ξ 0 the decomposition of ξ with respect to (3.36), and dene the cones C f (x) and C i (x), for i ∈ N by

C 0 (x) = {ξ ∈ T x M : |ξ u | x ≤ γ |ξ s | x and |ξ 0 | x ≤ ˜ s |ξ s | x } , C i (x) = ξ ∈ T x M : |ξ s | x ≤ γ i |ξ u | x and |ξ 0 | x ≤ λ -(i-1)t 0 4 ˜ u |ξ u | x (3.37) for i ∈ N * and C f (x) = {ξ ∈ T x M : |ξ 0 | x ≥ s |ξ s | x and |ξ 0 | x ≥ u |ξ u | x } .
Notice that all these cones depend Hölder-continuously on x. We will see that our choice of parameter ensures that if x ∈ M , then Θ (x) = (C 0 (x) , . . . , C r (x) , C f (x)) is a system of r + 2 cones with respect to the direction X (x). Indeed:

(i) if ξ ∈ T x M \ C f (x), since γγ 1 > 1, we have either |ξ u | x < γ |ξ s | x or |ξ s | x < γ 1 |ξ u | x .
In the rst case, either

|ξ 0 | x < ˜ s |ξ s | x , in which case ξ ∈ • C 0 (x), or |ξ 0 | x ≥ ˜ s |ξ s | x , which implies ξ ∈ • C f (x), since s < ˜ s and |ξ 0 | x > ˜ s γ |ξ u | x > u |ξ u | x . Similarly, we can see that in the second case either ξ ∈ • C 1 (x) or ξ ∈ • C f (x); (ii) if ξ ∈ C f (x) then |ξ 0 | x ≥ 1 √ 1+ -2 u + -2 s |ξ|, which implies that C f (x) is one dimen- sional; (iii) if ξ ∈ C 0 (x) then |ξ| x ≤ 1 + γ 2 + ˜ 2 s |ξ s | and thus C 0 (x) is d s -dimensional, where d s is the dimension of E s
x , for the same reason C i (x) is d u -dimensional for i ∈ {1, . . . , r};

(iv) C i+1 (x) C i (x) for i ∈ {1, . . . , r -1} because γ i+1 < γ i and λ -it 0 2 ˜ u < λ -(i-1)t 0 2 ˜ u ; (v) C 0 (x) ∩ C 2 (x) = {0} because γγ 2 < 1 and C f (x) ∩ C 2 (x) = {0} because we have λ t 0 4 u /˜ u = λ t 0 8 > 1. Let us set Λ := λ t 0 min((˜ 2 u +γ 2 r-1 +1) -1 2 , (˜ 2 s +γ 2 +1) -1 2 ) > 1.
Our choice of parameter ensures that for t ≥ t 0 and x ∈ M :

• for all i ∈ {1, . . . , r} we have (D

x φ t ) tr (C i (φ t (x))) ⊆ C i+4 (x) because λ -2t γ i ≤ γ i+4 and λ -t λ -(i-1)t 0 4 /λ -(i+3)t 0 4 < 1; • (D x φ t ) tr (C f (φ t (x))) ∩ C 0 (x) = {0} because s λ t > ˜ s ; • for all ξ ∈ C r-1 (φ t (x)) we have (D x φ t ) tr (ξ) x ≥ Λ |ξ| φt(x) ; • if ξ ∈ T φt(x) M and (D x φ t ) tr (ξ) ∈ C 0 (x) then (D x φ t ) tr (ξ) x ≤ Λ -1 |ξ| φt(x) holds.
Then, for every x ∈ M , we may choose a C κ,υ chart κ x : V x → W x = B (0, δ x )×]-t x , t x [ such that κ x (x) = 0, the map D x κ x : T x M → R d+1 is an isometry and, for every y ∈ V x , we have D y κ x (X (y)) = e d+1 (we can require the last two points simultaneously because |X (x)| x = 1). For every x ∈ M , choose a system of r + 2 cones Θ x = (C 0,x , . . . , C r,x , C f,x )

such that D x κ tr x (C f,x ) C f (x),D x κ tr x (C 0,x )
C 0 (x), and, for every i ∈ {1, . . . , r},

we have D x κ tr x -1 (C i+1 (x)) C i,x D x κ tr x -1 (C i (x)).
Here we recall that the denition (3.37) of C i (x) is valid for any i ≥ 1. Up to making V x smaller, we may ensure that for all y ∈ V x we have

D y κ tr x (C f,x ) C f (y) , D y κ tr x (C 0,x ) C 0 (y) , (3.38)
for all i ∈ {1, . . . , r} we have

D y κ tr x -1 (C i+1 (y)) C i,x D y κ tr x -1 (C i (y)) , (3.39)
and, in addition,

D y κ x ≤ 1 + and (D y κ x ) -1 -1 ≥ 1 -, (3.40)
where > 0 is small enough so that

1 - 1 + Λ > 1.

CHAPTER 3

By compactness of M , there are x 1 , . . . , x n such that M is covered by the open sets κ -1

x i B 0, δx i 2 × - tx i 100 ,
tx i 100 for i = 1, . . . , n. Let t 1 = min i=1,...,n tx i 100 . By cutting the charts into pieces and translating them, we may assume that for every i = 1, . . . , n we have t x i = 100t 1 (this could make us lose the fact that D x i κ x i is an isometry, but this is of no harm since (3.40) remains true and that is all we need). Notice that for such a t 1 there is no periodic orbit of (φ t ) t∈R of length less than 3t 1 . If necessary, we reduce the value of t 1 so that t 1 < t 0 . Set t 2 = 30t 1 and let N = 2t 0 t 2

. Choose χ : R d → [0, 1] Gevrey, compactly supported and such that χ (y) = 1 if |y| ≤ 1.

If i, j ∈ {1, . . . , n}, if k ∈ {0, . . . , N }, and if y ∈ B 0,

δx i 2
are such that the point

φ t 0 +kt 2 κ -1 x i (y, 0) lies in κ -1 x j B 0, δx j 2 × [-t 2 , t 2 ],
and η > 0 is small enough dene

F i,j,k,y,η : R d → R d+1 by (here we see R d R d × {0} as a subset of R d+1 ) F i,j,k,y,η (z) = χ z -y η κ x j • φ t 0 +kt 2 • κ -1 x i (z) + 1 -χ z -y η κ x j • φ t 0 +kt 2 • κ -1 x i (y) + D y κ x j • φ t 0 +kt 2 • κ -1 x i (z -y) Notice that F i,j,k,y,η coincides with κ x j • φ t 0 +kt 2 • κ -1
x i on B (0, η), and that it can be made arbitrarily close in the C 1 topology to the ane map z → κ

x j • φ t 0 +kt 2 • κ -1 x i (y) + D y κ x j • φ t 0 +kt 2 • κ -1
x i (z -y) by taking η = η i,j,k,y small enough. In particular, F i,j,k,y,η denes a cone-hyperbolic family of dieomorphisms T i,j,k,y,η,t t ∈R from Θ x i to Θ x j (the cone-hyperbolicity follows from the properties of the dierential of φ t 0 +kt 2 proven above and the quasi-isometry property (3.40) of the charts, to see that the T i,j,k,y,η,t 's are diffeomorphisms just notice that they are proper local dieomorphism and hence covering of R d+1 by itself ). Dene ηi,y = min k=0,...,N j=1,...,n

η i,j,k,y (if j is such that φ t 0 +kt 2 κ -1 x i (y, 0) / ∈ κ -1 x j B 0, δx j 2 ×[-t 2 , t 2 ], i.e.
there is no allowed transitions from i to j at the considered time, set η i,j,k,y = ∞ and take for F i,j,k,y,η i,y any C κ,υ map that denes a cone-hyperbolic family of dieomorphisms 9 from Θ x i to Θ x j ) .

Notice also that for all (z, z ) ∈ B (0, ηi,y ) × ]-t 2 , t 2 [ and all t, t ∈ ]-t 2 , t 2 [ we have

κ x j • φ t 0 +kt 2 +t+t • κ -1 x i z, z = κ x j • φ t 0 +kt 2 +t+t +z • κ -1 x i (z, 0) = F i,j,k,y,η i,y (z) + z e d+1 + t + t e d+1 = T i,j,k,y,t t z, z , where T i,j,k,y,t t t ∈R
denotes the family of cone-hyperbolic dieomorphisms associated with F i,j,k,y,η i,y + te d+1 . 9 There always is a linear such map.

By compactness of B 0, δx i 2 , we may nd y i,1 , . . . , y i,m i ∈ B 0,

δx i 2 such that B 0, δ x i 2 ⊆ m i =1 B y i, , ηi,y i, 2 .
Finally, set Ω = {(i, ) : i ∈ {1, . . . , n} , ∈ {1, . . . , m i }} ,

and, for all ω = (i, ) ∈ Ω,

V ω = B 0, ηi,y i, 2 × ]-t 1 , t 1 [ , U ω = κ -1 x i B 0, ηi,y i, 2 × ]-t 1 , t 1 [ , κ ω = κ x i | Uω , Θ ω = Θ x i . If ω = (j, ) ∈ Ω and t ∈ [t 0 + kt 2 , t 0 + (k + 1) t 2 ] let F ω,ω ,t = F i,j,k,y i, ,η i,y i, + (t -kt 2 ) e d+1 .
From what is above, these data satisfy the statement of the lemma.

Choose a Gevrey partition of unity (ϕ ω ) ω∈Ω subordinated to the open cover (U ω ) ω∈Ω .

Fix α ∈ υ-1 υ , 1 (if υ < 2, we choose α < 1 2 ) and choose υ ∈ υ, 1 1-α . Then dene

Φ : D υ (M ) → ⊕ ω∈Ω S υ u → (ϕ ω u) • κ -1 w and S : ⊕ ω∈Ω S υ → D υ (M ) (u ω ) ω∈Ω → ω∈Ω (h ω u ω ) • κ ω ,
where h ω : R d+1 → [0, 1] is Gevrey, supported in W ω , and takes value 1 on κ ω (supp ϕ ω ).

Notice that S •Φ is the identity of D υ (M ) . It can be veried that Φ and S are continuous.

We may now dene the rst version of the global Hilbert space (the nal one will be introduced in 3.5). Dene

H Ω = ⊕ ω∈Ω H Θω,α and
H = u ∈ D υ (M ) : Φ (u) ∈ H Ω , endowed with the norm u H = Φ (u) H Ω = ω∈Ω (ϕ ω u) • κ -1 ω 2 H Θω ,α . CHAPTER 3 Proposition 3.30. H is a separable Hilbert space (equivalently, Φ H is closed in H Ω )
that does not depend on the choice of υ. The inclusion of H in D υ (M ) is continuous, and

C ∞,υ (M ) is continuously contained in H. Proof. To see that Φ H is closed in H Ω , just notice that Φ H = {u ∈ H Ω : ΦSu = u} ,
and that the inclusion of H Ω in ω∈Ω S υ is continuous. The inclusion of H in D υ (M ) may be written as the composition of Φ, the inclusion of H Ω in ω∈Ω S υ and S. It is thus

continuous. Finally, Φ sends C ∞,υ (M ) continuously into ω∈Ω S υ, which is continuously contained in H Ω , thus C ∞,υ (M ) is contained in H, the inclusion being continuous.
Let H 0 be the closure 10 of C ∞,υ (M ) in H. Recall from 3.1 that for each t ∈ R we may dene the operator L t from (9) as an operator from D υ (M ) to itself. We start by proving that, for t ≥ t 0 , the operator L t is bounded from H to H 0 .

Proposition 3.31. For all t ∈ [t 0 , +∞[ the operator L t is bounded from H to H 0 . Moreover, as an operator from H to H 0 , the operator L t depends continuously on t ∈ [t 0 , +∞[ in the strong operator topology.

Proof. We only need to prove the result for t ∈ [t 0 , 3t 0 ], and then use the group property of (L t ) t∈R . Recall indeed that H 0 is a closed subspace of H. For all t ∈ R dene

L t : ω∈Ω S υ → ω∈Ω S υ by L t = Φ • L t • S.
The operator L t may be described via a matrix of operators L ω,ω ,t ω,ω ∈Ω , that is, we have

L t (u ω ) ω∈Ω = ω ∈Ω L ω,ω ,t u ω ω∈Ω . (3.41) Now, if t ∈ [t 0 , 3t 0 ] and t ∈ ]-t 1 , t 1 [, then the operator L ω,ω ,t+t for ω, ω ∈ Ω may be described as L ω,ω ,t+t u (x) = ϕ ω • κ -1 ω (x) e t+t 0 V •φτ (κ -1 ω (x))dτ × h ω • κ ω • φ t+t • κ -1 ω (x) u • κ ω • φ t+t • κ -1 ω (x) = G ω,ω ,t,t (x) u • T ω ,ω,t t (x) , (3.42) where T ω ,ω,t t t ∈R
is the family of dieomorphisms associated to F ω ,ω,t by (3.14), and

G ω,ω ,t,t (x) = ϕ ω • κ -1 ω (x) e t+t 0 V •φτ (κ -1 ω (x))dτ h ω • κ ω • φ t+t • κ -1 ω (x)
properly extended by zero. We can then apply Proposition 3.17 to prove that L ω,ω ,t+t is bounded from H Θ ω ,α to H Θω,α . Then L t+t is bounded from H Ω to itself thanks to (3.41).

Notice that if u ∈ ω∈Ω S υ then We want now to prove Schatten properties for operators dened in term of the L t 's for t ≥ t 0 . To do so, it is convenient to introduce (ψ ) ∈Z , a t 1 3 Z invariant smooth partition of unity on R (that is, we have

L t+t u = Φ (L t+t • Su) ∈ Φ H 0 . Thus,
ψ = ψ 0 • -t 1 3 ) such that ψ 0 is supported in -t 1 2 , t 1 2 .
Proposition 3.32. Assume υ < 2. There is ∈ R with the following property: if

h : R * + → C and k ∈ N satisfy (i) h is supported in [t 0 , +∞[;
(ii) h is kth time dierentiable and its kth derivatives has bounded variations;

(iii) there is a constant C > 0 such that for every ∈ N we have

ψ 0 h • + t 1 3 C k-1 + ψ 0 h • + t 1 3 (k) BV ≤ Ce -;
then the operator

+∞ 0 h(t)L t dt : H → H 0 (3.43)
is in the Schatten class S p for every p ≥ 1 such that p > d+1 k+1 . Moreover, if k > d and if we see (3.43) as an operator from H 0 to itself, we have

tr +∞ t 0 h (t) L t dt = γ T # γ h (T γ ) |det (I -P γ )| exp γ V ,
where the sum on the right-hand side runs over closed periodic orbits 11 γ of the ow (φ t ) t∈R .

11 Recall that Tγ is the length of γ, while T # γ denotes its primitive length and Pγ is a linearized Poincaré map. We will see during the proof of the proposition that this sum converges.

Finally, if υ ≥ 2, it remains true (under the same assumptions) that the operator (3.43) is compact.

Remark 3.33. Notice that if h is a C ∞ function supported in [t 0 , +∞[, then h clearly satises the conditions (i)-(iii) from Proposition 3.32. This will be the main application of Proposition 3.32 in order to prove the trace formula (see Lemma 3.44). However, we will also need to consider other functions h in the proof of Lemma 3.43 and in the Appendix C.

For the sake of the proof, we split Proposition 3.32 into Lemmas 3.34, 3.35 and 3.36. Lemma 3.34. Under the assumption of Proposition 3.32, the operator (3.43) is in the Schatten class S p for every p ≥ 1 such that p > d+1 k+1 . If υ ≥ 2, it remains true that (3.43) is compact.

Proof. Let p ≥ 1 be such that p > d+1 k+1 . Choose N large enough so that N t 1 ≥ t 0 , and write for all

≥ N R ψ (t) h (t) L t dt = L q N t 1 3 R ψ 0 (t) h t + t 1 3 L rt 1 3 +t dt,
where = qN + r with q, r ∈ Z and N ≤ r < 2N (notice that q ≥ 0). Applying Proposition 3.17 as in the proof of Proposition 3.31, we see that the operator Finally, when υ ≥ 2 it remains true that the operator (3.44) is compact according to Proposition 3.17, and the convergence in the operator norm topology ensures that (3.43) is compact.

R ψ 0 (t) h t + t 1 3 L rt 1 3 +t dt (3.
We need now to compute the trace of this operator when k > d and υ < 2. We will deduce the global formula for the trace from the local formula from Proposition 3.17.

Lemma 3.35. Under the assumptions of Proposition 3.32 and if

∈ Z we have tr R ψ (t) h (t) L t dt = γ ψ (T γ ) exp γ V h (T γ ) T # γ |det (I -P γ )| , (3.46)
where the sum runs over periodic orbits γ of the ow (φ t ) t∈R . Here, we recall that (ψ ) ∈Z is a t 1 3 Z invariant smooth partition of unity on R such that ψ 0 is supported in

-t 1 2 , t 1 2 .
Proof. If is such that t 1 3 < t 0 -t 1 2 then (3.46) is immediate: both sides vanish. Otherwise, choose an integer m ≥ 0 such that t 1 3 -mt 0 ∈ t 0 -t 1 2 , 2t 0 (one can for instance take m to be the largest integer such that

t 1 3 -mt 0 ≥ t 0 -t 1 2 ) and dene t 3 = max t 0 , t 1 3 -mt 0 . This ensures that the support of ψ is contained in mt 0 +t 3 +]-t 1 , t 1 [ and that t 3 ∈ [t 0 , 2t 0 ]. For all - → ω = (ω 1 , . . . , ω m ) ∈ Ω m dene U-→ ω = m j=1 φ -jt 0 (U ω i ) .
Then choose a renement U-→ ω ,w ( -→ ω ,w)∈Ω m ×W of (U-→ ω ) ω∈Ω m whose elements are small enough such that, if γ is a periodic orbit of (φ t ) t∈R of length T γ less than t 3 + mt 0 + t 1 , and ( -→ ω , w) ∈ Ω m × W, then the intersection of γ with U-→ ω ,w is an interval (i.e. connected, while possibly empty). This can be done because there are a nite number of such orbits. Choose a Gevrey partition of unity θ-→ ω ,w ( -→ ω ,w)∈Ω m ×W adapted to the open cover

U-→ ω ,w ( -→ ω ,w)∈Ω m ×W of M . For t ∈ t 3 + mt 0 + ]-t 1 , t 1 [, recall from the proof of Proposition 3.31 the operators L t = Φ • L t • S, and L ω,ω ,t , for ω, ω ∈ Ω, dened by the formula, L ω,ω ,t u (x) = ϕ ω (κ -1 ω (x))e t 0 V •φτ (κ -1 ω (x))dτ h ω (κ ω φ t κ -1 ω (x))u(κ ω φ t κ -1 ω (x)).
Then write L ω,ω ,t as a sum of operators

L ω,ω ,t = ( -→ ω ,w)∈Ω m ×W A ω,ω , -→ ω ,w,t CHAPTER 3
where, for -→ ω = (ω 0 , . . . , ω m-1 ) and w ∈ W,

A ω,ω , -→ ω ,w,t u (x) = θ-→ ω ,w κ -1 ω (x) L ω,ω ,t u (x) = θ-→ ω ,w ϕ ω • κ -1 ω (x) e t 0 V •φτ (κ -1 ω (x))dτ h ω • κ ω • φ t • κ -1 ω (x) × u • T ωm,ω ,t 3 t-t 3 -mt 0 • T ω m-1 ,ωm,t 0 0 • • • • • T ω 1 ,ω 2 ,t 0 0 • T ω,ω 1 ,t 0 0 (x) .
Consequently, we can use Proposition 3.17 to see that A ω,ω , -→ ω ,w,t : H Θ ω ,α → H Θω,α is bounded (here, we recall that α has been xed after the proof of Lemma 3.29, when dening the space H). Then, working as in the proof of Lemma 3.34, the operator

R ψ (t) h (t) L t dt (3.47)
is trace class, sends H Ω into Φ H 0 and the induced operator is conjugated to the operator dened by (3.47) without the tilde. Consequently, using Lidskii's trace theorem, we get tr

R ψ (t) h (t) L t dt = tr R ψ (t) h (t) L t dt = ω∈Ω tr R ψ (t) h (t) L ω,ω,t dt = ω∈Ω ( -→ ω ,w)∈Ω m ×W tr R ψ (t) h (t) A ω,ω, -→ ω ,w,t dt .
Next, we x ω and ( -→ ω , w) and we compute tr R ψ (t) h (t) A ω,ω, -→ ω ,w,t dt using Proposition 3.17. To do so, recall the family of cone-hyperbolic dieomorphisms

T ω, -→ ω t t∈R := T ωm,ω ,t 3 t-t 3 -mt 0 • T ω m-1 ,ωm,t 0 0 • • • • • T ω 1 ,ω 2 ,t 0 0 • T ω,ω 1 ,t 0 0 t∈R
and denote by F ω, -→ ω : R d → R d+1 the associated immersion. By Proposition 3.17, we have tr

R ψ (t) h (t) A ω,ω, -→ ω ,w,t dt = p•F ω, - → ω (x)=x h T ω, -→ ω (x) ψ T ω, -→ ω (x) det I -p • D x F ω, -→ ω R G ω, -→ ω ,w,T ω, - → ω (x) (x, y) dy ,
where, as in Proposition 3.17, if x ∈ R d , then T ω, -→ ω (x) denotes the opposite of the last coordinate of F ω, -→ ω (x), and

G ω, -→ ω ,w,t (x) = θ-→ ω ,w ϕ ω • κ -1 ω (x) e t 0 V •φτ (κ -1 ω (x))dτ h ω • κ ω • φ t • κ -1 ω (x) ,
properly extended by zero.

Now, denote by Q 0 the (nite) set of x ∈ R d such that p • F ω, -→ ω (x) = x and D (x) := h T ω, -→ ω (x) ψ T ω, -→ ω (x) det I -p • D x F ω, -→ ω R G ω, -→ ω ,w,T ω, - → ω (x) (x, y) dy = 0,
and by Q 1 the (nite) set of periodic orbits γ for (φ t ) t∈R such that

E (γ) := h (T γ ) ψ (T γ ) |det (I -P γ )| e γ g γ # θ-→ ω ,w ϕ ω = 0.
We will construct a bijection x → γ (x) between Q 0 and Q 1 such that, for all x ∈ Q 0 , we have D (x) = E (γ (x)). This will immediately imply that tr

R ψ (t) h (t) A ω,ω, -→ ω ,w,t dt = γ h (T γ ) ψ (T γ ) |det (I -P γ )| e γ V γ # θ-→ ω ,w ϕ ω
and we can then end the proof by summing over ω ∈ Ω and ( -→ ω , w) ∈ Ω m × W.

Let x ∈ Q 0 . Since D (x) = 0, there is ỹ ∈ R such that G ω, -→ ω ,w,T ω, - → ω (x) (x, ỹ) is non- zero. Set z = (x, ỹ), and notice that z ∈ V ω , so that κ -1 ω (z) make sense. Moreover, since G ω, -→ ω ,w,T ω, - → ω (x) (z) = 0, we must have φ jt 0 (z) ∈ U ω j for j ∈ {1, . . . , m}, and φ T ω, - → ω (x) (z) ∈ U ω . In addition, since ψ T ω, -→ ω (x) = 0, we know that T ω, -→ ω (x) ∈ t 3 + mt 0 + ]-t 1 , t
κ ω • φ T ω, - → ω (x) • κ -1 ω (z) = T ωm,ω ,t 3 t-t 3 -mt 0 • T ω m-1 ,ωm,t 0 0 • • • • • T ω 1 ,ω 2 ,t 0 0 • T ω,ω 1 ,t 0 0 (z) = T ω, -→ ω T ω, - → ω (x) (z) = F ω, -→ ω (x) + T ω, -→ ω (x) e d+1 + ỹd+1 e d+1 = p • F ω, -→ ω (x) -T ω, -→ ω (x) e d+1 + T ω, -→ ω (x) e d+1 + ỹd+1 e d+1 = z.
Consequently, there is a periodic orbit γ (x) of length T γ(x) = T ω, -→ ω (x) for (φ t ) t∈R passing through the point κ -1 ω (z). Notice that, while the point κ -1 ω (z) depends on the choice of ỹ, the orbit γ (x) does not (another choice of ỹ would only change κ -1 ω (z) into another point of the orbit γ (x)).

The map D κ -1 ω (z) φ T γ(x) is conjugated via D κ -1 ω (z) κ ω to D z T ω, -→ ω T ω, - → ω (x) .
However, in a base adapted to the decomposition of the tangent space into the stable and unstable directions and the direction of the ow, the matrix of the map

D κ -1 ω (z) φ T γ(x) is P γ(x) 0 0 1 , while the matrix of D z T ω, -→ ω T ω, - → ω (x) in the canonical basis of R d+1 is of the form p • D x F ω, -→ ω 0 * 1 .

CHAPTER 3

Thus, the linear maps P γ(x) and p • D x F ω, -→ ω have the same spectrum, which implies that det I -P γ(x) = det I -D x F ω, -→ ω . Denote by I x the set of y ∈ R such that G ω, -→ ω ,w,T ω, - → ω (x) (x, y) = 0. Then for all y ∈ I x , we have exp

T ω, - → ω (x) 0 V • φ τ κ -1 ω (x, y) dτ = exp γ(x)
g .

Moreover, the map I x y → κ -1 ω (x, y) = φ y-ỹ κ -1 ω (z) is injective (the length of I x is at most 2t 1 , and there is no periodic orbit of (φ t ) t∈R of length less than 3t 1 ), and its image is γ ∩ U-→ ω ,w (thanks to our assumption on the renement), so that a change of variable gives

R G ω, -→ ω ,w,T ω, - → ω (x) (x, y) dy = exp γ(x) V γ # (x) θ-→ ω ,w ϕ ω ,
and thus we have

E (γ (x)) = D (x) = 0, in particular γ(x) ∈ Q 1 . It remains to prove that Q 0 x → γ (x) ∈ Q 1 is a bijection. If x ∈ Q 0 then the intersection of γ (x) with U-→ ω ,
w is an interval, and thus the set κ ω γ (x) ∩ U-→ ω ,w is contained in a line perpendicular to R d × {0} (recall that κ ω is a ow box) and this line projects on x ∈ R d . Thus γ (x) determines x and consequently the map x → γ (x) is injective.

Reciprocally, if γ ∈ Q 1 then γ must intersect U-→ ω ,w on a non-empty interval that is sent by κ ω into a line perpendicular to R d × {0}, that projects on a point x ∈ R d . Choose y ∈ R such that (x, y) is the image by κ ω of some point z ∈ γ such that θ-→ ω ,w (z) ϕ ω (z) = 0.

Working as in the other case, we see that T ω, -→ ω Tγ (x, y) = (x, y), and thus p • F ω, -→ ω (x) = x and T γ = T ω, -→ ω (x). The same calculation as above implies that D (x) = E (γ) = 0, so that x ∈ Q 0 . Finally, it is clear that γ = γ (x) from the construction of γ (x): these two periodic orbits pass through the point z. Thus, the map x → γ (x) is surjective, and the proof is over. Lemma 3.36. Under the assumptions of Proposition 3.32, the series

γ h (T γ ) T # γ |det (I -P γ )| e γ V (3.48)
converges absolutely and

tr +∞ 0 h(t)L t dt = γ h (T γ ) T # γ |det (I -P γ )| e γ V .
Proof. First, use Lemma 3.35 and (the proof ) of Lemma 3.34, with the weight V replaced by Re (V ) ∞ h replaced by 1 + |h| 2 , to get that (this can also be seen using an estimates on the number of periodic orbit for (φ t ) t∈R such as [DZ16, Lemma 2.2]):

∈Z γ ψ (T γ ) exp (T γ Re (V ) ∞ ) |h (T γ )| T # γ |det (I -P γ )| < +∞.
We can then use the FubiniTonelli and monotone convergence theorems to get that

∈Z γ ψ (T γ ) exp (T γ Re (V ) ∞ ) |h (T γ )| T # γ |det (I -P γ )| = γ ∈Z ψ (T γ ) exp (T γ Re (V ) ∞ ) |h (T γ )| T # γ |det (I -P γ )| = γ exp (T γ Re (V ) ∞ ) |h (T γ )| T # γ |det (I -P γ )| < +∞.
This proves that the sum (3.48) converges absolutely and provides integrability and domination which allow us to apply Fubini's theorem and the dominated convergence theorem

to get tr +∞ 0 h(t)L t dt = ∈Z tr +∞ 0 ψ (t) h(t)L t dt = ∈Z γ ψ (T γ ) exp γ V h (T γ ) T # γ |det (I -P γ )| = γ ∈Z ψ (T γ ) exp γ V h (T γ ) T # γ |det (I -P γ )| = γ exp γ V h (T γ ) T # γ |det (I -P γ )| .
We end this section with the proof of two merely technical lemmas that will be useful in the following section to construct and study the anisotropic Hilbert spaces from Theorem 3.1.

Lemma 3.37. For all u ∈ D υ (M ) , the map R t → L t u H is measurable (with the

convention that u H = ∞ if u / ∈ H).
Proof. Let us prove rst that the map D υ (M ) u → u H is measurable. Since the inclusion of H in D υ (M ) is continuous (hence measurable) and • H is continuous on H, we only need to check that H is a measurable subset of D υ (M ) . Keeping track of the dierent steps in the denition of H, we see that it is enough to prove that L 2 loc is a CHAPTER 3 measurable subset of S υ , which is clear with the following characterization of L 2 loc :

L 2 loc = {u ∈ S υ : ∀ compact K ⊆ R d+1 , ∃C > 0, ∀ϕ ∈ S υ supported in K, | u, ϕ | ≤ C ϕ 2 }. Finally, recall that, if u ∈ D υ (M ) , the map R t → L t u ∈ D υ (M ) is measurable
(and even C ∞ ) according to Lemma 3.7 to end the proof.

Lemma 3.38. There is a continuous semi-norm N on C ∞,υ (M ) such that for all u ∈ C ∞,υ (M ) and t ∈ [-t 0 , t 0 ] we have

L t u H ≤ N (u) .
The same is true replacing L t by (L -t ) * and C ∞,υ (M ) by D υ (M ).

Proof. Since the inclusion of C ∞,υ (M ) in H is continuous and (L t ) t∈R is a group, we only need to prove that there is > 0 such that for every continuous semi-norm N 1 on

C ∞,υ (M ) there is a continuous semi-norm N 2 on C ∞,υ (M ) such that for all u ∈ C ∞,υ (M ), and t ∈ [-, ] we have N 1 (L t u) ≤ N 2 (u) . (3.49)
In fact, we only need to achieve (3.49) for N 1 of the form • κω,ϕω,κ,ν for ω ∈ Ω and κ ∈ R * + (because these semi-norms generate the topology of C ∞,υ (M )). But then, it becomes clear that (3.49) can be achieved, since the κ ω are ow boxes. The proof for the adjoint is similar.

Global space: second step

In this section, we end the proof of Theorem 3.1. We keep using the notations that we introduced at the beginning of 3.4.

Given the spaces H and H 0 and Proposition 3.32 from the previous section, the proofs of Theorem 3.1, Proposition 3.3, and Proposition 3.4 are now reduced to functional analysis, and we deal with these proofs in this last section. These proofs are split into several lemmas as follow: as far as Theorem 3.1 and Proposition 3.4 are concerned, (i) is contained in Lemma 3.40, (ii) is in Lemma 3.41, (iii) is a consequence of Lemma 3.41 and Lemma 3.8, (iv) is in Lemma 3.43, and (v) is in Lemma 3.44 (with 2t 0 instead of t 0 ). We end the section with the proof of Proposition 3.3. First of all, we dene the space H.

Denition 3.39. Thanks to Lemma 3.37, we may dene for all u ∈ D υ (M ) ,

u 2 H = t 0 0 L t u 2 H dt, (3.50)
and then dene the space

H = u ∈ D υ (M ) : u 2 H < ∞
endowed with the norm • H . Let H be the closure of C ∞,υ (M ) in H (for some υ ∈ υ, 1 1-α , where α has been dened in 3.4, we recall in particular that if υ < 2 then α < 1 2 ). Lemma 3.40. H and H are separable Hilbert spaces. The inclusion of H and H in D υ (M ) are continuous, and C ∞,υ is contained in H and H, and the inclusion is continuous.

Proof. We only need to prove the lemma for H (the statements for H immediately follows).

Notice that the map

H → L 2 [0, t 0 ] , H u → (L t u) 0≤t≤t 0 (3.51)
is an isometry. To show that H is a separable Hilbert space, we only need to prove that the image of the map (3.51) is closed . Let (u n ) n∈N be a sequence in H such that the sequence

(L t u n ) 0≤t≤t 0 n∈N converges to (v (t)) 0≤t≤t 0 in L 2 [0, t 0 ] , H .
Then there is a subset A of N and a Borel subset B of full measure in [0, t 0 ] such that, for all t ∈ B, the sequence

(L t u n ) n∈A converges to v (t) in H (in particular, it converges in D υ (M ) ). Choose t ∈ B and set u = L -t v (t ) ∈ D υ (M ) .
Then, for all t ∈ B and n ∈ A, we have

L t u n = L t-t (L t u n ) .
Letting n tend to innity, we have

v (t) = L t-t v t = L t L -t v t = L t u.
Since v ∈ L 2 [0, t 0 ] , H , this implies that u ∈ H, and thus the image of H under the map (3.51) is closed, so that H is a Hilbert space.

To prove that the inclusion of We now prove that H has the property that H 0 missed: the operator L t for t ≥ 0 is bounded from H to itself.

H in D υ (M ) is continuous, just notice that if ϕ ∈ C ∞,υ (M ) then u, ϕ = 1 t 0 t 0 0 L t u, (L -t ) * ϕ dt,
Lemma 3.41. For every t ≥ 0, the operator L t is bounded from H to itself. Moreover, (L t ) t≥0 is a strongly continuous semi-group of operators on H.

CHAPTER 3

Proof. If u ∈ H and t ≥ t 0 then we have

L t u 2 H = t 0 0 L τ L t u 2 H dτ = t 0 0 L t L τ u 2 H dτ ≤ L t 2 H→ H u 2 H .
If 0 ≤ t ≤ t 0 then we have Notice that, according to Lemma 3.8, the generator of the semi-group (L t ) t≥0 is P = X + V . We prove now a lemma that allows us to go from H to H 0 and back, in order to prove that the properties that we stated for H 0 in Proposition 3.32 may be extended to H. But recall that the norm of u in H is the norm of (L t u) 0≤t≤t 0 in the space L 2 [0, t 0 ] , H .

L t u 2 H = t 0 L t 0 L τ u 2 H dτ + t 0 t L τ u 2 H dτ ≤ 1 + L t 0 2 H→ H u 2 H . Thus L t is bounded from H to itself, but since L t sends C ∞,υ (M ) into C ∞,υ (
Thus, for all n ∈ N, the norm of (L t u) nt 0 ≤t≤(n+1)t 0 in the space L 2 [0, t 0 ] , H is smaller than L t 0 n H→H u H . Thus if Re (z) > ln ( L t 0 H→H ), then, by CauchySchwarz inequality, there is a constant C > 0 such that the L 1 norm of t → e -zt L t u is smaller than C u H→H . Thus (z -P ) -1 is bounded from H to H.

We are now ready to prove that the spectrum of P = X + V acting on H is discrete.

Lemma 3.43. The spectrum of P acting on H is made of isolated eigenvalues of nite multiplicity and it coincides with the Ruelle spectrum of P (multiplicity taken into account).

Proof. According to Lemmas B.3 and 3.41, it is enough to prove that the spectrum of P acting on H is made of isolated eigenvalues of nite multiplicity (recall Denition 7 of the Ruelle spectrum). Let z ∈ C be such that Re z 1. Since P is the generator of a strongly continuous semi-group, z belongs to the resolvent set of P . From [Kat66, Problem 6.16 p.177], we see that we only need to prove that the essential spectral radius of (z -P ) -1 is zero (see Denition 2).

To do so, let χ : R * + → [0, 1] be a compactly supported C ∞ function such that χ(t) = 1 if t ≤ 2t 0 . Then, according to [Kat66, Problem 1.15 p.487], for all n ≥ 1 we have, provided that Re z 1:

(z -P ) -n = 1 (n -1)! +∞ 0 e -zt t n-1 L t dt = 1 (n -1)! +∞ 0 χ(t)e -zt t n-1 L t dt + 1 (n -1)! +∞ 0 h n (t)L t dt, (3.52)
where the function h n : R * + → C is dened by h n (t) = (1 -χ(t)) e -zt t n-1 . Set also hn (t) = zh n (t + t 0 ) + h n (t + t 0 ), so that for all t ∈ R * + we have (n-1)! for some constant C > 0. With (3.52), it follows then from Hennion's argument [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux lipchitziens[END_REF] based on Nussbaum formula [START_REF] Roger | The radius of the essential spectrum[END_REF] that the essential spectral radius of (z -P ) -1 on H is less than

lim inf n→+∞ C(2t 0 ) n (n -1)! 1 n = 0.
We can now give the proof of the most interesting property of the Hilbert space H. Proof.

h (t) L t dt = L t 0 • +∞ t 0 h (t) L t dt • (z -P ) -1 : H → H is compact.
In order to identify the non-zero spectrum of (3.54), we denote by f the function dened by f (z) = Lap(h)(-z) and by A the operator (3.54). If λ ∈ C denotes by E λ the generalized eigenspace of P associated to λ and, if λ = 0, by F λ the generalized eigenspace of A associated to λ. We want to prove that for all µ ∈ C \ {0} we have

F µ = λ∈C f (λ)=µ E λ ,
which is a more precise statement that our claim on the eigenvalues of A. Let λ ∈ σ (X) be such that f (λ) = 0. Since P commutes with L t for t ≥ 0, it commutes with A so that E λ is stable by A. We denote by P and A the endomorphisms of E λ induced respectively by P and A. Since E λ is nite-dimensional (according to Lemma 3.43), the operator P is bounded, and we may dene for t ≥ 0 the operator e t P on E λ . Then, e t P is nothing else where f ( P ) is meant in the sense of the holomorphic calculus of bounded operators (we may develop e t P in power series). Since σ( P ) = {λ} by denition of E λ , it follows that σ( A) = {f (λ)}, which gives

E λ ⊆ F f (λ) .
Reciprocally, let µ ∈ σ (A) \ {0}. From the equality

P A = - +∞ 0 h (t)L t dt,
we see that the range of A is included in the domain of X. In particular, F µ is contained in the domain of P and thus P induces a bounded operator on the nite dimensional space F µ . Applying as above the holomorphic calculus of bounded operators, we get that Finite order of the dynamical determinant for Gevrey Anosov ows This last chapter is dedicated to the proof of Theorem 10. In 4.1, we gather some facts about Gevrey classes and analytic FBI transforms. This is basically an abridged version of Chapters 1 and 2 from [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]. In order to keep the length of this thesis reasonable, most of the results in 4.1 will be stated without proof. However, we will always give precise references to [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF] where the interested reader can nd all the required proofs or references. The section 4.2 is dedicated to the detailed proof of Theorem 10 using the tools introduced in 4.1. Finally, in 4.3, we expose some other applications of FBI transforms method to the study of Anosov ows. For the same reasons as above, we will refer to [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF] for proofs. The content of 4.2 and 4.3 correspond to the third chapter of [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF].

F µ ⊆ λ∈C f (λ)=µ
Mind that in this chapter we will use, as it is common in the semi-classical analysis literature, an implicit parameter h > 0, thought of as tending to 0. Whenever needed, we will assume that h is small enough.

Crash course: Analytic FBI transform and Gevrey differential operators

In this rst section, we recall the results from the rst two chapters of [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF] that are needed for the proof of Theorem 10.

In 4.1.1, we recall some basic facts about Gevrey and analytic classes of regularity.

These results may be found in 1.1 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]. We will not need the other results from the rst chapter of [BJ20] since we will not discuss Gevrey pseudo-dierential operators here and because we will admit all the required results from the second chapter of [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF] such as the existence of a global analytic FBI transform on a compact manifold.

In 4.1.2, we dene a global analytic FBI transform on a compact analytic manifold.

Admitting that such an object exists, we recall then its basic properties. We also investigate CHAPTER 4

the notion of complex Lagrangian deformation which is central in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF].

In 4.1.3, we explain how a global analytic FBI transform may be used to study dierential operators with Gevrey coecients, stating the results that will be fundamental in the proof of Theorem 10.

Remark 4.1. The FBI transform methods that we expose here are extensions of the work of Heler and Sjöstrand [START_REF] Heler | Résonances en limite semi-classique[END_REF][START_REF] Sjöstrand | Density of resonances for strictly convex analytic obstacles[END_REF] in the analytic category. These methods had already been investigated in the Gevrey category by Lascar and Lascar [START_REF] Lascar | FBI transforms in Gevrey classes[END_REF]. However, all these versions of this method contain limitations that we had to overcome in order to prove Theorem 10. The main problem was that they only consider compactly supported

Lagrangian deformations (which amounts to take G compactly supported in Denition 4.14 below). Consequently, an important part of [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF] is dedicated to the extension of the methods of Heler and Sjöstrand and to the investigation of certain micro-local techniques in the Gevrey category. For more details, see the introduction of [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF].

As we were working on this project, Galkowski and Zworski were elaborating a very similar extension of the work of Heler and Sjöstrand [START_REF] Galkowski | An introduction to microlocal complex deformations[END_REF][START_REF] Galkowski | Viscosity limits for 0th order pseudodierential operators[END_REF]. However, their work is restricted to the real-analytic case and they only work on tori.

Gevrey functions and ultradistributions on manifolds

In 2.1, we introduced the notion of Gevrey regularity see in particular Example 2.1. Let us now explore a bit further Gevrey classes.

Denitions

Let s ≥ 1 be xed. Let U be an open subset of R n . We recall from Example 2.1 that a function f : U → C is said to be s-Gevrey (or, for short, G s ) if f is C ∞ and if, for every compact subset K of U , there are constants C, R > 0 such that, for all α ∈ N n and x ∈ K, we have

|∂ α f (x)| ≤ CR |α| α! s . (4.1)
The constant R in (4.1) may be interpreted as the inverse of a (Gevrey) radius of convergence. Recall that when s = 1, this describes the class of real-analytic function on U . When s > 1, the class of G s functions is non-quasi-analytic according to Theorem 2.3: it contains compactly supported functions. We will denote by G s (U ) the space of G s functions on U and by G s c (U ) the space of compactly supported elements of G s (U ). The non-quasi-analyticity of G s when s > 1 implies in particular that there are G s partitions of unity.

The denition above extends immediately to the case of Banach valued functions. A function f from U to some Banach space B is said to be G s if (4.1) holds with the modulus replaced by the norm of B. With this denition, the class of G s functions is closed under composition, as was proved by Gevrey in his original paper [START_REF] Gevrey | Sur la nature analytique des solutions des équations aux dérivées partielles[END_REF]. In fact, the class of G s functions is very well-behaved and, when s > 1, quite exible: for instance, it is Cartesian closed, closed under dierentiation, solving ODEs, Implicit Function Theorem, etc, and there are versions of Borel's and Whitney's theorem for G s functions (see [START_REF] Kriegl | The convenient setting for non-quasianalytic Denjoy-Carleman dierentiable mappings[END_REF] and references therein for details).

Since the class G s is stable by composition and inversion, we have a natural denition of a G s structure on a manifold: a G s manifold is a Hausdor topological space with countable basis endowed with a maximal G s atlas. Here, a G s atlas is dened to be an atlas with G s change of charts (notice that we retrieve the usual notion of real-analytic manifold when s = 1). As usual, if M and N are two G s manifolds, then a map f : M → N is said to be G s if it is G s in charts. Since the class G s satises the Implicit Function Theorem, most elementary results from dierential geometry are easily checked to be true in the G s category. In particular, there is a well-dened notion of G s (vector) bundle, and each usual bundle associated with M (tangent, cotangent, etc) admits a natural G s structure. As a consequence, it makes sense to say that a vector eld over M is G s . Remark 4.2. Of course, a real-analytic manifold has a natural structure of G s manifold, since real-analytic maps are G s . As pointed out in [START_REF] Lascar | FBI transforms in Gevrey classes[END_REF], all G s manifolds may be described in this way. Indeed, there is a Gevrey version of the famous Whitney's Embedding Theorem [START_REF] Whitney | Dierentiable manifolds[END_REF] : every G s compact manifold is G s -dieomorphic to a real-analytic submanifold of an Euclidean space. The adaptation of the proof of Whitney's Theorem to our setting is straightforward. Since the Inverse Function Theorem holds in the G s category, it suces to follow the lines of the proof of [Hir94, Theorem 7.1], replacing C ∞ by G s at every step. This gives an analytic structure on a compact G s manifold, compatible with the G s structure.

We want now to dene ultradistributions on a G s manifold M . To do so, we need to give a structure of topological vector space to the space G s c (M ) of compactly supported G s functions on M . If (U, κ) is a G s chart for M , and K a relatively compact subset of U , then we dene for every R > 0 and function f , innitely dierentiable on a neighbourhood of K, the semi-norm (n denotes the dimension of M )

f s,R,K := sup x∈κ(K) α∈N n ∂ α f • κ -1 (x) R |α| α! s . (4.2)
We extend this denition to any relatively compact subset K of M by covering K by a nite number K 1 , . . . , K N of compact sets included in some domains of charts and setting

f s,R,K := N j=1 f s,R,K j .

CHAPTER 4

Then, if K is a relatively compact subset of M and R > 0, we dene E s,R (K) to be the Banach space of functions f ∈ C ∞ (M ), supported in K, such that f s,R,K < +∞, endowed with the norm • s,R,K . For s = 1, consider this denition as temporary, as we will give a more practical but equivalent scale of spaces of real analytic functions below.

Notice that these spaces heavily depend on the choices of charts that we made above.

It is, however, not the case of the inductive limit, when U is an open subset of M ,

G s c (U ) := lim -→ K U lim -→ R>0 E s,R (K) .
Here, the rst limit is taken over compact subsets K of U , and the inductive limit is taken in the category of locally convex topological vector spaces. Notice that the underlying set of this limit is indeed the set of compactly supported G s functions on U . In particular, when s = 1, if U is not compact itself then G s c (U ) is trivial. In the case that M itself is compact, we will write G s (M ) for G s c (M ). We dene the space U s (U ) of ultradistributions on U to be the strong dual of G s c (U ). Then, by [START_REF] Grothendieck | Sur la complétion du dual d'un espace vectoriel localement convexe[END_REF], we see that U s (U ) identies with the projective limit

U s (U ) = lim ← - K U lim ← - R>0 E s,R (K) .
In particular, U s (U ) is a Fréchet space. It is a Roumieu-type space of ultradistributions.

More details on Roumieu and Beurling spaces, can be found in [START_REF] Bonet | Whitney's extension theorem for nonquasianalytic classes of ultradierentiable functions[END_REF]. The support of an ultradistribution is dened as in the case of usual distributions, and the spaces of compactly supported distributions on U is denoted by U s c (U ). The topology of U s c (U )

only matters for Theorem 2.27, and we can take for instance the weak-star topology. An element of U 1 (M ) or more generally of E 1,R (M ) for some large R ≥ 1 will be referred to as a hyperfunction.

Remark 4.3. The fact that the limit (4.1.1) does not depend on the choices of charts in the denition of the norms • s,R,K follows from the stability by composition of the class G s . Indeed, if we denote by ( • s,R,K ) R>0 the family of norms that is obtained with another choice of charts, one sees (applying Faa di Bruno's formula or adapting the original proof of Gevrey [START_REF] Gevrey | Sur la nature analytique des solutions des équations aux dérivées partielles[END_REF]) that there is a constant C > 0 such that for every R > 0 we have

• s,C max(1,R),K ≤ C • s,R,K and • s,C max(1,R),K ≤ C • s,R,K . (4.3)
In particular, when the manifold M is an open subset of R n , we can (and will) choose the chart κ in the denition (4.2) to be the inclusion in R n .

Remark 4.4. Let s ≥ 1. In order to discuss perturbations of G s Anosov ows in 4.3, we need to dene a topology on the space of G s sections of a G s vector bundle. Let p : F → M denote a real G s vector bundle on M (the case of complex vector bundle is similar). Let (U, κ) be a G s chart for M and (κ, Ψ) : p -1 (U ) → κ (U ) × R d be a trivialization for p : F → M . Then, if K is a relatively compact subset of U and R > 0, we dene for every

C ∞ section f of F the semi-norm f s,R,K = sup x∈κ(K) α∈N n ∂ α Ψ f • κ -1 (x) R |α| α! s ,
where • denotes any norm on R d . Using this semi-norm to replace (4.2), the case of sections of the vector bundle F is dealt with as the case of the trivial line bundle over M .

In particular, when M is a compact manifold, we have a denition the spaces E s,R (M ; F ), for R > 0, and a topology on the space G s (M ; F ) of G s sections of F .

The particular case of real-analytic functions

We want now to rewrite the denitions of the previous paragraph in a way that may be more intuitive, in the case s = 1. Indeed, it may seem more natural to describe real analytic functions as restrictions of holomorphic functions. If K is a compact subset of R n , then we see that a smooth function f dened on a neighbourhood of K such that f 1,R,K < +∞ admits a holomorphic extension to a complex neighbourhood of K of size (CR) -1 (for some C > 0 that does not depend on R), the L ∞ norm of this extension being bounded by C f 1,R,K . Reciprocally, if f admits a bounded holomorphic extension to a complex neighbourhood of size CR -1 , then f 1,R,K is nite and controlled by the L ∞ norm of the extension (independently on R). We will now explain how this remark generalizes to the case of compact manifolds.

Let then M be a compact real-analytic manifold of dimension n. By a result of Bruhat and Whitney [START_REF] Whitney | Quelques propriétés fondamentales des ensembles analytiques-réels[END_REF], the manifold M admits a complexication M . That is, M is a holomorphic manifold of complex dimension n endowed with a real-analytic embedding M ⊆ M , such that M is a totally real submanifold of M . This means that at each p ∈ M , we have T p M ∩iT p M = {0}. It follows then that if N is a complex manifold and f : M → N is a real-analytic map, then f extends to a holomorphic map from a neighbourhood of M in M to N . In particular, if M is another complexication for M then the identity of M extends to a biholomorphism between a neighbourhood of M in M and a neighbourhood of M in M . Here, the inductive limit (in the category of locally convex topological vector spaces) is CHAPTER 4 over all the complexications 1 M of M . This coincides with (4.1.1) when s = 1. This denition may be quite appealing because it is very intrinsic. However, we will rather use a more concrete description of G 1 (M ), that boils down to choosing a basis of complex neighbourhoods for M .

We will use particular complexications of M called Grauert tubes. The notion of Grauert tube rst appeared in [START_REF] Grauert | On Levi's problem and the imbedding of real-analytic manifolds[END_REF], but we will rely on the exposition from [START_REF] Guillemin | Grauert tubes and the homogeneous Monge-Ampère equation[END_REF].

First, according to [START_REF] Morrey | The analytic embedding of abstract real-analytic manifolds[END_REF], there is a real-analytic embedding of M into an Euclidean space. Hence, we may choose a real-analytic Riemannian metric g on M . According to [START_REF] Whitney | Quelques propriétés fondamentales des ensembles analytiques-réels[END_REF], there exists a complexication M of M endowed with an anti-holomorphic involution z → z such that M is the set of xed point of z → z. Then, since the square of the distance induced by g on M is real-analytic near the diagonal, it extends to a holomorphic function on a neighbourhood of the diagonal of M × M in M × M . Following [START_REF] Guillemin | Grauert tubes and the homogeneous Monge-Ampère equation[END_REF], we dene ρ on M (up to taking M smaller) by

ρ(z) = - 1 4 d(z, z) 2 .
From [START_REF] Guillemin | Grauert tubes and the homogeneous Monge-Ampère equation[END_REF], we know that ρ denes a strictly plurisubharmonic function on M such that M = {z ∈ M : ρ(z) = 0}. Then, if > 0 is small, we dene the Grauert tube (M ) as the sublevel set of ρ:

(M ) := z ∈ M : ρ(z) < 2 .

(4.4)

Notice that, since ρ is strictly plurisubharmonic, the Grauert tube (M ) is strictly pseudoconvex. Moreover, the real (1, 1)-form i∂ ∂ρ is Kähler and the associated hermitian form coincides with g on M . We shall consequently still denote this hermitian form by g.

Using the notion of Grauert tube, we can replace the spaces E 1,R (M ) dened above with a more convenient scale. If R > 0 is large enough, denote by E 1,R (M ) the space of bounded holomorphic functions on (M ) 1/R (endowed with the sup norm). The spaces E 1,R (M ) dened in this way do not need to coincide with the previous version, but they give rise to the same inductive limit (in the category of locally convex topological vector spaces):

G 1 (M ) = lim -→ R→+∞ E 1,R (M ) .
Hence, we will always assume that we use this denition of the spaces E 1,R (M ) in the case s = 1. To see that we get the same topology on G 1 (M ) as previously, work as in Remark 4.3 (using Cauchy's and Taylor's formulae to prove an estimate similar to (4.3)).

1 We get rid of the set-theoretic complications here in the usual way: the cardinal of a complexication for M is at most 2 ℵ 0 .

Let us point out that there is another way to describe Grauert tubes. Indeed, since the Riemannian metric g is real-analytic, so is its exponential map. Consequently, if x is a point in M , the exponential map exp x : T x M → M extends to a holomorphic map, still denoted by exp x , from a neighbourhood of 0 in T x M ⊗ C to M . Then, the map

(x, v) → exp x (iv) (4.5)
denes a real analytic dieomorphism between a neighbourhood of the zero section in T M and a neighbourhood of M in M . For > 0 small enough, under the map in (4.5), the Grauert tube (M ) is the image of

(x, v) ∈ T M : g(v, v) < 2 .
With this description of the Grauert tube (M ) , we see that the projection T M → M induces a real-analytic projection from (M ) to M , that we shall denote by Re. We dene also the function |Im| : (M ) → R + as the square root of ρ. We will sometimes, slightly abusively, write |Im z| instead of |Im| (z).

Since M is compact, we could have chosen any decreasing basis of neighbourhoods for M in M to dene the spaces E 1,R (M ). However, we will need to consider real-analytic functions dened on T * M (for instance symbols) or more generally on products of the type M N 1 × (T * M ) N 2 . Since these manifolds are non-compact, the choice of a complex neighbourhood for T * M becomes non-trivial. Since we want to consider symbols on T * M , it seems natural to introduce the Grauert tubes for the KohnNirenberg metric g KN .

Recall that g KN is dened in the following way. The LeviCivita connexion associated with g gives a splitting T T * M = V ⊕ H into vertical and horizontal bundles, where both subbundles are identied with T M , so that we can dene

g KN (x, ξ) := g H (x, ξ) + 1 1 + |ξ| 2 x g V (x, ξ).
In charts, it is uniformly equivalent to its at version

g at KN = dx 2 + 1 1 + |ξ| 2 dξ 2 .
The curvature of g KN is bounded, and so are all its covariant derivatives, and one can check that it admits Grauert tubes, that roughly looks like cones at innity. More precisely, the cotangent space T * M of M is a complexication of T * M . Notice that there is a natural inclusion of T * M ⊗ C into T * M and that the anti-holomorphic involution that xes T * M is given on T * M ⊗ C by (x, ξ) → (x, ξ). As above, we nd a strictly plurisubharmonic function ρ KN dened on a neighbourhood of T * M in T * M .

CHAPTER 4

Then, mimicking (4.4), we set for > 0 small enough

(T * M ) := α ∈ T * M : ρ KN (α) < 2 .
As in the compact case, |Im α| will denote the square root of ρ KN (α). This gives a rough but tractable idea of the shape of Grauert tubes in local coordinates.

Let us discuss some others notations. Since x → g x is a Kähler metric, the map α → g αx (α ξ , α ξ ) is real analytic and non-negative. On the other hand, we can consider the holomorphic extension g of g, so that α → gαx (α ξ , α ξ ) is a holomorphic map. With the determination of the square root positive on the reals, we dene for α = (α x , α ξ ) in T * M the Japanese brackets α = 1 + gαx (α ξ , α ξ ) and |α| = 1 + g αx (α ξ , α ξ ).

(4.6) Hence, α is holomorphic in α, while |α| is not. However, notice that on a Grauert tube (T * M ) , for > 0 small enough, the positive quantities |α| , | α | and Re α are uniformly equivalent. Notice also that we may dene a KohnNirenberg metric on T * M (since T * M identies with the cotangent bundle of M seen as a real-analytic manifold).

Almost analytic extensions

Let (M, g) be a compact real-analytic Riemannian manifold. To study deformations in the Grauert tube of M (or of T * M ), we will make extensive use of almost-analytic extensions of smooth functions on M . The notion of almost analytic extension was introduced by Hörmander [Hör69] and then by Nirenberg [START_REF] Nirenberg | A proof of the Malgrange preparation theorem[END_REF]. It has become a very common notion in microlocal analysis, and are essential for instance in [START_REF] Melin | Fourier integral operators with complex-valued phase functions[END_REF].

Recall that if f is a C ∞ function on M then an almost-analytic extension for f is a compactly supported C ∞ functions f on some (M ) that coincides with f on M and such that ∂ f vanishes at all orders on M . It is classical that such a f exists [Zwo12, Theorem 3.6]. While this is hardly surprising, it will be crucial in our analysis that if f is G s then f may be chosen G s as well. This will allow us to make the atness of ∂ f near M quantitative with Lemma 4.7 below.

It seems to be folklore that the fact that G s functions admit a G s almost analytic extension may be deduced from results of Carleson on universal moment problems [START_REF] Carleson | On universal moment problems[END_REF].

Since we were not aware of any existing proof in the literature, we provided one in [BJ20, Lemma 1.1].

Lemma 4.6. Given a Grauert tube (M ) ⊃ M , for each s > 1, there exists a constant C s and a compact subset K ⊆ (M ) so that for all R > 0, there exists a bounded map f → f from E s,R (M ) to E s,Cs max(1,R) (K), such that f is an almost-analytic extension for f .

In order to apply Lemma 4.6, we need to investigate the way a Gevrey function can be at. To do so, we can apply the sommation au plus petit terme, a method for regularizing certain divergent series that is particularly well suited for Taylor series of Gevrey functions.

The interested reader may refer to [START_REF] Ramis | Séries divergentes et théories asymptotiques[END_REF] for details and historical references and to [BJ20, Lemma 1.3] for a proof of the following lemma.

Lemma 4.7. Let U be an open subset of R n and K a compact and convex subset of U .

Then for every s > 1, there is a constant C > 0 such that for every R ≥ 1, x ∈ K and f ∈ C ∞ (U ), if the quantity f s,R,K dened by (4.2) is nite, and all derivatives of f at x vanish, then for every y ∈ K we have

|f (y)| ≤ C f s,R,K exp - (R |x -y|) -1 s-1 C . (4.7)
Remark 4.8. Let us explain how Lemma 4.7 allows to control the size of an almost analytic extension of a Gevrey function. Let s > 1 and R ≥ 1. Let M be a compact real-analytic manifold. According to Lemma 4.6, for some > 0, R ≥ 1 and compact subset K ⊆ (M ) , there is a continuous map f → f from E s,R (M ) to E s,R (K) such that f is an almost analytic extension for f . Hence, if f ∈ E s,R (M ) the 1-form ∂ f vanishes at innite order on M . Then, applying Lemma 4.7 in charts, we nd that for z ∈ (M ) we have

∂ f (z) ≤ C s,R f s,R,M exp - 1 C s,R |Im z| 1 s-1 . (4.8)
Recall that |Im z| has been dened on (M ) . Here, we use any metric on the cotangent bundle of (M ) to measure the size of ∂ f . Moreover, since the derivatives of f are Gevrey too, one can easily see that (4.8) can in fact be improved to a Gevrey estimates (in charts).

In G s micro-local analysis, the error terms that are allowed are those that decay like exp(-1/(Ch) 1/s ), for some C > 0, when the small parameter h tends to 0. Hence, we see that it only makes sense to consider the value of the almost analytic extension of a G s function at a distance at most C -1 h 1-1/s of the real if we do not want to produce uncontrolled errors.

CHAPTER 4 Analytic and Gevrey symbols, admissible phase

In order to dene an analytic FBI transform in 4.1.2, we need the notion of analytic symbols on a compact real-analytic manifold M . To do so, we use the notations that we introduced above. Denition 4.9. Let M be a compact real-analytic manifold. Let m ∈ R. An analytic symbol of order m on T * M is a real-analytic function a : T * M → C such that there are > 0 and C > 0 such that a admits a holomorphic extension to (T * M ) that satises for

every α ∈ (T * M ) |a(α)| ≤ C |α| m .
We denote by S 1,m (T * M ) the set of analytic symbols of order m on T * M . We say that

a symbol a ∈ S 1,m (T * M ) is elliptic in S 1,m (T * M ) if there is C > 0 such that for every α ∈ T * M we have |a(α)| ≥ 1 C α .
Denition 4.11 of real-analytic symbols extends mutatis mutandis to product of the form (T * M ) N 1 × M N 2 , or reasonable subsets of such a product. Since we will not discuss here Gevrey pseudo-dierential operators, we do not need to dene the most general Gevrey symbols. We only need to consider symbols for Gevrey dierential operators, that is polynomials on the bers of T * M with Gevrey coecients. More concretely, such a symbol corresponds to a Gevrey section of the vector bundle m k=0 (T M ⊗ C) ⊗k for some m ∈ N, and it has a G s almost analytic extension which is a section of the complex analytic vector bundle m k=0 T M ⊗k where M is a complexication for M . Let us put it in more concrete terms in the case of dierential operators of order 1, our principal interest here. If X is a G s vector eld on M and V : M → C be a G s function, let P = X + V . Then the principal symbol of the semi-classical dierential operator hP may be dened by p(α) = iα ξ (X(α x ))

for α = (α x , α ξ ) ∈ T * M . Then X admits a G s almost analytic extension X which is a section of T M (where we recall that M denotes a complexication for M ). Then the almost analytic extension for p may be dened on T * M by p(α) = iα ξ X(α x ) . This is a function on T * M , polynomial in every ber, with G s almost analytic coecients.

Before being able to dene what an analytic FBI transform is, we need a last denition. Denition 4.10. Let M be a compact real-analytic manifold. A phase on M is a holomorphic symbol Φ of order 1 dened for (α, y)

∈ (T * M ) × (M ) with d(α x , y) < δ (for some , δ > 0) such that (i) if (α, y) ∈ T * M × M then the imaginary part of Φ(α, y) is non-negative; (ii) Φ(α, α x ) = 0 for α = (α x , α ξ ) ∈ T * M ; (iii) for α ∈ T * M , we have d y Φ(α, α x ) = -α ξ .
We say that Φ is an admissible phase if it satises in addition the coercivity condition:

(iv) there is a constant c > 0 such that if α, y are real and Φ(α, y) is dened, then Im Φ(α, y) ≥ c α d (α x , y) 2 .

Global analytic FBI transform on a compact manifold Denition of an FBI transform

We explain now what we mean by a global analytic FBI transform on a manifold and give a short introduction to the notion of complex Lagrangian deformation of the cotangent bundle. The interested reader may refer to [BJ20, 2.1] and to the introduction of the same paper for historical references.

We x a compact real-analytic manifold M of dimension n ∈ N. Without loss of generality, we may endow M with a Riemannian metric g, so that the machinery described in 4.1.1 is available. We denote by M a complexication for M . We dene now the notion of analytic FBI transform.

Denition 4.11. An analytic FBI transform is an operator T .

, from C ∞ (M ) to D (T * M ), such that for some C, 0 , 1 , η > 0, the Schwartz kernel K T of T is holomorphic in (M × T * M ) 0 and for (x, α) = (x, (α x , α ξ )) therein satises (a) if d(x, α x ) > 1 /2, then |K T (α, x)| ≤ Ce -η |α| h . (b) for d(x, α x ) ≤ 1 , K T (α, x) -e i Φ T (α,x) h a(α, x) ≤ Ce -η |α| h , ( 4
We can also express the Liouville 1-form:

θ = ξ • dx -η • dy + i(ξ • dy + η • dx).
Following Sjöstrand, we will denote with an I objects of symplectic geometric dened through the use of ω I . For example, the I-Hamiltonian (i.e. w.r.t. ω I ) vector eld of a C 1 function f is given in the coordinates above by Here, we assume that G is a real-valued function so that G 0 := h -1+1/s G is a symbol (in the usual KohnNirenberg class of symbol) on (T * M ) 0 of order 1/s, supported in some (T * M ) 1 with 1 < 0 . Additionally, we require that (we use the covariant derivatives associated to the KohnNirenberg metric to measure the derivatives of G 0 )

H ω I f = ∇ η f • ∂ ∂x -∇ x f • ∂ ∂η + ∇ ξ f • ∂ ∂y -∇ y f • ∂ ∂ξ , (4.10) so that df = ω I (•, H ω I f ). One nds directly that T * M is a I-Lagrangian submanifold of (T * M ) 0 . The idea of
sup α∈(T * M ) 0 k≤3 ∇ k G 0 (α) KN |α| 1 s ≤ τ 0 .
(4.12) Remark 4.15. If G is as in Denition 4.14, then one easily sees that the vector eld H ω I G is complete, so that we can dene a (τ 0 , s)-adapted Lagrangian Λ by the formula (4.11) (the manifold Λ is then I-Lagrangian since exp(H ω I G ) is an I-symplectomorphism). Notice also that the assumptions on the symbol G impose that it depends on h, but in a uniform fashion as h → 0 (in particular, Λ is uniformly smooth with respect to the Kohn CHAPTER 4

Nirenberg metric when h tends to 0). In the applications, the dependence of G on Λ will be fairly explicit since G will be of the form τ 0 h 1-1/s G 0 with G 0 of order 1/s satisfying the assumptions of Denition 4.14 with τ 0 = 1.

One may notice that if τ 0 ≥ τ 1 and s ≥ s then any (τ 0 , s)-adapted Lagrangian is also (τ 1 , s )-adapted. The notion of (τ 0 , s)-adapted Lagrangian is tailored so that it makes sense to restrict the almost analytic extension of the principal symbol of a dierential operator with Gevrey coecients (as dened in 4.1.1, see also Remark 4.8) to a (τ 0 , s)-adapted Lagrangians when τ 0 is small. More precisely, if G is as in Denition 4.14, it follows from the local expression 4.10 for H ω I G that the norm of H ω I G for the KohnNirenberg metric is O τ 0 h 1-1/s |α| 1/s-1 .

This essentially proves

Lemma 4.17. Let s ≥ 1 and T ≥ 0. There is a constant C > 0 such that, for every

τ 0 ∈ [0, T ], if Λ is a (τ 0 , s)-adapted Lagrangian and α ∈ Λ then | Im α| ≤ Cτ 0 h 1-1 s |α| 1 s -1
In particular, for every > 0, there is a τ 1 > 0 such that, for every τ 0 ∈ [0, τ 1 ], any (τ 0 , s)-adapted Lagrangian is contained in (T * M ) .

In particular, we see that if T is an analytic FBI transform and Λ is a (τ 0 , s)-adapted

Lagrangian manifold with τ 0 small enough, then, according to Remark 4.12, if u is a hyperfunction on M then T u is dened on Λ. We dene then the transform T Λ associated by restriction: T Λ u = (T u) |Λ .

In order to dene weighted L 2 spaces on adapted Lagrangians, we need to dene a volume form on such Lagrangians. To do so, we use the following lemma, which is proven by a perturbative argument (see [BJ20, Lemma 2.2]) Lemma 4.18. Let s ≥ 1. There exists T > 0 and a constant C > 0 such that, for every

τ 0 ∈ [0, T ],
if Λ is a (τ 0 , s)-adapted Lagrangian then the restriction of ω R to Λ (that we shall also denotes by ω R ) is symplectic.

From now on, if Λ is an adapted Lagrangian, we will just denote by dα the 2n-form ω n R /n!, which induces a volume form on Λ. We will denote as usual the corresponding duality pairing f, g Λ = Λ f g dα.

(4.13)

Using the volume form dα, we may also dene adjoint analytic FBI transforms associated to Λ. If S is an adjoint analytic transform with kernel K S and Λ is a (τ 0 , s)-adapted

Lagrangian manifold with τ 0 small enough, then we dene the operator S Λ by

S Λ v(x) = Λ K T (x, α)v(α)dα,
where x ∈ M and v is a measurable function on Λ that decays fast enough we will extend later this denition to more general v.

The natural space in our setting will not be L 2 (Λ, dα) but rather L 2 (Λ, e -2H/h dα),

where H is an action associated with Λ, solving dH = -Im θ |Λ .

Analytic FBI transform and Gevrey regularity

Since an FBI transform is some kind of generalization of the Fourier transform, we expect that regularity of a hyperfunction u on M may be described in terms of decay properties of the transform T u. This is indeed true and we explain it now.

Let T be an analytic FBI transform, S be an adjoint analytic transform and Λ be a (τ 0 , 1)-adapted Lagrangian with τ 0 small enough. We recall the associated transforms T Λ and S Λ that we introduced above. We need spaces to describe the decay or growth of a

function on Λ. Let s ≥ 1 and r ∈ R. If f is a function from Λ to C, let f Λ,s,r := sup α∈Λ |f (α)| e -r |α| 1 s ∈ R + ∪ {+∞} .
While the statements of Lemmas 4.21 and 4.22 may seem technical, their proofs are relatively easy. Lemma 4.22 follows from bounds on the growth of Gevrey norms of y → K T (α, y) and y → K S (y, α) when α tends to +∞ along Λ. These bounds may be obtained for instance by applying Cauchy's Formula on a polydisc with shrinking radius (see [BJ20, Lemmas 2.5 and 2.6]). The rst point in Lemma 4.21 may be proved by an application of Gevrey non-stationary phase method and the second point follows by a duality argument (see [BJ20, Lemma 2.4 and Proposition 2.2]).

Inversion formula

It follows from Lemmas 4.21 and 4.22 that if T is an analytic FBI transform and S is an adjoint analytic FBI transform, then the composition2 ST makes sense as an operator from E 1,R 0 (M ) to E 1,R 1 (M ) for any R 1 , and R 0 large enough depending on R 1 . It would be useful to be able to retrieve an ultradistribution u from its FBI transform T u if possible in an explicit way. We will use the following result to do so. Proposition 4.23 (Theorem 6 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Let M be a compact real-analytic manifold and h > 0 be small enough. Then there is an analytic FBI transform T on M such that T * T is the identity3 .

The proof of Proposition 4.23 is given in [BJ20, 2.1.3]. Let us give a short glimpse of the ideas behind this demonstration. First of all, from Proposition 4.13, we know that there is an analytic FBI transform T 0 on M . Then, from an application of the stationary phase method, we may prove that the self-adjoint operator T * 0 T 0 is an elliptic analytic pseudo-dierential operator of order 0 (in a sense dened in [BJ20, 1.3]). Then, we apply a suited functional calculus to construct an inverse square root for T * 0 T 0 : this is a selfadjoint analytic pseudo-dierential operator P that satises P T * 0 T 0 P = I (the operator P is a priori only dened for h small enough). It remains to see by another application of the holomorphic stationary phase method that T = T 0 P is an analytic FBI transform to end the proof of Proposition 4.23.

From now on, we x a compact real-analytic manifold M and let T be an analytic FBI transform given by Proposition 4.23 assuming that h > 0 is small enough. We will also denote by S the adjoint T * of T . Using Stokes' Formula to shift a contour, we may generalize Proposition 4.23 to FBI transforms associated to Lagrangian deformations of T * M . Proposition 4.24 (Lemma 2.7 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Let Λ be a (τ 0 , 1)-adapted Lagrangian with τ 0 small enough. Then, for every R 1 large enough, for R 0 large enough, the composition CHAPTER 4 S Λ T Λ is well-dened from E 1,R 0 (M ) to E 1,R 1 (M ) and is the inclusion between these spaces.

Remark 4.25. Beware that S Λ is only a left inverse for T Λ . Indeed, the image of T Λ only contains smooth function and the operator Π Λ := T Λ S Λ is a projector on the image of T Λ .

The study of this projector and of related operators is the object of [BJ20, 2.2].

Spaces associated to Lagrangian deformation

Let us x a (τ 0 , 1)-adapted Lagrangian Λ with τ 0 small enough. We want to associate a scale of Sobolev-like spaces to Λ. In 4.2, we will see that for a relevant choice of Λ these spaces are suited to the spectral analysis of Gevrey Anosov ows. First of all, recalling the action H on Λ given by (4.15) and satisfying the equation (4.14), we dene a scale of weighted L 2 spaces on Λ by

L 2 k (Λ) := L 2 Λ, |α| 2k e -2H h dα ,
where k ∈ R and we recall that dα denotes the volume form ω n R /n! on Λ. Then we dene the associated scale of spaces of hyperfunctions on the manifold by

H k Λ := u ∈ E 1,R (M ) : T Λ u ∈ L 2 k (Λ) , (4.16)
where k ∈ R, and R ≥ 1 is large enough so that Proposition 4.24 applies. The space H k Λ is endowed with the norm

u H k Λ := T Λ u L 2 k (Λ) .
Finally, it is also useful to introduce the notations

H ∞ Λ = k∈R H k Λ and H ∞ Λ,FBI = k∈R H k Λ,FBI .
Let us now list the basic properties of these spaces.

Proposition 4.26. Let Λ be a (τ 0 , 1)-adapted Lagrangian with τ 0 small enough. Then, for every k ∈ R, the space H k Λ is a separable Hilbert space and (equivalently)

H k Λ,FBI is a closed subspace of L 2 k (Λ).
Proposition 4.27. Let s ≥ 1 and R 0 be large enough. Then, if τ 0 is small enough depending on R 0 and Λ is a (τ 0 , s)-adapted Lagrangian, we have for every

k ∈ R E s,R 0 ⊆ H k Λ ⊆ (E s,R 0 ) . (4.17)
Under the same assumptions, for 1 ≤ s < s and every k ∈ R, we have

G s (M ) ⊆ H k Λ ⊆ U s (M ) .
All these inclusions are continuous. Proposition 4.28 (Corollary 2.3 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Assume that τ 0 is small enough and that Λ is a (τ 0 , 1)-adapted Lagrangian. Then, assuming that h > 0 is small enough, there is R > 0 such that, for all k ∈ R, the space

E 1,R (M ) is dense in H k Λ . Moreover, if u ∈ L 2 (M ) ∩ H k Λ , there is a sequence (u n ) n∈N in E 1,R (M ) such that (u n ) n∈N converges to u both in L 2 (M ) and in H k Λ .
It may also be insightful to notice that in the absence of deformations, we just gave an equivalent denition of the usual (semi-classical) Sobolev spaces.

Proposition 4.29 (Corollary 2.4 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Let h > 0 be small enough. Then, for every k ∈ R, the space H k T * M is the usual semi-classical Sobolev space of order k on M , with uniformly equivalent norms as h tends to 0. Some properties of the undeformed case remain true for general Λ's, for instance we have: Proposition 4.30 (Lemma 2.24 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Let Λ be a (τ 0 , 1)-adapted Lagrangian. Assume that τ 0 and h are small enough and let

k ∈ R. Then, if u ∈ H -k Λ and v ∈ H k Λ the pairing u, v := T Λ u, T Λ v L 2 0 (Λ) (4.18)
is well-dened and induces an (anti-linear) identication between H -k Λ and the dual of H k Λ (inducing equivalent, but a priori not equal norms).

Estimates on singular values of certain operators will be crucial in the proof of Theorem 10. To this end, we will use the following result which is also reminiscent of the undeformed case.

Proposition 4.31 (Proposition 2.13 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Assume that Λ is a (τ 0 , 1)-adapted Lagrangian with τ 0 small enough. Assume that h > 0 is small enough. Let m > 0 and q ∈ R, then the inclusion of H m+q Λ into H q Λ is compact. In addition, if (µ k ) k∈N denotes the singular values of this inclusion then we have

µ k = k→+∞ O 1 k m n . CHAPTER 4
In particular, the inclusion of H m+q Λ into H q Λ belongs to the Schatten class S p for every p > n/m.

Analytic FBI transform and Gevrey dierential operators

It is now time to explain how dierential operator with Gevrey coecients acts on the spaces H k Λ 's that we just dene, which is the object of [BJ20, 2.2 and 2.3]. We start by stating a boundedness result. Proposition 4.32 (Proposition 2.4 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Let s ≥ 1. Let P be a (semi-classical) dierential operator of order m ∈ N with G s coecients. Let Λ be a (τ 0 , s)-adapted Lagrangian with τ 0 small enough depending on P . Assume that h > 0 is small enough. Then, for every k ∈ R the operator P is bounded from

H k Λ to H k-m Λ
with uniform bound when h tends to 0.

Actually, we will rather consider dierential operators as unbounded operators on H 0 Λ , so that the following result is more suited to our needs. This is a straightforward generalization of [BJ20, Lemmas 3.2 and 3.4] Proposition 4.33. Let s ≥ 1. Let P be a dierential operator of order 1 with G s coecients. Let Λ be a (τ 0 , s)-adapted Lagrangian with τ 0 small enough and assume that h > 0 is small enough. Then the operator P acting on H 0 Λ with domain

D (P ) = u ∈ H 0 Λ : P u ∈ H 0 Λ
is a closed operator. Moreover, if u ∈ D (P ), there is a sequence (u n ) n∈N of elements of H ∞ Λ such that (u n ) n∈N and (P u n ) n∈N converge respectively to u and P u in H 0 Λ . The same approximation property holds when P is replaced by its adjoint 4 P * .

Finally, the main tool that we will use to study Anosov ows is the following multiplication formula [BJ20, Proposition 2.11].

Proposition 4.34 (Multiplication formula). Let s ≥ 1 and P be a dierential operator of order m with G s coecients. Assume that Λ is a (τ 0 , s)-Gevrey adapted Lagrangian with τ 0 small enough. Let p Λ denotes the restriction to Λ of a G s almost analytic extension of the principal symbol of P . Assume that h is small enough. Let f ∈ S m KN (Λ) be a symbol of order m on Λ, uniformly in

h. Then, if m 1 , m 2 ∈ R are such that m 1 + m 2 = m + m -1, there is a constant C > 0 such that for any u, v ∈ H ∞ Λ , we have (the scalar product is in L 2 0 (Λ)) | f T Λ P u, T Λ v -f p Λ T Λ u, T Λ v | ≤ Ch u H m 1 Λ v H m 2 Λ . (4.19)
4 Here, we consider the adjoint of P with respect to the Hilbert structure on H 0 Λ and not its formal adjoint.

[BJ20], we prove additionally a statement on the Gevrey wave front set of resonant states (that we recall in 4.3, see Proposition 4.41) and we need additional properties on the escape function G 0 to do so.

Before proving Lemma 4.35, let us explain why we need our escape function to satisfy these properties. The point (i) will be used in the proof of Lemma 4.37 to deduce the hypoellipticity of the operator P acting on H 0 Λ from the multiplication formula, Proposition 4.34. The point (ii) will be used in the proof of Proposition 4.36 to show that the Koopman operator (9) denes a continuous semi-group on H 0 Λ . This property is what ensures that the spectrum of P on our spaces has a dynamical meaning. Even though point (i) seems to be the most crucial one, since it is the one that allows us to enter the world of Schatten operators and nally prove Theorem 10, the importance of (ii) could not be overestimated.

Finally, notice that, in order to apply Proposition 4.34 in the most favorable case for us, it will be natural in the following to choose δ = 1/s.

Proof of Lemma 4.35. We want to understand {G 0 , Re p} in order to control how the real part of p evolves under the ow of H ω I G 0 . However, since {G 0 , Re p} = -{Re p, G 0 }, we may understand {G 0 , Re p} by controlling how G 0 evolves under the ow of -H ω I Re p. Hence, we need to understand the dynamics of this ow. To do so, we may multiply X by a bump function identically equals to 1 near M (since we only claim properties for G 0 near T * M and we will not use the high regularity of X in this proof ). Then, it follows from the formula (4.10) that the ow of -H ω I Re p is complete. We denote this ow by (Θ t ) t∈R and write Θ t (α) = (Θ t,x (α), Θ t,ξ (α)) .

Using (4.10), we see that H ω I Re p is given in coordinates (x + iy, ξ + iη) by

-H ω I Re p = n j=1 Re X j ∂ ∂x j + Im X j ∂ ∂y j -ξ ∂ Im X ∂y j + η ∂ Re X ∂y j ∂ ∂ξ j -ξ ∂ Im X ∂x j + η ∂ Re X ∂x j ∂ ∂η j . (4.23)
Indeed, it follows from (4.21) that in such coordinates we have Re p = -ξ Im X -η Re X .

From (4.23), we see that the projection Θ t,x of Θ t is in fact given by the formula

Θ t,x (α) = φt (α x ),
where ( φt ) t∈R denotes the ow of X (in particular, the restriction of φt to M is φ t ). Then, we notice that in (4.23), the component of -H ω I Re p along ∂/∂ξ and ∂/∂η is linear in (ξ, η).

It implies that

Θ t,ξ (α) = L t (α x )(α ξ ), where L t (α x ) is a R-linear application from T * αx M to T * φt(αx)
M (that depends smoothly on t and α x ). Now, since X satises the CauchyRiemann equations and is tangent to M on M , we nd that, for y = 0, in the same system of coordinates than (4.23), we have

∂ Re X ∂y = - ∂ Im X ∂x = 0 and ∂ Im X ∂y = ∂ Re X ∂x = ∂X ∂x . (4.24)
By uniqueness in the CauchyLipschitz Theorem, we nd by plugging (4.24) in (4.23) that, for x ∈ M and t ∈ R we have6

L t (x) = Dφ t (x) -1 tr .

(4.25)

Hence, the hyperbolicity of (φ t ) t∈R will have important consequences on the dynamics of (Θ t ) t∈R . Let us complexify the bundles E * 0,u,s . For x ∈ M , we denote by E C, * 0 , E C, * u and E C, * s the complexication of E * 0 , E * u and E * s , considering linear forms valued in C instead of R. For instance, for x ∈ M , we write E C, * 0,x for the subspace of T * x M ⊗ C consisting of R-linear maps from T x M to C that vanish on E u ⊕ E s (or, under a natural identication, of C-linear forms on T *

x M that vanish on E u ⊕ E s ). From the fact that T x M = E 0

x ⊕ E u x ⊕ E s x is a totally real subspace of maximal dimension of T x M , we deduce that T *

x M = E C, * 0,x ⊕ E C, * u,x ⊕ E C, * s,x . Since E 0 x , E u x and E s
x depends in a Hölder-continuous fashion on x ∈ M , so does E C, * 0,x , E C, * u,x and E C, * s,x . Consequently, we may extend continuously

E C, * 0 , E C, * u and E C, * s to M . Then, if M is small enough, we have T x M = E C, * 0,x ⊕ E C, * u,x ⊕ E C, * s,x for all x ∈ M . A priori, this decomposition is only invariant under L t for t ∈ R when x is real. If σ ∈ T * x M then we write σ = σ 0 + σ u + σ s for the decomposition of σ under T x M = E C, * 0,x ⊕ E C, * u,x ⊕ E C, * s,x .
Then, we put a real Riemannian metric on M and dene for γ > 0 the cone elds C γ u and C γ s by setting for

x ∈ M C γ u (x) = σ ∈ T * x M : |σ 0 | + |σ s | ≤ γ |σ u | and C γ s (x) = σ ∈ T * x M : |σ 0 | + |σ u | ≤ γ |σ s | .
Without loss of generality, we may assume that C 0 is closed and does not intersect E * u ⊕ E * s \ {0}. We may also assume that there is a closed conic neighbourhood C C,0 of E C, * 0 in T * M such that C C,0 ∩ T * M = C 0 . Choose then a small closed conic neighbourhood of C 0s of E * s ⊕ E * 0 in T * M . From (4.25) and standard arguments in hyperbolic dynamic, there are a large T 0 > 0, small 0 < γ < γ, some λ > 1 and a constant c > 0 such that if α = (α x , α ξ ) ∈ T * M , and T 1 ≥ T 0 then:

a. either Θ T 1 ,ξ (α) ∈ C γ u ( φT 1 (α x )) or Θ -T 0 ,ξ (α) ∈ C γ s ( φ-T 0 (α x )) or α ∈ C C,0 ; b. if Θ -T 0 ,ξ (α) ∈ C γ u ( φ-T 0 (α x )) then Θ T 1 ,ξ (α) ∈ C γ u ( φT 1 (α x )) and we have the bound |Θ T 1 ,ξ (α)| ≥ λ |Θ -T 0 ,ξ (α)|; c. if Θ T 1 ,ξ (α) ∈ C γ s ( φT 1 (α x )) then Θ -T 0 ,ξ (α) ∈ C γ s ( φ-T 0 (α x )) and we have the bound |Θ T 1 ,ξ (α)| ≤ λ -1 |Θ -T 0 ,ξ (α)|; d. if α ∈ T * M does not belong to C 0s , then, for t ≥ 0, we have that Θ t,ξ (α) does not belong to C γ s ( φt (α x )), and, for t ≥ T 0 , we have Θ t,ξ (α) ∈ C γ u ( φt (α x )) and |Θ t,ξ (α)| ≥ c |α ξ |.
Since we ask here for x ∈ M , these are consequences of the hyperbolicity of (φ t ) t∈R , that is it only relies on the dynamic on M . We want to apply a perturbation argument to show that b and c remain true on a small complex neighbourhood of M , but we need rst to x the value of T 1 . Hence, we x the value of T 1 , large enough such that we have

sup α∈T * M \{0} 1 |α ξ | δ 0 -T 0 |Θ t,ξ (α)| δ dt < c δ (T 1 -T 0 ) 2 . (4.26)
Now that T 1 is xed, it follows from a perturbation argument that, up to taking a smaller λ, a smaller γ, a larger γ and a smaller c, the properties b and c above remain true when α ∈ T * (M ) 0 , for some small 0 > 0 (and (4.26) remains true since we asked for a strict inequality). Then, we choose a symbol m ∈ S 0 KN T * (M ) 0 of order 0 on T * (M ) 0 , valued in [-1, 1], with the following properties:

• if x ∈ (M ) 0 and σ ∈ T * x M \ (C γ u (x) ∪ C γ s (x)) or σ is near 0 then m(x, σ) = 0; • there is C > 0 such that if x ∈ (M ) 0 and σ ∈ C γ s (x) satises |σ| ≥ C then m(x, σ) = 1; • there is C > 0 such that if x ∈ (M ) 0 and σ ∈ C γ u (x) satises |σ| ≥ C then m(x, σ) = -1;
• m is non-positive on C u γ and non-negative on C s γ .

CHAPTER 4

Then, we may dene G 0 near T * M ⊗ C by the formula

G 0 (α) = T 1 T 0 m(Θ t (α)) |Θ t,ξ (α)| δ dt. (4.27)
We multiply G 0 by a bump function to satisfy the claim on the support. It does not change the value of G 0 near T * M and hence it will not interfere with the properties (i) and (ii).

We may consequently use the formula (4.27) to prove (i) and (ii).

We start by proving (i). To do so we compute

{G 0 , Re p} = -{Re p, G 0 } = -H ω I Re pG 0 = m(Θ T 1 (α)) |Θ T 1 ,ξ (α)| δ -m(Θ -T 0 (α)) |Θ -T 0 ,ξ (
m(Θ T 1 (α)) |Θ T 1 ,ξ (α)| δ -m(Θ -T 0 (α)) |Θ -T 0 ,ξ (α)| δ ≤ -|Θ T 1 ,ξ (α)| δ + |Θ -T 0 ,ξ (α)| δ ≤ -λ δ -1 |Θ -T 0 ,ξ (α)| δ ≤ - 1 C |α ξ | δ ,
for some C > 0. If Θ -T 0 ,ξ (α) does not belong to C u γ , then the situation is even simpler since the term -m(Θ -T 0 (α)) |Θ -T 0 ,ξ (α)| δ is non-positive. Hence, the right-hand side in (4.28) is negative and elliptic of order δ outside of C 0 .

It remains to prove (ii). The analysis is based on (4.28) again. Let α ∈ (T * M )

2 . If Θ T 1 ,ξ (α) ∈ C γ u or Θ -T 0 ,ξ (α) ∈ C γ
s , then the analysis from the proof of (ii) applies, and we see that the right-hand side in (4.28) is non-positive for α ξ large enough. Otherwise, Θ -T 0 ,ξ (α) / ∈ C γ u and Θ T 1 ,ξ (α) / ∈ C γ s and both terms in the right-hand side of (4.28) are non-positive. Thus, the right-hand side of (4.28) is always non-positive when |α ξ | is large enough. Hence, by compactness, the right hand side of (4.28) is bounded from above, proving (ii).

Now, that we are equipped with a good escape function, we are in position to apply the tools from 4.1 to study the spectral theory of P = X + V .

Spectral theory for the generator of the ow

With the notations of the previous section, we set δ = 1/s and let G 0 be an escape function given by Lemma 4.35 (for arbitrary C 0 , we only assume that C 0 is closed an does not intersect E * u ⊕ E * s \ {0}). Then we dene G = τ G 0 and Λ = e H ω I G T * M with τ = cτ 0 h 1-1 s , where c and τ 0 are small. Notice that if c is small enough, then Λ is a (τ 0 , s)-adapted Lagrangian (in the sense of Denition 4.14), and hence the results from 4.1 will apply to the G s semi-classical pseudor hP . We will not consider the asymptotic h → 0, and hence we shall assume that h is xed, small enough so that the results from 4.1 apply. We shall also assume that τ 0 is small enough for the same reason.

We want now to study the spectral theory of the operator P = X + V on the space H 0 Λ dened by (4.16). Notice that, according to Proposition 4.32, if h and τ 0 are small enough, then the operator hP (and hence P ) is bounded from H k Λ to H k-1 Λ for every k ∈ R. From Proposition 4.33, we see that P denes a closed operator on H 0 Λ . We will start by proving that P is the generator of a semi-group.

Proposition 4.36. The operator P is the generator of a strongly continuous semi-group (L t ) t∈R on H 0 Λ . Moreover, if t ≥ 0 and u ∈ H 0 Λ ∩L 2 (M ) then L t u is given by the expression (9).

We will then prove the following key lemma that will be used with Proposition 4.31 to prove that the resolvent of P is in a Schatten class.

Lemma 4.37 (Hypo-ellipticity of P ). There is a constant C > 0 such that for every u ∈ D(P ) we have u ∈ H δ Λ and

u H δ Λ ≤ C u H 0 Λ + P u H 0 Λ ,
where we recall that we set δ = 1/s.

With Proposition 4.31, we deduce then from Lemma 4.37 that P has a good spectral theory on H 0 Λ . More precisely, we have: Theorem 4.38. If z is any element in the resolvent set of P , then the resolvent (z -P ) -1 :

H 0 Λ → H 0 Λ is compact and if (σ k ) k≥0 denotes the sequence of its singular values, we have

σ k = k→+∞ O k -1 sn .
In particular, the operator (z -P ) -1 is in the Schatten class S p for any p > ns. Consequently, P has discrete spectrum on H 0 Λ , and this spectrum is the Ruelle spectrum of P . The eigenvectors of P acting on H 0 Λ are also the resonant states for P . If N (R) denotes the number of Ruelle resonances of modulus less than R, we have

N (R) = R→+∞ O (R ns ) .
Theorem 10 will then be proved in the following section as a corollary of Theorem 4.38.

P is dense since it contains E 1,R 0 ), and we know that P is the generator of a strongly continuous semi-group.

Denote by ( L t ) t≥0 the semi-group generated by P on H 0 Λ and (L t ) t≥0 the semi-group on L 2 (M ) dened by (9). We want to prove that for t ≥ 0 and u ∈ H 0 Λ ∩L 2 (M ) we have L t u = L t u. Thanks to the semi-group property, we only need to prove it for t ∈ [0, t 0 ] for some small t 0 > 0. Then, since elements of L 2 (M ) ∩ H 0 Λ may be simultaneously approximated in L 2 (M ) and in H 0 Λ by elements of E 1,R 0 (according to Proposition 4.28), we only need to prove the equality for u ∈ E 1,R 0 . Now, there is a t 0 > 0 and a R 1 > 0 such that for u ∈ E 1,R 0 , the curve γ : [0, t 0 ] t → L t u is C 1 in E s,R 1 with γ (t) = P γ(t). Provided that τ 0 is small enough, E s,R 1 is continuously included in H 0 Λ (see Proposition 4.26) and hence the curve γ has the same property in H 0 Λ . Consequently, we have γ(t) = L t u for t ∈ [0, t 0 ],

according to [ABHN11, Proposition 3.1.11], ending the proof of the proposition.

We turn now to the proof of the hypo-ellipticity of the operator P .

Proof of Lemma 4.37. Assume rst that u ∈ H ∞ Λ . Let χ + , χ -and χ 0 be C ∞ functions from R → [0, 1] such that χ + + χ 0 + χ -= 1, and, for some small η > 0, the function χ 0 is supported in [-η, η], the function χ -is supported in -∞, - Hence, either e -τ H G 0 (α) is small, either it does not belong to C 0 (provided that η is small enough, we use here the assumption that C 0 does not intersect E * u ⊕ E * s ). We deal now with the general case u ∈ D(P ). Let (u n ) n∈N be a sequence of elements of H ∞ Λ as in Proposition 4.33. Since we already dealt with the case of elements of H ∞ Λ we know that for some C > 0 and all n ∈ N we have

u n H δ Λ ≤ C u H 0 Λ + P u H 0 Λ .
In addition, T Λ u n converges pointwise to T Λ u and hence the result follows by Fatou's Lemma.

We are ready to prove Theorem 4.38.

Proof of Theorem 4.38. Let z be any element of the resolvent set of P . If u ∈ H 0 Λ then we have that P (z -P ) -1 u = z(z -P ) -1 u -u.

Hence, (z -P ) -1 and P (z -P ) -1 are both bounded from H 0 Λ to itself and, consequently, Lemma 4.37 implies that (z -P ) -1 is bounded from H 0 Λ to H δ Λ . Hence, Proposition 4.31 implies that (z -P ) -1 , as an operator from H 0 Λ to itself, is compact, with the announced estimates on its singular values.

We prove the estimates on the number of eigenvalues of P before proving that these eigenvalues are indeed the Ruelle resonances. Let z ∈ C be any point in the resolvent set of P and denote by N (R) the number of eigenvalues of (z -P ) -1 of modulus larger than R -1 . Then let (µ k ) k∈N denote the sequence of eigenvalues of (z -P ) -1 and (σ k ) k∈N the sequence of its singular values, and choose p > 0 such that δp/n < 1. According to [GGK00, Corollary IV.3.4], we have then for every R > 0 that

N (R) R p ≤ N (R)-1 k=0 |µ k | p ≤ N (R)-1 k=0 σ p k ≤ C N (R)-1 k=0 (1 + k) -δp n ≤ C N (R) 1-δp n .
Here, we applied the estimates on singular values that we just proved, and C may vary It remains to prove that the eigenvalues of P acting on H 0 Λ are indeed the Ruelle res- onances of P . The situation here is a bit complicated due to the use of very irregular hyperfunctions, so that we cannot rely directly on Lemma B.3. We detail here the argument, but the basic ideas are the same than those that we explain in Appendix B. Let R 0 > 0 be large enough, and assume that τ 0 is small enough, so that (4.17) holds and that E 1,R 0 is dense in H 0 Λ (see Proposition 4.28). We also assume that R 0 is large enough so that E 1,R 0 (M ) is dense 7 in C ∞ (M ). Denote by i 0 the inclusion of E 1,R 0 in H 0 Λ and by i 1 the inclusion of H 0 Λ in E 1,R 0 . Then, we dene

R(z) = i 1 • (z -P ) -1 • i 0 : E 1,R 0 → E 1,R 0 ,
where (z -P ) -1 denotes the resolvent of P on the space H 0 Λ . We just saw that it is a meromorphic family of operator on C.

Then, if we denote by i the inclusion of E 1,R 0 into C ∞ (M ) and by j the inclusion of 7 This is possible since, if ∆ is an analytic Laplacian on M , then E 1,R 0 (M ) contains all the eigenvectors for ∆ when R0 is large enough. See [Zel17, 14.5] and references therein. CHAPTER 4

Thus, we only need to prove that a 1 = a 2 = 0 to end the proof of the lemma. By dominated convergence, when z is real and tends to +∞ the rst term in the right-hand side of (4.38) tends to 0. Hence, we only need to prove that the trace of (z -X) -m tends to 0 when z is real and tends to +∞. However, this fact follows from Lidskii's Trace Formula tr (z -X) -m = Lemma 4.40. Let z be a complex number with large real part and m be the smallest integer strictly larger than sn. Let Q z be the polynomial of order at most m -1

Q z (λ) = - m-1 =0 γ T # γ e γ V e -zTγ T -1 γ |det (I -P γ )| (z -λ) ! .
Then for every λ ∈ C we have

Further applications of the FBI transform

In order to lighten the exposition, we focused in this chapter on the proof of Theorem 10. However, [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF] contains other results on Anosov ows that we sum up now. Let consequently s ≥ 1 and (φ t ) t∈R be a G s Anosov ow on a G s compact manifold M of dimension n. Let also V : M → C be a G s function. It is classical [START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov ows[END_REF] that the C ∞ wave front sets of the resonant states for P = X + V are contained in the stable 8 direction E * s dened in 4.2.1. In [BJ20, Denition 2.3], we dene a Gevrey wavefront set for an ultradistribution, using an analytic FBI transform. With this denition, we are able to prove the following.

Proposition 4.41 (Proposition 3.2 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). The G s wave front sets of the resonant states for P = X + V are contained in the stable direction E * s .

Certain perturbations of the operator P are also studied in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]. The rst kind of perturbations that we consider are of the form (for ≥ 0 small)

P = P + ∆, (4.48)
where ∆ is a self-adjoint elliptic non-positive dierential 9 operator of order m > 1 with G s coecients for instance the LaplaceBeltrami operator associated to a G s Riemannian metric. On a dynamical level, the operator (4.48) appears when studying stochastic perturbations of the ow (φ t ) t∈R . We see that when > 0 the operator P is elliptic of order m and has bounded from above real part, so that it has discrete spectrum on L 2 (M ). We will denote by σ L 2 (P ) this spectrum. It has been proven by Dyatlov and Zworski [DZ15,

Theorem 1] that σ L 2 (P ) converges to the Ruelle spectrum of P = P 0 when tends to 0.

In our G s context, we are able to give a global version of this result. To do so, we need Then our result on stochastic perturbations of G s Anosov ow reads:

Proposition 4.42 (Theorem 9 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Let p > ns and z ∈ R + be large enough. Then 8 Beware that most of the references in the literature, in particular [START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov ows[END_REF], do not work with the Koopman operator (9) but with its adjoint, the transfer operator. It amounts to reverse the direction of the time for the ow (φt) t∈R , and consequently the stable and unstable direction are inverted. That is why it is often stated that the wave front sets of resonant states are contained in the unstable rather than stable direction.

9 Actually, certain pseudo-dierential operators are allowed in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF].

CHAPTER 4

there is C > 0 such that for > 0 small enough we have Here, d z,H denotes the Hausdor distance associated to the distance d z .

The proof of Proposition 4.42 is based on an investigation of the Schatten properties of the resolvent of the operator (4.48) and on resolvent bounds from [START_REF] Bandtlow | Estimates for norms of resolvents and an application to the perturbation of spectra[END_REF]. The convergence in Proposition 4.42 seems to be very weak. However, we think that it is not reasonable to expect too fast a convergence in such a global result. Indeed, when we add the dierential operator ∆ to P in order to form P , since ∆ has higher order, we can expect that the spectrum of P looks globally like the spectrum of ∆, rather than like the Ruelle spectrum of P . Indeed, the higher order operator will be predominant at higher frequencies this fact can be made rigorous looking for instance at the symbol of P and using the ellipticity of ∆. Furthermore, the spectrum of ∆ is contained in R -while we expect some kind of vertical structure for the Ruelle spectrum of P (see for instance [START_REF] Jin | A local trace formula for Anosov ows[END_REF][START_REF] Faure | Band structure of the Ruelle spectrum of contact Anosov ows[END_REF]). Hence, we may expect σ L 2 (P ) to be some kind of attened version of the Ruelle spectrum of P , and its global structure is thus very dierent from the actual Ruelle spectrum of P .

We also consider in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF] deterministic perturbations of the Anosov ow (φ t ) t∈R or linear response. To do so, we consider a perturbation → X of our vector eld X = X 0 . Here, the perturbation is dened for in a neighbourhood of 0 and is assumed to be (at least) C ∞ from this neighbourhood of zero to a space of G s sections of the tangent bundle of M (see Remark 4.4). We can also consider a perturbation → V with the same features and then form for near zero the operator P = X + V . Let us consider the most simple example: in the C ∞ case, if P 0 has a simple resonances λ 0 , then it will extend to a C ∞ family of simple resonances → λ (provided that the perturbation is C ∞ ). However, even if the perturbation → X is real-analytic in the C ∞ category, then we do not know that the family → λ is real-analytic (in fact, it is reasonable to expect that it is not). We shall see below that if the perturbation → X is real-analytic in the real-analytic category, then the family → λ is real-analytic (this is an immediate consequence of Theorem 4.44). Using the notations from 4.2, we are able to prove the following result. Proposition 4.43 (Proposition 3.4 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Assume that δ > 1/2 (i.e. s < 2). Let ∈ R + \ N. Assume that h and τ are small enough. Then for small enough, the spectrum of P acting on H 0 Λ is the Ruelle spectrum of P . Moreover, if k = ( + 1)δ -and r is a large enough positive real number, then the the map

→ (r -P ) -1 ∈ L H 0 Λ , H k Λ is C on a neighbourhood of 0.
The interesting point in Proposition 4.43 is that the loss of regularity when dierentiating the resolvent of P is mitigated by its smoothing property (given by Lemma 4.37).

This new feature of our high regularity setting is even more striking when s = 1.

Theorem 4.44 (Theorem 11 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Assume that s = 1 and that the perturbations → X and → V are real-analytic. Assume that h and τ are small enough. Then, for r ∈ R + large enough, the map

→ (r -P ) -1 ∈ L H 0 Λ , H 1 Λ
is real-analytic on a neighbourhood of zero.

Theorem 4.44 allows to apply Kato theory on analytic perturbations of operators [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF]. We can prove in particular the following result, which is hardly surprising, but we are not aware of any proof in the literature devoted to this subject (see [KKPW89,

Corollary 1] however for a related statement).

Theorem 4.45 (Theorem 2 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Let → X be a real-analytic family of realanalytic vector elds on a real-analytic manifold M , dened for near 0. We assume that X 0 generates an Anosov ow that admits a unique SRB measure 10 µ 0 . Let µ denote the unique SRB measure of the Anosov ow generated by X , for near zero. Then the map

→ µ ∈ U 1 (M )
is real-analytic on a neighbourhood of zero.

Remark 4.46. In Proposition 4.43, we need to make the additional assumption s < 2.

This is for technical, but denitely not anecdotal, reasons. A natural rst step in the proof of Proposition 4.43 would be to prove that Proposition 4.36 and Lemma 4.37 still holds for near 0. This is true for Lemma 4.37 but a priori not for Proposition 4.36. This is because unfortunately the property (ii), from Lemma 4.35, of the escape function G 0 is not stable under small perturbations of the vector eld X. This is quite an issue since Proposition 4.36 is what ensures that the resolvent set of P acting on H 0 Λ is non-empty and that its spectrum coincides with the Ruelle spectrum. To prove that the resolvent set 10 See for instance [START_REF] Young | What are SRB measures, and which dynamical systems have them?[END_REF] for denition and discussion of the notion of SRB measure. replace the operator L by L ψ in the proof of Lemma 2.14. Lemma A.2. Lemma 2.14 remains true when L is replaced by L ψ .

Proof. Recall the dierential operator L a k, introduced in the proof of Lemma 2.14 and notice that for all m ∈ N we have L ψ e , e k L 2 = S 1 e 2iπ( x-kT (x)) ψ(x)dx = (1-x) 2m+1 , and this ends the proof.

thus closed in E λ,i , so that E λ,i = P λ,i (B 1 ∩ B 2 ) . We give now the results that we will use to identify Ruelle resonances for Anosov ows.

This is just an abstract version of the argument based on the meromorphic extension of the resolvent presented in the introduction. However, since we will be working with nonstandard spaces of ultradierentiable functions and generalized distributions, it will be useful to have a general result at our disposal that do not make reference to a particular class of regularity.

Lemma B.3. Let B 1 and B 2 be Banach spaces. Let B be a Hausdor topological vector space. Assume that there are continuous inclusions of B 1 and B 2 into B, and that the intersection B 1 ∩ B 2 is dense in both B 1 and B 2 . For i = 1, 2, let (L i,t ) t≥0 be a strongly continuous semi-group of operator on B i whose generator is denoted by X i . Assume that for every t ≥ 0 the restrictions of L 1,t and L 2,t to B 1 ∩ B 2 coincide. Let U be a connected open subset of C that contain points with arbitrarily large positive real parts. Assume that the intersections of the spectra of X 1 and X 2 with U only contain isolated eigenvalues of nite algebraic multiplicities.

Then, the intersections of the spectra of X 1 and X 2 with U coincide and so do the associated generalized eigenspaces (that are consequently contained in B 1 ∩ B 2 ). has a unique pole at λ whose order is 1 and whose residue is 1. Thus there is an entire function G λ,t 0 such that for all z ∈ C G λ,t 0 (z) G λ,t 0 (z) = -n≥d+1 z n λ n+1 e -(z-λ)t 0 = e -(z-λ)t 0 z -λ

+ d n=0
z n λ n+1 e -(z-λ)t 0 and G λ,t 0 (0) = 1. Choose for G 0,t 0 any logarithmic primitive of z → e -t 0 z z . Now, let R > 0 and assume that λ ∈ σ R (P ) is such that |λ| ≥ 2R. Notice that for all z ∈ D (0, R) we have G λ,t 0 (z) G λ,t 0 (z) ≤ 2e Rt 0 R d+1 e Re(λ)t 0 |λ| d+2 and using the fact that G λ,t 0 has a logarithm on D (0, R) that vanishes at 0 (since G λ,t 0 vanishes only at λ) we get that, for some constant C depending only on R and all z ∈ D (0, R), G λ,t 0 (z) .

(C.3)

In order to make this factorization more explicit, let us describe the G λ,t 0 's. For all λ ∈ C \ {0}, dene the polynomial

Q λ,t 0 = - d k=0 d n=k k! n! (t 0 -λ) n-k-1 λ k+1 X k ,
and notice that Q λ,t 0 (z) e -z(t 0 -λ) = d n=0

z n λ n+1 e -(z-λ)t 0 .

Thus we have for all λ ∈ C \ {0} and z ∈ C G λ,t 0 (z) = 1 -z λ exp Q λ,t 0 (z) e -(z-λ)t 0 -Q λ,t 0 (0)e λt 0 × exp z 1 0 e -(zu-λ)t 0 -1 zu -λ du .

The last factor is a logarithmic primitive of z → e -(z-λ)t 0 -1 z-λ . Indeed, if p = q, then 2 p -1 and 2 q -1 are relatively prime 2 . Then let g = p∈P b 2 p -1 e p .

Since √ b n n≥1 decays faster than any geometric sequence, we see that g belongs to E. 2 More generally, if m, n ∈ N * , then the greatest common divisor of 2 m -1 and 2 n -1 is 2 m∧n -1, where m ∧ n denotes the greatest common divisor of m and n.

1

  Alice et Jean) et tou.te.s les ami.e.s qui ont pu passer par chez nous. À sa demande, je remercie Louise pour toutes [s]es contributions à [m]a thèse . Je remercie Vincent pour son compagnonnage cinématographique. Enn, je remercie mes parents et mes frères pour leur soutien inconditionnel. Résumé An d'étudier la validité d'une formule de trace pour les ots d'Anosov C ∞ proposée par Dyatlov et Zworski, nous développons des outils qui permettent de comprendre le spectre de Ruelle des dynamiques hyperboliques inniment dérivables. Un rôle central est joué dans cette étude par la notion d'ultradiérentiabilité (principalement via le langage des classes de DenjoyCarleman). On donne ainsi un analogue ultradiérentiable des méthodes développées originellement par Ruelle, Rugh et Fried (basées sur les résultats de Grothendieck sur les opérateurs nucléaires) pour étudier les dynamiques hyperboliques analytiques. En particulier, une analyse détaillée des opérateurs de transfert associés aux applications dilatantes du cercle ultradiérentiables est menée. La formule de trace est ensuite démontrée pour une grande classe de ots d'Anosov ultradiérentiables. Enn, une transformée de FBI analytique est utilisée pour prouver que l'ordre du déterminant dynamique associé à un ot d'Anosov de régularité Gevrey est ni. Mots-clés Dynamiques hyperboliques, spectre de Ruelle, formule de trace, déterminant dynamique, Trace formulae, dynamical determinants and counter-examples 1.1

INTRODUCTIONFigure 1 :

 1 Figure 1: Example of spectrum of a quasi-compact operator. The radius of the dashed disc is the essential spectral radius. The blue dots are eigenvalues of nite multiplicities.

  if z ∈ C belongs to the spectrum of L acting on C k and |z| > max(r ess (L | C k ), r ess (L | C k )),then z also belongs to the spectrum of L acting on C k , and the associated generalized eigenspaces coincide (see Appendix B and in particular Lemma B.1 and Example B.2).

[ BKL02 ,

 BKL02 [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF][START_REF] Gouëzel | Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties[END_REF], Liverani in[START_REF] Liverani | On contact Anosov ows[END_REF] and then Butterley and Liverani in[START_REF] Butterley | Smooth Anosov ows: Correlation spectra and stability[END_REF] (with necessary details provided in[START_REF] Butterley | Robustly invariant sets in ber contracting bundle ows[END_REF]) gave geometric constructions of spaces adapted to Anosov ows. Then, Faure and Sjöstrand adapted in[START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov ows[END_REF] the semi-classical approach from[START_REF] Faure | Semi-classical approach for Anosov dieomorphisms and Ruelle resonances[END_REF] to the continuous-time case. A slightly dierent point of view on this approach is presented in[START_REF] Dyatlov | Dynamical zeta functions for Anosov ows via microlocal analysis[END_REF] (using radial estimates and propagation of singularities). This construction has then been adapted to some open systems (including Axiom A systems) by Dyatlov and Guillarmou in[START_REF] Dyatlov | PollicottRuelle resonances for open systems[END_REF][START_REF] Dyatlov | Afterword: Dynamical zeta functions for Axiom A ows[END_REF] and to certain MorseSmale ows by Dang and Rivière in[START_REF] Viet | Spectral analysis of MorseSmale ows I: Construction of the anisotropic spaces[END_REF][START_REF] Viet | Spectral analysis of Morse-Smale ows, II: Resonances and resonant states[END_REF]. The equivalent of Theorem 3 for Anosov ow reads:

  {u ∈ B : P u ∈ B} ; INTRODUCTION (iii) the intersection of the spectrum of P with {z ∈ C : Re z > -A} is a discrete set of eigenvalues of nite (algebraic) multiplicity.

Figure 2 :

 2 Figure 2: Spectrum of P acting on the space B from Theorem 5. The abscissa of the green line is -A. The blue dots depict eigenvalues of nite multiplicity. Their imaginary parts may be arbitrarily large but their real parts are bounded from above.

d

  (z) dened by (12) has a holomorphic extension to C whose zeros are exactly the Ruelle resonances for P = X + V (counted with multiplicities). Remark 5. Dynamical determinants are often outshined by the closely related Ruelle zeta functions. The most simple example of Ruelle zeta function associated to an Anosov ow

  for z in a neighbourhood of zero, extends to an entire function whose zeros are exactly the e 2iπmθ ln(m) -1

  For every n ∈ N, denote by k (n) the unique integer such that n ∈ I k(n) . Then set for all integers n 1 a n = +∞ m=0 e 2iπmθ ln (k (m) + 2) n . CHAPTER 1 Then we use Lemma 1.8 as in the proof of (a) to show that f (z) = exp -+∞ n=1 1 n a n z n extends to an entire function whose zeros are exactly the z m = e 2iπmθ ln(k(m)+2) -1

1 2 .

 2 Then for every k ∈ N and n 1, denote by N (n) k the number of those m ∈ I k such that mnθ ∈ [0, ] mod 1, and choose a permutation σ (n) k of I k which puts these elements rst. Equidistribution of the mnθ, for n xed and m 0, implies that

  Hence, the innite product (1.11) does converge uniformly on all compact subsets of C to an entire function d. That d has same genus as f is straightforward from the Denition 1.2 and Hadamard's factorization Theorem (we use the positivity of the c k 's to ensure that no unwanted cancellation happens). Let us point

.

  and thus the series k 0 u k converges in the space of converging sequences equipped with the sup norm. But its sum is clearly the sequence of partial sums of Thus this series converges, and its sum is a n +∞ k=0 c n k , as wanted.

  U ) and D c (U ), the spaces respectively of C ∞ functions and of distributions that are compactly supported in U . Denition 5 of Ruelle resonances, Ruelle spectrum and resonants states adapt then naturally to this case. As in the case of Anosov dieomorphisms the Ruelle resonances are the zeros of a dynamical determinant, the analytic continuation of

  Using (1.23) again, one gets m (x, y) = -) -g (y)| d (x, y) r Cd (x, y) -ln(4 r θ) ln 4

  2] gives a better bound on the number of Ruelle resonances, using only the local trace formula from [JZ17]. The main interests of these results is that they give lower bounds on the number of Ruelle resonances, which are not very easy to obtain in general. Considering Ruelle resonances for a constant time suspension of a cat map, we see that the bounds from Corollary 1.28 and [JZ17, Theorem 2] are quite accurate as general lower bounds on Ruelle resonances. However, we expect that for a generic Anosov ows, there are much more Ruelle resonances than predicted by those results. Proof of Proposition 1.27. Let us rst recall the folklore proof of (i) ⇒ (ii) see [MPM15, Theorem 17] for another approach. Let α denote the order of the dynamical determinant d. The estimate (1.25) for any ρ > α is a consequence of Jensen's formula see [Boa54, 2.5.12]. Let ϕ ∈ C ∞ c R * + and recall that its Laplace transform is dened for z ∈ C by Lap (ϕ) (z) = +∞ 0 e -zt ϕ(t)dt.

Figure 1

 1 Figure 1.1: The contour Γ R .

  22)and notice that g(N ) → N →+∞ 0 by Lemma 2.6. The operator L b is morally strictly upper triangular, the following proposition uses the function g to quantify the fact that L b is quasi-nilpotent.

  then for every N ∈ N, the image of e k by L n b,N belongs to the span of the e 's such that | | < θ m-n . In particular, L n b,N e k is orthogonal to e k . Since, we also have L n b,N e 0 = 0, we nd that, for every k ∈ Z, we have L n b,N e k , e k L 2 = 0, and, letting N tends to innity, that L n b e k , e k L 2 = 0. Hence, (2.31) givesa n = k∈Z L n e k , e k L 2 = k∈Z S 1 e 2ikπ(x-T n (x)) dx = lim m→+∞ S 1sin ((2m + 1)π(x -T n (x))) sin (π (x -T n (x))) dx.

  that f and g have been dened respectively in (2.23) and (2.22). Notice that F has genus zero. Thus if n(r) denotes the number of integers m such that f (m) -1 ≤ r or m = 0, we have the following estimate [Boa54, Lemma 3.5.1] for r > 0: log F (r) ≤ be used with Proposition 2.23 to control the growth of the dynamical determinant, see 2.2.3 for examples. Proposition 2.23. There is a constant C such that for all z ∈ C we have |d(z)| ≤ F (C |z| G (C |z|)) . Proof. Let z ∈ C. Denote by (c k ) k∈N the sequence of singular values of the operator -z (I -zL b ) -1 L c and by (λ k ) k∈N the sequence of its eigenvalues. By Lidskii's Theorem we have

  denotes the sequence of singular values of L c then we have for k ≥ 1 (replace f (k) by 1 in the case k = 0)c k ≤ |z| (I -zL b ) -1 H→H s k ≤ C |z| (I -zL b ) -1 H→H f (k),for some constant C > 0, and thus |d(z)| ≤ F C |z| (I -zL b ) -1 H→H . But from Lemma 2.16, we get that, up to taking larger C, we have (I -zL b ) -1 H→H ≤ G (C |z|) , which ends the proof of the proposition.

  Theorem 2.27 suggests that the dynamical determinant (12) associated to a Gevrey Anosov ow should have nite order. Let us explain why. In general, we expect an exponential scaling between results for Anosov dieomorphisms and ows. For instance, if d denotes the dynamical determinant (4) associated to a transitive Anosov dieomorphism T and d the dynamical determinant (12) associated to the time 1 suspension of T , then an elementary computation ensures that d(z) = d(e z ) for every z ∈ C. Under this exponential scaling, the almost polynomial bound from (v) in Theorem 2.27 deteriorates into nite order for the dynamical determinant d.

µ

  ∈ D ν (M ) positive and fully supported, it induces an isomorphism between D υ (M ) and C ∞,υ (M ) , ν → dν dµ . Then notice that d((φt) * µ) dµ satises for all x ∈ M and t, t ∈ R the cocycle equation

•Y

  and Y . If C and C are two cones in an Euclidean space, we write C C for C ⊆ • C ∪ {0}. The dimension of a cone C in an Euclidean space E is by denition the maximum dimension of a linear subspace of E contained in C.

  d and C r-1 ∩ S d form an open cover of S d . Fix α ∈ ]0, 1[ for the remaining of the section. Choose a Gevrey function χ : R → [0, 1]

  Since in addition we have |ξ d+1 | ≥ c |ξ| and |η d+1 | ≥ c |η|, we can conclude in this case (discussing whether |ξ| or |η| is larger).

0h

  (t) M t dt and structure of the local Koopman operator.

  and use Lemma 3.38. The continuous inclusion of C ∞,υ (M ) in H is an immediate consequence of Lemma 3.38.

  h n (t + t 0 ) = e -zt )L τ dτ • (z -P ) hn (τ )dτ L t 0 +u du = +∞ 0 h n (t 0 + u)L t 0 +u du = +∞ 0 h n (t)L t dt. (3.53) Moreover, if Re z 1 , then, for every n ≥ 1, the function hn satises the assumptions of Proposition 3.32 and consequently the operator +∞ 0 hn (t)dt : H → H is compact. It follows then from (3.53) and Lemma 3.42 that the operator 1 (n -1)! +∞ 0 h n (t)L t dt : H → H CHAPTER 3 is compact. On the other hand, we see that the operator norm of 1 (n -1)! +∞ 0 χ(t)e -zt t n-1 L t dt : H → H is less than C(2t 0 ) n

  Lemma 3.44. Let h be a C ∞ function supported on a compact subset of [2t 0 , +∞[. Then the operator +∞ 0 h (t) L t dt : H → H (3.54) is compact. Its non-zero spectrum is the intersection of C \ {0} with the image of the spectrum of P by λ → Lap(h)(-λ), where Lap denotes the Laplace transform. If υ < 2, the operator (3.54) is trace class and tr +∞ 0 h (t) L t dt = γ h (T γ ) T # γ |det (I -P γ )| exp γ V .

  than the operator induced by L t on E λ (they solve the same Cauchy problem). It follows that we have A = +∞ 0 h(t)e t P dt = f P , (3.55)

  E λ , and (3.55) is proven. If υ < 2, we may replace compact by trace class in the argument above. Then, using Lemma B.1, we see that the operator ∞ 0 h (t) L t dt has the same non-zero spectrum when acting on H or on H 0 and thus, by Lidskii's trace theorem, the same trace. This ends the proof with Proposition 3.32.

S

  p , its spectrum is in p (see [GGK00, Corollary 3.4 p.54]), so that λ∈σ R (P ) e λt 0 z -λ p < +∞.Since t 0 > 0 and p > d + 1 are arbitrary, Proposition 3.3 follows.Chapter 4

  Remark 4.5. If M is a complexication for M , let B( M ) denote the Banach space of bounded holomorphic functions on M . Then, we may give a new denition of the space of real-analytic functions on M by G 1 (M ) = lim -→ M B M .

  To describe these tubes in more concrete terms, we may examine them in local coordinates. Given a realanalytic chart for M , it extends holomorphically to a chart for M . It hence denes a holomorphic trivialization for T * M , mapping T * M on T * R n . If we denote these coordinates by x = x + iy, ξ = ξ + iη, then for a point α ∈ T * M that writes (x, ξ) in local coordinates, the quantity |Im α| is uniformly equivalent to |Im α| |y| + |η| ξ .

  .9) that (T * M ) 0 may be seen as an open subset in the cotangent bundle T * M of a complexication M of M . Then set ω R = Re ω and ω I = Im ω. Notice that ω R and ω I are real symplectic forms on (T * M ) 0 . In local charts with x = x + iy, ξ = ξ + iη, the expression for ω is given by ω = dξ ∧ dx -dη ∧ dy =ω R + i(dη ∧ dx + dξ ∧ dy)=ω I

[ HS86 ]

 HS86 is to replace it by another I-Lagrangian submanifold of (T * M ) 0 . However, we will not work with all I-Lagrangian subspaces of (T * M ) 0 , we shall only consider adapted I-Lagrangians as we dene now. Denition 4.14. Let s ≥ 1 and τ 0 ≥ 0. Let Λ be an I-Lagrangian in (T * M ) 0 . We say that Λ is a (τ 0 , s)-adapted Lagrangian if it takes the form Λ = e H ω I G (T * M ) .

.

  Remark 4.16. Let m ∈ R and Ω be a manifold on which there is a notion of Kohn Nirenberg metric and of Japanese bracket (the main examples are T * M, (T * M ) 0 and an adapted Lagrangian Λ), then we dene as usual the KohnNirenberg class of symbol S m KN (Ω) as the space of C ∞ functions a : Ω → C such that for every k ∈ N we have (using the covariant derivative associated to the KohnNirenberg metric):sup α∈Ω ∇ k a(α) KN |α| m < +∞.For instance, in Denition 4.14 we ask for G 0 ∈ S 1/s KN (T * M ) 0 Since the adapted Lagrangians are uniformly smooth submanifolds with respect to the KohnNirenberg metric, and image of T * M under a uniformly smooth ow, the symbol class S m KN (Λ) is well dened, and to check that a ∈ S m KN (Λ), we can compute the derivatives either directly on Λ, or through the pullback by exp(H ω I G ), with covariant derivatives or with partial derivatives in coordinates.

  Λ is I-Lagrangian, we deduce that there are local solutions to this equation. However, since Λ is assumed to be of the form (4.11), we can nd an explicit global solution, given by Im θ(H ω I G ))dτ.

Propositions 4 .

 4 26 and 4.27 are consequences of Proposition 4.24 (see [BJ20, Corollary 2.2]). Further analysis leads to a density statement.

η 2 and the function χ + is supported in η 2 ,

 2 +∞ . Then dene for σ ∈ {+, -, 0} the symbol f σ on Λ by f σ (α) = χ σ Then notice that if α is in the support of f + then we have Im p Λ (α) = Im p e -τ H ω I G 0 (α) + O (τ |α| ) τ is small enough (depending on η). And similarly, if α belongs to the support of f -we have Im p Λ (α) ≤ -η 2 |α| .

(4. 31 )

 31 If α belongs to the support of f 0 then we have p e -τ H G 0 (α) ≤ η |α| ≤ Cη e -τ H G 0 (α) .

  is trivial when u = 0, we may divide by u H δ Λ in (4.34) to end the proof of the lemma when u ∈ H ∞ Λ .

  from one line to another. It follows that N (R) ≤ C n δp R n δ . The estimates on N (R) follows since, if (λ k ) k∈N denotes the sequence of eigenvalues of P , we have the relation µ k = 1 z-λ k (up to reordering, and recall that δ = 1/s).

  One can also refer to[START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]: the density statement is proved there under the name of Corollary 2.1.If f ∈ B m,b then the Laplace transform Lap(f )(z) = +∞ 0 e -zt f (t)is well-dened for Re z > -b. Moreover, successive integration by part ensures that|Lap(f )(z)| = 1 z m Lap(f (m) )(z) ≤ 1 |z| m f B m,b b + Re z .Hence, if b is greater than the real part of all Ruelle resonances for P , then, according to the counting bound in Theorem 4.38, we may dene a continuous linear form on B m,b by l : f → λ∈σ(P ) Lap(f )(-λ). However, if Re z > b, the function f z : t → e -zt t m+1 (m+1)! belongs to B m,b and, by Lidskii's Trace Theorem,l(f z ) = λ∈σ(P ) 1 (z -λ) m+2 = tr (z -X) -m-2 .Then, since X and V are Gevrey, the global trace formula (TFF) holds according to Theorem 9: for every f ∈ C ∞ c R * + we have l(f ) = γ T # γ e γ V |det (I -P γ )| f (T γ ).

(4. 37 )

 37 Notice that the right-hand side in (4.37) denes a continuous linear form on B m,b , provided that b is large enough (apply for instance Margulis' bound). Then, if Re z > b, the functionf z belongs to the adherence of C ∞ c R * + in B m,b (use the fact that f (m)z (0) = 0 to see so). Hence, an approximation argument provides that, for Re z > b, tr (z -X)-m = tr (z -X) -m-2 V |det (I -P γ )| e -zTγ T m-1 γ .Consequently, there are constants a 1 , a 2 ∈ C such that for Re z1 we have tr (z -X) -m = 1 (m -1)! γ T # γ e γ V |det (I -P γ )|e -zTγ T m-1 γ + a 1 + a 2 z.

  λ) m , the counting bound in Theorem 4.38 and dominated convergence. With Lemma 4.39, we are ready to relate the dynamical determinant d(z) dened by (12) with a regularized determinant. See [GGK00, Chapter XI] for the general theory of regularized determinants.

d

  (λ) = det m I + (λ -z) (z -P ) -1 exp (Q z (λ)) ,where det m denotes the regularized determinant of order m.Proof. By analytic continuation principle, we only need to prove this result for λ close to z. For such a λ the regularized determinant is dened bydet m I + (λ -z) (z -P ) -1 = exp   -≥m (z -λ) tr (z -P ) - γ V e -zTγ |det (I -P γ )| e (z-λ)Tγ --λ)T γ ) ! = d(λ)e -Qz(λ) .The applications of Fubini Theorem are justied when Re z 1 and |z -λ| is small enough by Margulis' bound.We are now ready to prove Theorem 10.Proof of Theorem 10. Let m be as in Lemma 4.40. Recall the Weierstrass primary factor(1.3) E(λ, m -1) = (1 -λ) expthe second expression being valid when |λ| < 1. It follows from Lidskii's Trace Theorem that for Re z 1 and λ ∈ C we havedet m I -(z -λ)(z -P ) -1 = +∞ k=0 E λ -z λ k -z , m -1 , (4.39) where (λ k ) k∈N denotes the sequence of Ruelle resonances of P . We want to use this expression with Lemma 4.40 in order to prove Theorem 10, but let us make an observation rst. If λ is a complex number such that Re λ ≥ |λ| /2. Hence, using the expression (12) and dominated convergence, we see that when |λ| tends to +∞ while satisfying (4.40), the function d(λ) tends to 1. In particular, d(λ) remains bounded when λ satises (4.40). Hence, we may assume in the following that that |λ| is large of course. When |λ| is large enough, we may apply Lemma 4.40 with z = |λ|. Then, notice that Q |λ| (λ) tends to 0 when |λ| tends to +∞ and thus we may ignore the factor exp (Q z (λ)) from Lemma 4.40. The other factor is given by (4.39) (with z = |λ|). Notice that if |λ k | ≥ 5 |λ| then λ-|λ| λ k -|λ| ≤ 1 hand if |λ k | < 5 |λ|, we have, since we assume (4.41), λ -|λ| λ k -|λ| ≥ a constant C such that Re λ k ≤ C for all k ∈ N (such a constant exists because P is the generator of a strongly continuous semi-group) and notice that for |λ| large enough, we have λ -|λ| λ k

  (4.42) and (4.44), that are valid respectively when |λ k | ≥ 5 |λ| and |λ k | < 5 |λ|, we nd thatlog det m I -(|λ| -λ)(|λ| -P ) -1 |λ k | -m + 48 m 4 # {k ∈ N : |λ k | < 5 |λ|} .

  the counting bound in Theorem 4.38, we see that# {k ∈ N : |λ k | < 5 |λ|} = O (|λ| sn ) , |λ k | -m = O |λ| sn-m .

  proof of Theorem 10 by plugging (4.46) and (4.47) in (4.45).

  to introduce a new distance to compare spectrum. If z ∈ C, we dene the distance d z on C ∪ {∞} \ {z} by d z (x, y)

d

  z,H (σ R (P ) ∪ {∞} , σ L 2 (P ) ∪ {∞}) ≤ C |ln | -1 p .

S 1 e

 1 2iπ( x-kT (x) L m a k, (ψ)(x)dx, and thus we want to bound L m a k, (ψ)∞ instead of L m a k,(1)∞ . As in the proof of Lemma 2.14, we notice that there are natural integer coecients that do not depend on a k, nor ψ such thatL m a k, (ψ) = n 1 +•••+nm+k=m c n 1 ,...,nm,k ψ (k)as in the proof of Lemma 2.14 and using the fact that ψ is of class C A , we nd constants C, R > 0 that do not depend on m, k or such thatL m a k, (ψ) c n 1 ,...,nm,k k! m j=1 n j !.As in the proof of Lemma 2.14, we introduce now the operator L a obtained by replacing the function a k, in the denition of L a k, by a : x → 1 1-x . Since the coecients in (A.1)do not depend on a k, nor ψ we haveL m a (a)(0) = n 1 +•••+nm+k=m c n 1 ,...,nm,k k! m j=1 n j !but direct computation shows that L m a (a) : x → 2 m m!

(B. 3 )

 3 The inclusion of B 1 in B induce an inclusion of E λ,i in B. Recalling (B.2), we deduce from(B.3) that E λ,1 = P λ,1 (B 1 ∩ B 2 ) = P λ,2 (B 1 ∩ B 2 ) = E λ,2 .Consequently, the spectrum of L 1 in U is contained in the spectrum of L 2 and the associated eigenspaces coincide. Since the roles of L 1 and L 2 are symmetric here, the lemma is proven.Example B.2. Let T be a C ∞ expanding map of the circle and L denote the associated transfer operator (1). Let z ∈ C * and take k, k > 0 such that |z| > max(λ -k , λ -k ), where λ is a dilation constant for T . Recalling Theorem 1, we can then apply Lemma B.1 withB 1 = C k (S 1 ), B 2 = C k (S 1 ), B = C 0 (S 1 ) and U = {w ∈ C : |w| > max(λ -k , λ -k )}. Itimplies that z is an eigenvalue of L acting on C k (S 1 ) if and only if it is an eigenvalue of L acting on C k (S 1 ). Moreover, when z is actually an eigenvalue the associated generalized eigenspaces coincide. This argument justies Denition 3 and proves that the resonant states associated to a C ∞ expanding map of the circle are C ∞ . A similar reasoning can be made in the case of Anosov dieomorphism to justify Denition 5 (in that case we will take a space of distributions for B).

V

  |1 -G λ,t 0 (z)| ≤ C e Re(λ)t 0 |λ| d+2 .Using Proposition 3.3, this implies that the innite productd (z) = λ∈σ R (P ) G λ,t 0 (z)converges uniformly on all compact subsets of C. Notice that the zeros of d(z) are precisely the Ruelle resonances of P . Now, we nd that e zt 0 (ln G λ,t 0 (z)) (d+1) = (-1) d+1 (d + 1)! e λt 0 (z -λ) d+2 and thus, for Re z 1, e zt 0 ln d (z) (T γ -t 0 ) d+1 e -z(Tγ -t 0 ) |det (I -P γ |det (I -P γ )| (d+1) = e zt 0 (ln d (z)) (d+1) , (C.2) where d(z) is the usual dynamical determinant dened by (12). From (C.2), we deduce that there are a polynomial F of degree at most d and µ ∈ C such that, for all z ∈ C, we have the Hadamard-like factorization d (z) = µ exp F (z) e -t 0 z λ∈σ R (P )

Moreover, for every

  p ∈ P, we have a 2 p -1 (g) = b 2 p -1 . Now, for every m ∈ N * , we deneB m = {f ∈ E : ∀n ≥ m : |a n (f )| ≤ b n } .The B m 's are closed subsets of E. Letting A = E \ ∪ m≥0 B m , we only need to show that the B m 's have empty interior. The lemma then follows by an application of Baire category theorem. Let consequently m ∈ N * and f ∈ B m . Then, for p a prime integer greater than m and > 0 we have|a 2 p -1 (f + g)| ≥ |a 2 p -1 (g)| -|a 2 p -1 (f )| ≥ b 2 p -1 -b 2 p -1 ,and if p is large enough, we have b 2 p -1 > 2b 2 p -1 / , since (b n ) n≥1 tends to 0. Hence, for every > 0 we have f + g / ∈ B m , so that B m has empty interior.

  Denition 7 (Ruelle spectrum for Anosov ow). Let (φ t ) t∈R be a C ∞ Anosov ow on a compact manifold M with generator X and V : M → C be a C ∞ function. Let z ∈ C and m ∈ N * . We say that z is a Ruelle resonance for (φ t ) t≥0 and V (or just for P = X + V ) of multiplicity m if there is 12 a Banach space B satisfying the points (i)-(iii) from Theorem 5 for some A > -Re z such that z is an eigenvalue of P acting on B with multiplicity m. The associated generalized eigenvectors are called resonant states for P . As usual, the Ruelle spectrum is dened as the set of Ruelle resonances, and is denoted by σ R (P ).Remark 3. The notation σ R (P ) for the Ruelle spectrum of P may be slightly misleading: in general the Ruelle spectrum of -P is not made of the -λ's where λ runs over the Ruelle

X and V and called Ruelle resonances (see Lemma B.3). The denition of the Ruelle spectrum in this context is the following: resonances of P . This is a matter of convention: the Ruelle resonances of -P describe the statistical properties of (φ -t ) t∈R when t → +∞ rather than those of (φ t ) t∈R when t → -∞, hence a sign ip.

Remark 4. Let us mention that there is another (equivalent) denition of the Ruelle spectrum that is very popular in this context and maybe slightly easier to manipulate. If the real part of z ∈ C is positive and large enough, then formula (11) denes an operator

  Giulietti, Liverani and Pollicott also proved exponential decay of correlations for the measure of maximal entropy of geodesic ows on Riemannian manifold with negative curvature satisfying a pinching condition[START_REF] Giulietti | Anosov ows and dynamical zeta functions[END_REF]. Exponential decay of correlations has been proved recently by Tsujii and Zhang for topologically mixing

volume-preserving C ∞ Anosov ows

[START_REF] Tsujii | Smooth mixing Anosov ows in dimension three are exponential mixing[END_REF]

(based on previous work by Tsujii

[START_REF] Tsujii | Exponential mixing for generic volume-preserving Anosov ows in dimension three[END_REF]

). Let us mention that, if the question of the existence of a spectral gap is crucial, there are many others interesting results regarding the distributions of Ruelle resonances, see for instance

[FS11, DDZ14, JZ17, FT17].

Understanding the distribution of Ruelle resonances for P is consequently both dicult and very important. As in the discrete-time case, this question can be dealt with by the introduction of a dynamical determinant. Its denition is slightly more involved in that case, it is dened for Re z 1 by

13

  13 Once again, classical bounds on the number of periodic orbits justify the convergence for Re z 1,

such as Margulis' bound [Mar04, Theorem 5] or the more elementary estimate [DZ16, Lemma 2.2].

  (TFF) for C ∞ (rather than analytic) Anosov ows. As in the discrete-time case, notice that in the absence on a general upper bound on the number of Ruelle resonances, there is no reason for the left-hand side of (TFF) to converge in any sense. In[START_REF] Jin | A local trace formula for Anosov ows[END_REF], Jin and Zworski proved a local version of (TFF) for C ∞ Anosov ow and used it to show that,

for any C ∞ Anosov ow, there is a vertical strip that contains an innite number of Ruelle resonances 16 a bound that has been sharpened by Naud in an appendix to their paper. This is one of the reasons why trace formulae are interesting: they are one of the only tools available to prove lower bounds on the number of resonances. The already mentioned work of Adam

[START_REF] Adam | Generic non-trivial resonances for Anosov dieomorphisms[END_REF]

also illustrates this idea.

  deed, there is much more exibility when constructing open systems. The picture described above for transitive Anosov dieomorphisms adapt without much change to basic hyperbolic sets: we have notions of Ruelle spectrum, at trace, dynamical determinant, etc (see 1.2.1 for details). In 1.2, we will construct weights for a Smale's horseshoe [Sma67, I.5] such that the dynamical determinant of the associated Koopman operator is explicit. From this construction, we will deduce for instance the following result:

Proposition 2 (See Corollary 1.19). Let E be a subset of N * . Then there are a dieomorphism T : S 4 → S 4 with hyperbolic basic set K and a smooth function g : S 4 → R + , positive on K, such that, for every n ∈ N * , the trace formula (TFM) holds if and only if n ∈ E (see (1.18) for the denition of the at trace).

The relationship between dynamical determinants and trace formulae in the discretetime case is so explicit that we will construct counter-examples in 1.2 that are much more pathological than those of Proposition 2, see Corollary 1.20.

A question that is deeply related to the problem of trace formulae is the existence of an upper bound on the number of Ruelle resonances. The method that gives Proposition 2 also proves the following result, that asserts that there is no general upper bound on the number of resonances for a hyperbolic dieomorphism.

  It allows us to rewrite the dynamical determinant d(z) as a particular case of WeinsteinAronszajn determinant (see[START_REF] Kato | Perturbation Theory for Linear Operators[END_REF] IV.6] and references therein).

dened by (4) coincides with the Fredholm determinant det (I -zL) of L see Proposition 2.19. When L is not trace class, we are still able to use the space H from Theorem 7 to study the dynamical determinant d(z): the nuclear power decomposition from

[START_REF] Baladi | Dynamical determinants and spectrum for hyperbolic diemorphisms. Geometric and probabilistic structures in dynamics[END_REF]

may be implemented on H.

  s = 1, we retrieve the class of real-analytic regularity, while for s > 1 the class of s-Gevrey functions is non-quasianalytic: it contains compactly supported functions.

	The methods that we developed with Bonthonneau to study Gevrey Anosov ows are
	exposed in Chapter 4 in an abridged version. Indeed, the proof of Theorem 10 requires
	technical preparations that are not directly related to our topic. It was thus not possible
	to present here the full content of [BJ20] (available on the arXiv) without causing an
	ination in the length of this thesis. Consequently, we give a short introduction to the
	FBI transform methods from [BJ20] in 4.1 focusing on the results that are needed for the
	proof of Theorem 10 that we give in 4.2. Finally, we will mention other applications of
	our FBI transform methods to the spectral theory of Gevrey Anosov ows in 4.3.
	Content of the appendices
	Let us describe briey the content of the appendices of this thesis. In Appendix A, we
	explain how the results from 2.2 (in particular Theorems 7 and 8) extend to more general
	transfer operators than (1). In Appendix B, we give the technical results that prove that
	the Ruelle resonances are intrinsically dened. In Appendix C, we establish a Hadamard-
	like factorization for the dynamical determinant (12) under the hypotheses of Theorem 9.
	Gevrey classes are discussed in 4.1.1. Let us just mention for now that Gevrey classes
	are nice examples of DenjoyCarleman classes depending on a parameter s ∈ [1, +∞[.

When

  (φ t ) t∈R un ot d'Anosov (voir la Dénition 6) de régularité C ∞ sur une variété compacte M et V : M → C une fonction C ∞ . On note X le générateur de (φ t ) t∈R et on forme l'opérateur P := X + V . On cherche à décrire l'asymptotique, lorsque t tend vers +∞, de la famille d'opérateurs de Koopman dénis par L La somme en argument de l'exponentielle parcourt les orbites périodiques γ du ot (φ t ) t∈R .Pour une telle orbite, T γ dénote sa longueur, T γ sa longueur primitive (c'est-à-dire la longueur de la plus petite orbite périodique de (φ t ) t∈R de même image que γ). Dyatlov et Zworski s'interrogent sur la validité de la formule de trace (TFF) lorsque le ot (φ t ) t∈R et le poids V ne sont plus supposés analytiques (mais au moins C ∞ ), toujours dans le but de mieux comprendre la répartition des résonances de En particulier, l'ordre de d est plus petit que ns.

	INTRODUCTION
	(briey On a aussi 0 V (φ t (x)) dt pour x un point de l'image de γ, et P γ = Dφ Tγ (x) |E u Résumé en français posé γ V = Tγ x ⊕E s x dénit un opérateur compact. De plus, les polynômes trigonométriques forment un sous-espace dense de H. L'opérateur (18) joue ici un rôle similaire à l'adjoint de (16) dans le cas des ots ont été utilisées récemment par Galkowski et Zworski pour étudier des problèmes d'EDP en régularité analytique [GZ19a, GZ19b, GZ20a] (nos résultats s'appliquent à la fois en régularité Gevrey et analytique). Ces méthodes basées sur la transformée de FBI nous Dans [DZ16], Ruelle. Cette question constitue le l conducteur de notre thèse. Notre point de départ permettent alors de prouver la borne suivante sur le déterminant dynamique (17). d'Anosov. L'espace H permet également d'étudier un déterminant dynamique associé à T , est l'application de Poincaré linéarisée associée à γ. Ici, E u et E s sont les directions respectivement stable et instable de (φ t ) t∈R et x est toujours un point de l'image de γ (l'application P γ dépend du choix de x, mais pas sa classe de conjugaison, le déterminant pour comprendre ce problème est la suggestion par Sébastien Gouëzel de s'y intéresser dans et de discuter la validité d'une formule de trace pour L (un analogue à temps discret de Théorème C (Voir le Théorème 10). Soit s ≥ 1. Si M, (φ t ) t∈R et V sont s-Gevrey alors le contexte de la régularité Gevrey. Rappelons qu'il s'agit d'une hypothèse de régularité montrer que L est en fait un opérateur à trace, et une formule de trace s'en déduit par le intermédiaire entre C ∞ et analytique, introduite par Gevrey pour étudier la régularité (TFF)). En eet, si L ne dénit en général qu'un opérateur compact sur H, on peut parfois il existe C > 0 tel que pour tout z ∈ C on a
	exposed in Chapter 4) can be used to understand other problems than those discussed in dans (17) est donc bien déni). Un des intérêts du déterminant dynamique (17) est qu'il des solutions de certaines EDP [Gev18]. Cette suggestion nous a amené à développer théorème de Lidskii. Tous ces résultats sont énoncés de manière quantitative en utilisant |d(z)| ≤ C exp (C |z| ns ) .
	this thesis for which the Gevrey setting is relevant, maybe beyond dynamical systems. It seems natural to expect that our methods can be applied to some PDE problems. For Cette thèse est dédiée à l'étude de la théorie spectrale associée aux dynamiques hyper-des outils d'analyse fonctionnelle adaptés à l'étude de la théorie spectrale des dynamiques admet un prolongement holomorphe à C dont les zéros sont exactement les résonances de hyperboliques, non seulement de régularité Gevrey, mais aussi dans des classes de régularité Ruelle de P [GLP13, DZ16]. Les déterminants dynamiques sont également des éléments le langage des classes de DenjoyCarleman (voir en particulier le Théorème 2.9). Nos
	instance, we could probably use our tools to discuss in the Gevrey setting the questions that have been dealt with by Galkowski and Zworski using similar tools in the real-analytic category [GZ19b, GZ20a]. Another possible applications of these FBI transform methods boliques C ∞ . Notre approche se base sur les notions d'ultradiérentiabilité et de classe de DenjoyCarleman. On s'intéresse principalement au problème de la formule de trace pour les ots d'Anosov, proposées par Dyatlov et Zworski dans [DZ16]. Avant de présenter essentiels de la théorie des fonctions zêtas dynamiques. méthodes nous permettent également d'établir un résultat sur le problème de la conjugaison plus générales: les classes de DenjoyCarleman ou classes de fonctions ultradiérentiables. pour les applications dilatantes ultradiérentiables (le Corollaire 2.12). La dénition des classes de Gevrey est rappelée dans 4.1.1. Rappelons que 1-Gevrey est Il s'agit également de classes de régularité intermédiaire entre C ∞ et analytique, plus Lorsque le ot (φ t ) t∈R et le poids V sont analytiques réels, certains outils d'analyse variées que les classes de Gevrey, ce qui nous permet d'énoncer des résultats plus précis sur Enn, nous clôturons le Chapitre 2 en énonçant un résultat sur la théorie spectrale de un synonyme d'analytique réel, le Théorème C redonne ainsi dans le cas s = 1 une borne
	in the Gevrey setting is also mentioned in [GZ20b, Remark 4]. cette question, rappelons les bases de l'approche fonctionnelle des propriétés statistiques fonctionnelle puissants peuvent être utilisés pour étudier les propriétés du déterminant la validité de la formule de trace (TFF). Ces outils que nous développons sont en quelque l'opérateur de Koopman de certains diéomorphismes hyperboliques de régularité Gevrey connue sur le déterminant dynamique, obtenue par les méthodes mentionnées ci-dessus
	des ots d'Anosov. dynamique (17). Ces outils, qui se basent sur les travaux de Grothendieck sur la théorie sorte la généralisation au cas ultradiérentiable des méthodes mentionnées ci-dessus dans que nous avons obtenu par des méthodes similaires à celles détaillées pour les applications [Rug92, Rug96, Fri95] (aucune borne n'était connue dans la cas s > 1). Notre preuve est
	de Fredholm [Gro55], ont été utilisés d'abord par Ruelle [Rue76], puis par Rugh [Rug92, dilatantes du cercle (voir le Théorème 2.27). cependant plus directe: nous travaillons directement avec l'opérateur de Koopman (16) et le cas analytique. Rug96] et Fried [Fri95]. De ces travaux, il découle en particulier que lorsque (φ t ) t∈R et Les quatre chapitres de ce document présentent l'essentiel de nos travaux autour de la Le Chapitre 3 est consacré à la preuve du résultat principal de cette thèse : son générateur P = X + V . D'autres applications de la transformée de FBI à l'étude des V sont analytiques réels, alors le déterminant dynamique (17) est d'ordre ni (voir la formule de trace (TFF) et de questions proches liées à la théorie spectrale des dynamiques ots d'Anosov sont également esquissées à la n du Chapitre 4, nouvelles mêmes dans le Théorème B (Voir le Théorème 3.1 et le Corollaire 3.2). Il existe une grande classe Dénition 1.2). Il est classique (voir Proposition 1.27) que cette propriété du déterminant liens entre formules de traces et déterminants dynamiques. Cette étude est menée pour les alors la formule de trace (TFF) est vériée. dynamique implique la formule de trace: hyperboliques ultradiérentiables. Nous commençons par expliciter dans le Chapitre 1 les d'applications ultradiérentiables telle que si M, (φ t ) t∈R et V appartiennent à cette classe cas s = 1.
	(16) Ici u est par exemple une fonction C ∞ de M dans C. Pour comprendre cette asymptotique, discret). Les principales relations entre déterminants dynamiques et formules de traces λ∈σ R (P ) e λt = γ T γ ots d'Anosov, mais aussi pour des systèmes hyperboliques à temps discret (des analogues e γ V δ Tγ . La classe de régularité en question est décrite en détail dans 3.1 en utilisant le lan-(TFF) des questions que nous avons détaillées dans le cas continu se posent également à temps |det (I -P γ )| gage des classes de DenjoyCarleman introduit dans le chapitre précédent. Remarquons
	juste pour l'instant que cette classe de régularité est plus grande que toutes les classes on introduit [BL07, BL13, FS11] une notion de spectre pour P , le spectre de Ruelle σ R (P ). Ce spectre peut être déni de la manière suivante. Pour Re z 1, on dénit la résolvante de P par R(z) := +∞ 0 e -zt L t dt : C ∞ (M ) → D (M ) . sont données dans la Proposition 1.27 (pour les ots d'Anosov) et dans la Proposition 1 Ici, les notations sont les mêmes que dans (17) et l'égalité est à comprendre au sens des distributions sur R * de Gevrey. Nous prouvons également une borne sur le nombre de résonances de Ruelle + , c'est-à-dire que si ϕ est une fonction C ∞ à support compact sur R * (pour les systèmes à temps discret, voir aussi le Théorème 1.3). Nous utilisons ensuite ces + alors λ∈σ R (P ) Lap(ϕ)(-λ) = γ T # γ sous les hypothèses du Théorème B (voir la Proposition 3.3) qui implique en particulier liens pour construire des exemples de diéomorphisme hyperboliques C ∞ pour lesquels la que le membre de gauche dans (TFF) dénit bien une distribution sur R * + . De manière formule de trace est fausse (voir par exemple les Propositions 2 et 3). notable, le Théorème B n'est pas obtenu en prouvant que le déterminant dynamique (17) e γ V nous commençons à utiliser cette notion pour étudier la théorie spectrale des dynamiques |det (I -P γ )| Dans le Chapitre 2, nous rappelons la dénition des classes de DenjoyCarleman et ϕ(T γ ), est d'ordre ni (comme c'était le cas pour les ots analytiques). Nous ne nous attendons
	d'ailleurs pas à ce que les déterminants dynamiques associés aux ots considérés dans le On peut alors montrer que R(z) s'étend en une famille méromorphe sur C d'opérateurs de nous nous intéressons tout d'abord à des systèmes hyperboliques parmi les plus simples: C ∞ (M ) dans D (M ), et que ses résidus sont de rang ni. Les pôles de R(z) sont alors hyperboliques. An de rendre le plus clair possible le fonctionnement de nos méthodes, où Lap(ϕ) dénote la transformée de Laplace de ϕ dénie pour z ∈ C par Chapitre 3 soient d'ordre ni.
	appelés résonances de Ruelle de P (la multiplicité de la résonance étant le rang du résidu). Lap(ϕ)(z) = Si les résultats du Chapitre 3 sont relativement satisfaisants en ce qui concerne la +∞ les applications dilatantes du cercle. On prouve en particulier le résultat suivant, connu e -zt ϕ(t)dt. formule des traces, la question de l'ordre du déterminant dynamique nous semblait égale-L'ensemble des résonances de Ruelle forment le spectre de Ruelle. Les résonances de Ruelle contiennent de nombreuses informations sur les propriétés auparavant uniquement dans le cas analytique [Rue76]. 0 ment digne d'intérêt. Il s'agit du sujet du quatrième et dernier chapitre de cette thèse,
	Le membre de droite dans (TFF) correspond à une trace généralisée pour l'opérateur de Théorème A (Voir le Théorème 2.9). Soit T une application dilatante du cercle C ∞ (voir dans lequel nous présentons une version abrégée d'un travail en collaboration avec Yannick statistiques du ot (φ t ) t∈R , il est donc important d'en comprendre la distribution. Pour Koopman (16), dont la valeur est donnée par la formule de trace de Guillemin [Gui77] la Dénition 1). Il existe un espace de Hilbert H, continûment inclus dans l'espace des Guedes Bonthonneau. Le principal outil technique dans le Chapitre 4 est une transfor-cela, on peut introduire suivant Ruelle [Rue76] le déterminant dynamique déni pour Re z 1 par voir aussi [DZ16, Appendix B]. Le membre de gauche quant à lui joue le rôle de la somme fonctions C ∞ sur le cercle, sur lequel l'opérateur de transfert mée de FBI analytique dans l'esprit de [HS86, Sjö96]. An que la taille de cette thèse
	d(z) = exp -des valeurs propres (le spectre de Ruelle étant ici la notion pertinente pour dénir cette γ T γ T γ  reste raisonnable, nous admettrons les résultats techniques nécessaires an d'utiliser cette  e γ V |det (I -P γ )| e -zTγ . (17) somme). Le formule de trace (TFF) donne donc une interprétation spectrale de la formule de trace de Guillemin. L : u →  x → T y=x 1 u(y)  transformée de FBI pour étudier les ots d'Anosov de régularité Gevrey (les preuves de (18) |T (y)| ces résultats sont détaillées dans [BJ20] disponible sur arXiv). Des méthodes similaires

Soit t : u → exp t 0 V • φ τ dτ u • φ t .

  Denition 1.2. Let f be an entire function. The order of f may be dened as

		CHAPTER 1
	lim sup r→+∞	log + log + sup |z| r |f (z)| log r
	where log + x = log max (1, x). If f is non-zero and has nite order, let p be the smallest
	natural integer such that
	m 0

  1.14) Such a sequence exists since N 0 is locally bounded. Notice that if |z| 1 2 and p ∈ N then where E is the Weierstrass primary factor from (1.3). Consequently, we can choose an increasing sequence of integer (p m ) m∈N such that p m → m→+∞ +∞ and the innite product m 0 E z zm , p m converges uniformly on all compact subsets of C. For m 0 large enough, dene an entire function f m 0 by

	|E (z, p) -1|	1 2	sup 2 |w| 1	E (w, p)	1 2 p

  1 , and this last sum converges provided that N is large enough. Hence, we see that |h(z)|,

			(1.32)
	and
	sup + t∈R * n∈N	e tx ϕ (k) n (t) < +∞.	(1.33)
	Then, since the series (1.28) converges absolutely, we nd with (1.32) that
	γ

for z ∈ C Rn , grows at most polynomially with R n . Since C Rn remains in the domain where (1.29) holds, (1.31) follows.

We prove now the implication (ii) ⇒ (i). Let x > 0 be as above. Write k = ρ where ρ is from (1.25). Choose z ∈ C such that Re (z) > x. Then, we can nd a sequence (ϕ n ) n∈N of C ∞ functions, compactly supported in R * + such that lim n→+∞ sup t∈R e tx ϕ n (t) -t k e -zt = 0

  then the class of regularity C A is the class of s-Gevrey functions, that we will often denote by G s in this thesis. We will give more detailed information on this class of regularity in 4.1.1. Notice that in the case s = 1, See 3.1 for a more advanced discussion on this classes of regularity in particular, we will dene what it means for a ow on a compact manifold to be of regularity C κ,ν . Our use of DenjoyCarleman classes will be very basic, let us just recall a few wellknown facts for the sake of contextualizing. It follows from the logarithmic convexity of (A k ) k∈N that the class of functions C A is closed under multiplication and composition see[START_REF] Rudin | Real and Complex Analysis[END_REF] Chapter 19]. We say that the class C A is quasi-analytic if the following holds: for every function f of class C A on a connected open subset U of R n , if there is x 0 ∈ U such that all the derivatives of f vanish at x 0 then f is identically equal to 0. If the class C A is not quasi-analytic, then there are compactly supported functions of regularity C A and, consequently, C A partitions of unity. The original motivation behind the introduction of DenjoyCarleman classes is the DenjoyCarleman Theorem that characterizes quasianalytic classes among them.

	we retrieve the class of real-analytic functions (this is an easy consequence of Taylor's and
	Cauchy's Formulae).
	Example 2.2.

Let κ > 0 and υ ≥ 1. We denote by C κ,υ the class of regularity obtained by taking A = (exp (k υ /κυ)) k∈N . These are not standard classes of regularity. Actually, we are not aware of any appearance of these particular classes in the literature before our work, but the ultradierentiable classes that are used in

[START_REF] Teofanov | Ultradierentiable functions of class M τ ,σ p and microlocal regularity[END_REF]

look a bit alike. However, the classes C κ,υ are very interesting when considering hyperbolic dynamics: see 2.2.3 but especially Chapter 3. Indeed, when υ < 2, the trace formula (TFF) holds for Anosov ow of regularity C κ,ν . Theorem 2.3

[START_REF] Denjoy | Sur les fonctions quasi-analytiques de variable réelle[END_REF][START_REF] Carleman | Sur un théorème de M. Denjoy[END_REF][START_REF] Carleman | Sur les fonctions indéniment dérivables[END_REF]

). Let A = (A k ) k∈N be an increasing, logarithmically convex sequence of positive real numbers such that A 0 = 1. Then the DenjoyCarleman class C A is quasi-analytic if and only if

  Thi08, Theorem 1].Remark 2.5. Let us consider the classes C κ,υ from Example 2.2. For κ > 0 and υ ≥ 1, we deduce from Proposition 2.4 that the class C κ,υ is closed under dierentiation if and only if υ ≤ 2. When υ < 2, we have the estimate We will see in 2.2 below that, in the case of expanding maps of the circle, the condition (2.6) alone ensures that the trace formula (TFM) is satised by expanding maps of the circle of regularity C A . The case of DenjoyCarleman

					CHAPTER 2
	a stronger condition than (2.5). classes for which the weaker condition (2.5) holds but not (2.6) is more involved but very
	interesting (see Remark 2.22 and 2.2.3).
	We tend consequently to think that, when considering the question of trace formulae
	in DenjoyCarleman classes, it is relevant to distinguish classes that are closed under
	dierentiation or not (and, among those closed under dierentiation, that satisfy (2.6)
	or not). We will argue in favor of this approach in Remark 2.22, see also the examples
	from 2.2.3 for a more concrete approach of this point. Actually, being closed under
	dierentiation is a reasonable assumption to make on an ultradierentiable class if someone
	lim k→+∞	A k+1 A k	1 k	= 1,	(2.6)

wants to make dierential geometry in that class (see

[START_REF] Kriegl | The convenient setting for non-quasianalytic Denjoy-Carleman dierentiable mappings[END_REF]

for a discussion of that fact). Proposition 2.10 and Corollaries 2.11 and 2.12 also reinforce the idea that Denjoy Carleman classes closed under dierentiation are better-behaved than the others.

  Let us mention for further reference that it follows from Proposition 2.4 that there is γ > 1 such that the condition (2.8) is satised by the sequence (A k ) k∈N for every k ∈ N if and only if the class C A is closed under dierentiation.Proof of Lemma 2.6. Since w is dened as an inmum of increasing functions, w is increasing. Since w(x) is smaller than x k k!A k for all k, it is clear that w vanishes at all orders in 0.If x ∈ R * + , since x k k!A k → k→+∞+∞, the inmum in the denition of w(x) is attained by a nite number of integers k. Denote by k(x) the largest integer that realizes this inmum.

	Notice that if

  Assume that the DenjoyCarleman class C A is closed under dierentiation. Then the elements of the Hilbert space H from Theorem 2.9 belong to the Denjoy Carleman class C A . 2.11. Assume that the DenjoyCarleman class C A is closed under dierentiation. Then the resonant states of L are of class C A .Corollary 2.12. Assume that the DenjoyCarleman class C A is closed under dierentiation. Let T 1 and T 2 be expanding maps of the circle of regularity C A and assume that they are C 1 conjugated. Then the conjugacy is actually of regularity C A .Notice that Proposition 2.10 and Corollaries 2.11 and 2.12 are not from[START_REF] Jézéquel | Transfer operator for ultradierentiable expanding maps of the circle[END_REF]. Concerning Corollary 2.11, it is well-known that the resonant states of an expanding map of the circle T are C ∞ under the sole assumption that T is C ∞ (see Lemma B.1).

	We will then prove Theorem 8 in 2.2.2. Actually, this proof will also come with
	quantitative estimates see Propositions 2.16, 2.17 and 2.18. Finally, we will see how
	these results specify to the particular classes from Examples 2.1 and 2.2 in 2.2.3.

If Proposition 2.10 does not look that impressing, it admits the two following corollaries that may be more interesting.

Corollary

In Appendix A, we explain how the results from this section can be adapted to study weighted transfer operators that are more general than (2.11).

2.2.1 Compactness of the transfer operator

Let us start the proof of Theorem 2.9. Let θ ∈ ]1, λ[ be xed once for all. If n ∈ Z, we write e n for the function on the circle e n : x → e 2iπnx . Dene the family (π n ) n∈N of orthogonal projectors on L 2 S 1 by

  Proof of Corollary 2.11. Applying Lemma B.1 (as in Example B.2), we nd that the eigenvectors associated to non-zero eigenvalues of L acting on H are in fact the resonant states for L in the sense of Denition 3. Hence, the resonant states for L belong to H and are thus of regularity C A according to Proposition 2.10.Proof of Corollary 2.12. Any expanding map of the circle has at least one xed point (this is for instance a consequence of the classical result [KH95, Theorem 2.4.6]). Hence, without

  Notice that if T 1

	t t∈R	and T 2 t t∈R	are two families of dieomorphisms
	as above, then their composition may naturally be dened as T 1 t • T 2 0 t∈R	. Moreover,
	if there are systems of cones Θ, Θ and Θ such that T 1 t t∈R	is cone-hyperbolic from Θ
	to Θ and T 2 t t∈R

  The proof is similar to the proof of[START_REF] Baladi | Dynamical determinants and spectrum for hyperbolic diemorphisms. Geometric and probabilistic structures in dynamics[END_REF] Lemma 4.21]. Schatten class S p for every p ≥ 1 such that p > d+1 k+1 . Moreover, its norm in this Schatten class is bounded by

	We are now ready to prove Lemma 3.27.
	Lemma 3.27. Under the hypotheses of Lemma 3.25 and if in addition α < 1 2 , the operator
	h (t) M t dt	(3.31)
	R
	belongs to the

  since ω∈Ω S υ is dense in H Ω , the operator L t+t sends H Ω into Φ H 0 . Denote by Ψ : Φ H 0 → H 0 the inverse of the isomorphism induced by Φ, and notice that L t+t coincide on H with Ψ • L t+t • Φ, and is thus bounded from H to H 0 . Finally, from Proposition 3.17, we know that L t+t : H Ω → H Ω depends continuously on t ∈ ]-t 1 , t 1 [ in the strong operator topology, and consequently so does L t+t : H → H 0 .

  3.45)converges in S p provided that is large enough(there are a nite number of non-zero terms with k < N that are also in S p thanks to Proposition 3.17 since h (t) vanishes for t ≤ t 0 ).

	Now notice that, for every t ∈ R, the sum
		ψ (t) h (t) L t
		∈Z
	converges in operator norm topology to h (t) L t , and the convergence is uniform in t (pro-
	vided that	is large enough), so that the sum (3.45) is in fact the operator
		+∞
		h(t)L t dt,
		0

44) is in the Schatten class S p , with norm in this class an O e - (there is a nite number of possible values for r). Thus the sum ∈Z R ψ (t) h (t) L t dt (which is consequently in S p .

  1 [, and thus Lemma 3.29 ensures that

  M ) (and thus into H), the operator L t induces a bounded operator L t : H → H. Since (L t ) t≥0 is locally uniformly bounded and (L t u) t≥0 depends continuously on t as an element of H when u ∈ C ∞,υ (M ) (see Lemma 3.7), the semi-group (L t ) t≥0 is strongly continuous.

  Lemma 3.42. For all t ≥ t 0 , the operatorL t is bounded from H to H. If z ∈ C is such that Re (z) 1 then (z -P ) -1 is bounded from H to H. L t is bounded from H to H. Since it sends C ∞,υ (M ) into itself, L t sends H 0 into H.

	Proof. Let u ∈ H then
	L t u 2 H ≤ sup τ ∈[t,t+t 0 ]	L τ	2 H→ H u 2 H .
	Now, recall [Kat66, Problem 1.15 p.487] that if Re (z)	1 and u ∈ H then
	(z -P ) -1 u =	+∞	e -zt L t udt.
	0

Thus

  As in the proof of Lemma 3.43, dene the function h on R * h is C ∞ and compactly supported in [t 0 , +∞[, it satises the assumption of Proposition 3.32 and, working as in the proof of Lemma 3.43, we see that the operator

	+ by
	h (t) = zh (t + t 0 ) + h (t + t 0 ) .
	+∞
	0

Since

  Assuming that α does not belong to C 0 , we know that Θ T 1 ,ξ (α) belongs to C γ u or Θ -T 0 ,ξ (α) belongs to C γ s . Let us assume for instance that Θ T 1 ,ξ (α) belongs to C u γ (the other case is symmetric). Then again there are two possibilities: either Θ -T 0 ,ξ (α) belongs to C u γ or it does not. If it does then (for |α ξ | large enough)

	α)| δ	(4.28)

  In the second case, we may apply (i) in Lemma 4.35 to nd that Re p Λ (α) = Re p e (α) + τ {G 0 , Re p} + O τ 2 0 |α| 2δ-1 is small enough. Here, we added the constant C so that (4.32) remains true for any α in the support f 0 . Now, Proposition 4.34 and (4.32) give that (the constant C may vary from one line to another)-Re |α| δ f 0 T Λ P u, T Λ u ≥ -1 h Λ |α| δ f 0 (α) Re p Λ (α) |T Λ u(α)| 2 dα -C u H 0 (α) |α| 2δ |T Λ u(α)| 2 dα -C u H 0Applying CauchySchwarz formula, we nd then thatΛ f 0 (α) |α| 2δ |T Λ u(α)| 2 dα ≤ C u H δWorking similarly with (4.32) replaced by the better estimates (4.30) and (4.31), we nd that (4.33) still holds when f 0 is replaced by f + or f -. Summing these three estimates, we get

	ω I G 0 ≤ --τ H C |α| δ + C, 1				(4.32)
	provided that τ Λ	u H δ Λ
	≥	1 C Λ	f 0 Λ	P u H 0 Λ	+ u H 0 Λ	.	(4.33)

Λ u H δ Λ .

Since their introduction by Lasota and Yorke to study the existence of absolutely continuous invariant measures for piecewise expanding map of the interval [LY73], LasotaYorke inequalities have become a very common tool in the study of statistical properties for hyperbolic dynamical systems.

This is why we consider the transfer operator instead of its adjoint, the Koopman operator. Indeed, in order to make the Koopman operator quasi-compact, one has to make it act on a space of distributions of negative order.

We are mainly interested in this thesis in dynamics that are at least C ∞ , so that we only dened the Ruelle spectrum in that case. See [Bal18, Denition 1.1] for a more general denition.

The convergence of the right-hand side in (4) for z small enough is ensured by classical bounds on the number of periodic orbits for expanding maps, see for instance[START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems[END_REF] Theorem 2.46].

Basically, since the multipliers associated to the xed points of T n are dierent from 1, we can check that it makes sense to integrate the Schwartz kernel of L n on the diagonal of S 1 × S 1 using the general theory from[START_REF] Hörmander | The Analysis of Linear Partial Dierential Operators[END_REF] 8.2]. Taking this integral as a denition of the at trace, a computation yields (5).

It follows from Lemma B.1 that if L is compact when acting on a reasonable space of holomorphic functions then its non-zero spectrum on that space coincides with its Ruelle spectrum.

Other tools are available though, such as the specication property, a notion developed by Bowen[START_REF] Bowen | Periodic points and measures for Axiom A dieomorphisms[END_REF][START_REF] Bowen | Periodic orbits for hyperbolic ows[END_REF][START_REF] Bowen | Some systems with unique equilibrium states[END_REF] that has been fruitfully revisited recently, see for instance[START_REF] Climenhaga | Equilibrium states beyond specication and the Bowen property[END_REF][START_REF] Climenhaga | Intrinsic ergodicity via obstruction entropies[END_REF][START_REF] Climenhaga | Unique equilibrium states for ows and homeomorphisms with non-uniform structure[END_REF].

The convergence for z small enough is still ensured by classical bound on the number of periodic orbits, see for instance[START_REF] Bowen | Periodic points and measures for Axiom A dieomorphisms[END_REF] Theorem 4.5]

Actually, neither[START_REF] Butterley | Smooth Anosov ows: Correlation spectra and stability[END_REF][START_REF] Butterley | Robustly invariant sets in ber contracting bundle ows[END_REF] nor[START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov ows[END_REF] include the case of weighted Koopman operators. However, it is clear for specialists that their methods also apply in this case. The general case is dealt with in[START_REF] Dyatlov | PollicottRuelle resonances for open systems[END_REF].

measure for T associated to the potential log g. Consequently, we construct a Gibbs measure for which there are a lot of terms in the correlations asymptotics.

And even of any non-integral order according to footnote 1.

There is an error in the expression for ξ α ∂ β f (ξ) in the proof of [Jéz20a, Proposition 5.3]. However, the proof is easily xed by using the correct formula that we give here.

It makes easier to apply the results from Appendix B or to dene the norm • H in (3.50) for instance.

This ensures that it is C κ,υ for any κ > 0 and υ > 1, so that all the Fourier multipliers that appear later are automatically well-dened.

In order to lighten notations, we sometimes drop the index Λ when Λ = T * M . That is we use the short hand T = TT * M and S = ST * M .

With the notations above, it would be more precise to say that T * T is the inclusion of E 1,R 0 (M ) in E 1,R 1 (M ) .

We recall that A tr is our notation for the transpose of A.

Notice that the trace formula (TFF) may be deduced from this equality using residue's formula as in the proof of Proposition 1.27.

The estimate (E.3) may be established using the methods from[START_REF] Jézéquel | Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response[END_REF]. We asserted there that there was an issue in the presence of non-simple resonances, but we realized since then that it was not the case.

Remerciements

CHAPTER 3 is C ∞ from R to C ∞,υ (M ). Indeed, the multiplication is continuous from the product C ∞,υ (M ) × C ∞,υ (M ) to C ∞,υ (M ). The map (3.6) is easily seen to be C ∞ from R to C 0 (M ), and one may notice that its derivatives are valued in C ∞,υ (recall that the classes of regularity C κ,υ , and hence C ∞,υ , are closed by composition) with uniform bounds locally in t. Then, by successive applications of Taylor's formula at order 1 with integral remainder, one gets that the map (3.6) is C ∞ from R to C ∞,υ (M ), ending the proof of the lemma (we use the exact formula for the remainder in order to bound it in C ∞,υ (M )).

Lemma 3.8. Let B be a Banach space such that B ⊆ D υ (M ) , the inclusion being continuous. Assume that, for all t ∈ R + , the operator L t is bounded from B to itself, and that (L t ) t≥0 is a strongly continuous semi-group of operator of B. Then the generator of (L t ) t≥0 coincides with P on its domain which is {u ∈ B : P u ∈ B} .

Proof. Denote for now the generator of (L t ) t≥0 by X. Let u ∈ B be in the domain of X, then the map R + t → L t u ∈ B is dierentiable at 0 and its derivative at 0 is Xu (by denition of X). Since the inclusion B ⊆ D υ (M ) is continuous, the same is true for the map R + t → L t u ∈ D υ (M ) , whose derivative at 0 is Xu according to Lemma 3.7. Thus Xu = Xu ∈ B.

Reciprocally, if u ∈ B is such that Xu ∈ B, then we may dene a C 1 map c : R + → B by c (t) = u + t 0 L τ Xudτ for all t ∈ R + . Notice that c (0) = Xu. Since the inclusion B ⊆ D υ (M ) is continuous, the map c is still C 1 when seen as a map from R + to D υ (M ) and we have c (0) = u and c (t) = L t Xu for all t ∈ R + , so that c (t) = L t u for all t ∈ R + , using Lemma 3.7. This proves that u belongs to the domain of X.

Local spaces

We dene now local spaces H Θ,α that will be the basic pieces to construct the space H from Theorem 3.1. These spaces will depend on the choice of a system of cones Θ: this system encodes the three distinguished directions from Denition 6 of an Anosov ow.

These spaces are Sobolev-like spaces similar to the spaces from [Bal18, Denition 4.16] or from [START_REF] Baladi | Anisotropic Hölder and Sobolev spaces for hyperbolic dieomorphisms[END_REF] (for discrete-time systems) or [START_REF] Adam | Opérateurs de Transfert et Moyennes Horocycliques Sur Les Variétés Fermées[END_REF][START_REF] Adam | Horocycle averages on closed manifolds and transfer operators[END_REF] (even though the approach is a bit dierent, spaces in [START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov ows[END_REF] are also Sobolev-like spaces). As in [BT07, BT08, Bal18, Ada18b, Ada18a], we will use PaleyLittlewood decomposition to study these spaces and the action of Koopman operators on them. However, as in [START_REF] Jézéquel | Local and global trace formulae for smooth hyperbolic dieomorphisms[END_REF], we cannot use the usual dyadic PaleyLittlewood decomposition since the weights that we use to dene our Sobolev-like spaces have a growth faster than polynomial, so that we will introduce an adapted PaleyLittlewood-like decomposition. This approach is slightly dierent from the strategy exposed in 2.2 since the annuli that will appear in this decomposition are not of CHAPTER 3 In all other cases, we say that ( , j) → (n, i). Let us list the cases in which ( , j) → (n, i) in prevision of the proof of Lemma 3.21:

• i = j = 0 and < n + νn 1-α ;

• i ∈ {1, . . . , r -2} , j = f and n < -4 + log 2 a;

• i, j ∈ {1, . . . , r -2} , i ≥ j + 1 and n < -4 + log 2 a;

Lemma 3.21 is the main tool to use the hyperbolicity of the dynamics to rule out the transitions of frequencies that do not occur in the linear picture. Lemma 3.21. For i ∈ {0, . . . , r -1, f }, set α i = α if i = 0 or r -1, and α i = 1 otherwise.

There are c > 0 and N > 0 such that if ( , j) , (n, i) ∈ Γ we have: ( , j) → (n, i) or max (n, ) ≤ N or, for all x ∈ K and t ∈ R,

Proof. We will make massive use of the following fact in this proof : if C + and C -are two closed cones in R d+1 such that C + ∩ C -= {0} (we say that such cones are transverse) then for all ξ ∈ C + and η ∈ C -we have

Assume that (n, i) , ( , j) ∈ Γ are such that ( , j) → (n, i) and max (n, l) > N for some large N , and take ξ ∈ supp ψ Θ ,n,i , η ∈ supp ψΘ, ,j and t ∈ R. We go through the dierent cases in which ( , j) → (n, i) as listed above.

• If i = j = 0 and < n + νn

Global space: rst step

We are now ready to start the proof of Theorem 3.1 using the tools from 3.2 and 3.3.

So let M be a compact d + 1-dimensional C κ,υ manifold, let (φ t ) t∈R be a C κ,υ Anosov ow on M , and let V : M → C a C κ,υ function. We denote by X the generator of (φ t ) t∈R and write P = X + V . We also denote by (L t ) t∈R the associated Koopman operator, dened by (9). We x t 0 > 0 from now on.

We will construct in this section two auxiliary Hilbert spaces H and H 0 . The space H 0 almost satises the conclusions of Theorem 3.1 (this is the point of Proposition 3.32) but the Koopman operator L t is bounded from H 0 to itself only for large values of t a priori. This problem will be settled in 3.5. The rst thing that we need to do in order to construct the spaces H and H 0 is to show that, locally in space and for large times, the action of the ow (φ t ) t∈R behaves like the local model that we studied in 3.3, this is the point of Lemma 3.29. Indeed, we construct in Lemma 3.29 a system of admissible charts adapted to the dynamics of (φ t ) t∈R (this is a continuous-time analogue of [Jéz20a, Lemma 8.1]). We can then glue copies of the local spaces from 3.2 to dene the global spaces H and H 0 . Finally, we state and prove Proposition 3.32. Lemma 3.29. There are a nite set Ω, an integer r and t 1 ∈ ]0, t 0 [ such that:

(i) there is no periodic orbit of (φ t ) t∈R of length less than 3t 1 ;

(ii) for all ω ∈ Ω there is a

(iv) for all ω ∈ Ω, there is a system of r + 2 cones Θ ω = (C 0,ω , . . . , C r,ω , C f,ω ) in R d+1 (with respect to the direction e d+1 );

(v) for every ω, ω ∈ Ω and t ∈ [t 0 , 3t 0 ] there is a C κ,υ immersion F ω,ω ,t : R d → R d+1 such that the associated family T ω,ω ,t t t ∈R (dened by (3.14)) is indeed a family of dieomorphisms and is cone-hyperbolic from Θ ω to Θ ω ;

CHAPTER 3

Remark 3.45. As pointed out after the statement of Theorem 3.1, the point (v) of Theorem 3.1 proves trace formula (TFF) which was stated as an equality between distributions on R * + . However, it is clear from the proof that the equality in fact holds in the dual of the space of compactly supported C d+2 functions on R * + whose d + 2th derivative has bounded variation. In fact, using the same trick as in the proof of Proposition 3.3, we see that trace formula holds in the dual of the space of compactly supported C d+1 functions on R * + whose d + 1th derivative has bounded variations.

We end this section with the proof of Proposition 3.3.

Proof of Proposition 3.3. First of all, we need to prove that, when Re (z) 1, the essential spectral radius (see Denition 2) of the operator

acting on H is zero. From the proof of Lemma 3.43, we know that the essential spectral radius of (z -P ) -1 is zero. Then if r > 0 is such that (z -P ) -1 has no eigenvalue of modulus r we may dene the spectral projection

w -(z -P ) -1 -1 dw.

Then I -Π r has nite rank and the spectral radius of (z -P ) -1 Π r is less than r. Since L t 0 commutes with (z -P ) -1 , it also commutes with Π r and thus the spectral radius of L t 0 (z -P ) -1 Π r is less than L t 0 r. Then writing

and using Hennion's argument [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux lipchitziens[END_REF] as in the proof of Lemma 3.43 (notice that the second term of the right-hand side of (3.57) has nite rank), we see that the essential spectral radius of L t 0 (z -P ) -1 is less than L t 0 r. Since r > 0 may be chosen arbitrarily small, the essential spectral radius of L t 0 (z -P ) -1 is zero. Consequently, using functional calculus in nite dimension as in the proof of Lemma 3.44, we may prove that the spectrum of L t 0 (z -P ) -1 is made of the e t 0 λ z-λ where λ runs over the Ruelle spectrum of P . On the other hand, according to Proposition 3.32 (with h the characteristic function of [t 0 , +∞[ and k = 0), the right-hand side of (3.56) denes an operator on H 0 which is in the Schatten class S p for any p > d + 1 (in particular it is compact and has essential spectral radius zero). We may use Lemma B.1 to get that the spectrum of this operator is the same as the spectrum of the operator (3.56) acting on H, that we just described.

Consequently, for all p > d + 1, since the operator acting on H 0 is in the Schatten class CHAPTER 4 Φ T being an admissible phase (as dened in Denition 4.10), and a being a semi-classical analytic symbol, elliptic in the symbol class h -3n 4 S 1, n 4 (with the subtlety that a(α, x) is maybe only dened for α x and x close to each other). An adjoint analytic FBI transform is an operator S :

is the kernel of an analytic FBI transform.

Remark 4.12. While it does not appear in the notation, the symbol a is allowed to depend on the small implicit parameter h > 0 (but it has to satisfy estimates uniformly in h). We say that Φ T is the phase of T and that a is its symbol.

Recall that the fact that K T is the kernel of T means that if u is a smooth function on M then T u is dened by the formula

Actually, it follows from the holomorphy of K T that T u is well-dened as a holomorphic function on (T * M )

Notice that if T is an analytic FBI transform, its adjoint T * with respect to the L 2 spaces on M and T * M is an adjoint analytic FBI transform, since the kernel of T * is given by (x, α) → K T (α, x) (the cotangent bundle T * M is endowed with its canonical volume form).

Due to the holomorphy condition that we impose on the kernel of an FBI transform, it is not clear that such an object exists on a general compact real-analytic manifold.

However in [BJ20, Lemma 2.3] we prove: Proposition 4.13. Let M be a compact real-analytic manifold. Then there is an analytic FBI transform on M .

The proof of Proposition 4.13 is based on Hörmander's solution to the ∂ equation, following a suggestion by Maciej Zworski.

Complex Lagrangian deformations

As noticed in Remark 4.12, if u is a hyperfunction on M and T an analytic FBI transform, then T u denes not only a function on T * M but also a holomorphic function on (T * M ) 0 .

At the heart of [START_REF] Heler | Résonances en limite semi-classique[END_REF][START_REF] Sjöstrand | Density of resonances for strictly convex analytic obstacles[END_REF] is the idea to study the restriction of T u to a well-chosen submanifold Λ of (T * M ) 0 a priori dierent from T * M . However, in order to make the analysis works, some assumptions on Λ are necessary.

In order to state the assumptions that we need to make on the manifold Λ , we need to recall a few facts on the symplectic geometry of (T * M ) 0 . We let θ denotes the complex canonical 1-form on (T * M ) 0 and ω = dθ the associated symplectic form. Indeed, recall CHAPTER 4

Then we dene F s,r (Λ) to be the Banach space of continuous function f : Λ → C such that f Λ,s,r < +∞.

We dene then the following spaces of functions on the Lagrangian Λ. For s ≥ 1 introduce the space of functions decaying at least exponentially

and the space of functions diverging slower than any exponential

These spaces are endowed respectively with the inductive and projective limit structure (in the category of locally convex topological vector spaces). Notice that G s (Λ) is dense in U s (Λ) (just multiply by a bump function) and that the L 2 pairing (4.13) in Λ gives a natural duality bracket between U s (Λ) and G s (Λ). The spaces G s (Λ) and U s (Λ) are natural analogues of G s (M ) and U s (M ) on the FBI side. Indeed, we have the following results.

Proposition 4.19 (Proposition 2.1 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Let s > s ≥ 1 and assume that Λ is a (τ 0 , s )-Gevrey adapted Lagrangian with τ 0 small enough. Then, the transform T Λ is continuous from G s (M ) to G s (Λ) and from U s (M ) to U s (Λ).

Proposition 4.20 (Proposition 2.2 in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF]). Let s > s ≥ 1 and assume again that Λ is a (τ 0 , s )-Gevrey adapted Lagrangian with τ 0 small enough. Then, the transform S Λ is continuous from G s (Λ) from G s (M ) and admits a continuous extension from U s (Λ) to U s (M ).

If Propositions 4.19 and 4.20 give a good idea of the link between regularity on M and decay on the FBI side, they are not precise enough for what we intend to do. In particular, when dealing with G s Anosov ows, we want to consider (τ 0 , s)-adapted Lagrangians, in order to get the best results possible. Thus, we need more precise estimates that explains how (τ 0 , s)-adapted Lagrangians relate with G s functions and associated ultradistributions.

We can use for instance the following lemmas. Lemma 4.21. Let s ≥ 1 and R ≥ 1. Then there is C > 0 such that, if Λ is a (τ 0 , s)adapted Lagrangian with τ 0 small enough and 0 ≤ r ≤ C -1 h -1/s , then T Λ is bounded from E s,R (M ) to F s,-r (Λ) and S Λ extends continuously to an operator from F s,r (Λ) to

Lagrangian with τ 0 small enough and r ≥ h -1/s , then T Λ is bounded from E s,R (M ) to F s,r (Λ) and S Λ is bounded from F s,-r (Λ) to E s,R (Λ).

We recall that the notion of almost analytic extension for the principal symbol of a dierential operators with G s coecients has been dened in 4.1.1 and that the class of symbols S m KN (Λ) is dened in Remark 4.16.

Let us discuss briey the proof of Proposition 4.34, considering that this is one of the main results from [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF] and that it will be a key tool in the proof of Theorem 10 (see in particular the proof of Lemma 4.37). What we expose here rapidly is exposed in details in [BJ20, 2.2 and 2.3]. Recalling the inversion formula Proposition 4.24, in order to understand the action of a dierential operator P on the spaces H k Λ , one only needs to understand the action of the operator T Λ P S Λ on the spaces L 2 k (Λ). Using the Gevrey stationary phase method, we are able to identify the kernel of the operator T Λ P S Λ . Thanks to the introduction of the action H in the denition of the space L 2 0 (Λ), we see that the reduced kernel of T Λ P S Λ associated to its action on this space is the kernel of a certain Fourier Integral Operator with complex phase with positive imaginary part. We can then use MelinSjöstrand's version 5 of the stationary phase method with complex phase [START_REF] Melin | Fourier integral operators with complex-valued phase functions[END_REF] to study a certain class of Fourier Integral Operators that contains in particular T Λ P S Λ .

We prove in particular that the orthogonal projector B Λ on the image of T Λ in L 2 0 (Λ) is one of these Fourier Integral Operators. The understanding of the algebraic properties of these operators allow then to prove a formula of the form

up to negligible operators (this is some kind of Toeplitz representation for the operator P ).

Here, σ is a symbol on Λ that coincides at rst order with f p Λ . Proposition 4.34 follows then from the representation formula (4.20).

Finite order of the dynamical determinant

We turn now to the proof of Theorem 10. From now on, s ∈ [1, +∞[ is xed, X is a G s vector eld on a compact G s manifold M that generates an Anosov ow (φ t ) t∈R , and

We dene the dierential operator P = X + V , and the associated Koopman operator is given by (9). Without loss of generality, we may assume that M is endowed with a structure of real-analytic Riemannian manifold (coherent with its G s structure, see Remark 4.2).

The machinery from 4.1.1 is then available, in particular we denote by M a complex neighbourhood for M . According to Proposition 4.23, there is an analytic FBI transform T on M such that T * T = I. As above we set S = T * . In order to apply the results from 4.1 to the operator P , we need rst to nd a suitable (τ 0 , s)-adapted Lagrangian Λ.

5 The Gevrey regularity assumption on P is only used in order to justify the applications of the stationary phase method mentioned above. After that, we only need to do C ∞ micro-local analysis on Λ.

CHAPTER 4

The Lagrangian Λ will be dened by (4.11) where the symbol G is dened by G = τ G 0 where τ h 1-1 s and G 0 is a so-called escape function. The section 4.2.1 is devoted to the construction of G 0 (see Lemma 4.35). In 4.2.2, we will then describe the spectral theory of P on the related spaces dened in 4.1.2. Finally, the proof of Theorem 10 is given in 4.2.3.

Constructing an escape function

Recall that the decomposition T M = E 0 ⊕ E u ⊕ E s of the tangent bundle from Denition 6 induces a dual decomposition

Here,

We denote by p : T * M → C the principal symbol of the semi-classical dierential operator hP . We recall that for α = (α x , α ξ ) ∈ T * M , p(α) = iα ξ (X(α x )) . To apply the machinery presented in the previous part, we will need an almost analytic extension for p. We construct it as in 4.1.1: we take a G s almost analytic extension X for X, given by Lemma 4.6 if s > 1 (we just take X = X if s = 1), and then we set for α = (α x , α ξ ) ∈ (T * M ) 0 (for some small 0 > 0) p(α) = iα ξ X (α x ) . It will be important when constructing the escape function G 0 that this almost analytic extension is linear in α ξ . We are now ready to construct G 0 . Lemma 4.35. Let C 0 be a conical neighbourhood of E * 0 in T * M . Let δ ≥ 0. Then there are arbitrarily small 0 > 1 > 0 and a symbol G 0 of order δ on (T * M ) 0 , supported in (T * M )

1 with the following properties:

(i) the restriction of {G 0 , Re p} to T * M is negative and classically elliptic of order δ outside of C 0 , that is, for some C > 0 and α ∈ T * M \ C 0 large enough, we have

Here, the Poisson Bracket {G 0 , Re p} is the one associated with the real symplectic form

Notice that Lemma 4.35 is slightly simpler than the corresponding result in [START_REF] Guedes | FBI Transform in Gevrey Classes and Anosov Flows[END_REF] (Lemma 3.1 from the mentioned paper). This is because we focus on Theorem 10. In CHAPTER 4

The key tool in the proof of these results will be the multiplication formula Proposition 4.34. Let us start by proving that P is the generator of a semi-group.

Proof of Proposition 4.36. We will apply HilleYosida Theorem to prove that P is the generator of a strongly continuous semi-group. We denote by p Λ the restriction to Λ of the almost analytic extension p of the principal symbol of hP given by ( 4

By CauchySchwarz, we nd that

for u ∈ H ∞ Λ . By Proposition 4.33, this estimate remains true when u ∈ D(P ). This proves that if Re z > 2C, then the operator z -P is injective and its image is closed. To prove that the image of z -

and consequently (4.29) still holds when z -P is replaced by (z -P ) * (for u ∈ H ∞ Λ , but it implies the same result for u ∈ D(P * ) by Proposition 4.33). Hence, (z -P ) * is injective, and thus the image of z -P is closed.

Thus, z -P is invertible and from (4.29), we see that

for the operator norm on H 0 Λ . Hence, the HilleYosida Theorem applies (the domain of CHAPTER 4 is also satised by the residues of R(z) and R(z). Since these residues have nite rank

and since E 1,R 0 is dense in C ∞ (M ), it follows that the eigenvalues of P on H 0 Λ (the poles of R(z)) are the Ruelle resonances of P (the poles of R(z)) counted with multiplicity (the rank of the associated residues). For the same reason, the resonant spaces (the images of the residues) also coincide.

Proof of Theorem 10

In order to show that the dynamical determinant d(z) given by ( 12) has nite order (under our Gevrey assumption), we shall relate it to a regularized determinant associated with the resolvent of P . This will be based on the following version of the Guillemin trace formula:

Lemma 4.39. If the real part of z is large enough and m is an integer such that m > sn,

then the operator (z -P ) -m acting on H 0 Λ is trace class and

(4.36)

Here, we use the notations from ( 12) and (TFF).

The proof of Lemma 4.39 that we give here diers from the one from [BJ20, Lemma However, this construction is restricted to the case s < 2, hence the additional assumption in Proposition 4.43. To prove that the spectrum of P on H 0 Λ coincides with its Ruelle spectrum when is near 0, we use the characterization of the Ruelle spectrum as limit of stochastic perturbations [DZ15, Theorem 1].

Weighted transfer operators for ultradierentiable expanding maps of the circle

When studying an expanding map of the circle T , it is sometimes useful to consider more general transfer operators that the one dened by (1). We explain here how the methods from 2.2 may be adapted to deal with weighted transfer operators.

If ψ : S 1 → C is a function we may dene the weighted transfer operator L ψ by

We will assume in the following that ψ is in C A , the DenjoyCarleman class from 2.2. It is then easy to see that the analysis above remains true for the operator L ψ , so that we can state:

Proposition A.1. Theorem 2.9 remains true when L is replaced by L ψ . Moreover, we may also dene in this case the nuclear power decomposition (2.17). This decomposition satises Propositions 2.16 and 2.17. Propositions 2.18 and 2.23 and Corollary 2.21 remain true as well if we replace the dynamical determinant d, from (4), by d ψ which is obtained from (4) by replacing tr (L n ) by

.

To prove Proposition A.1, notice that the actual denition of L is only used in the proofs of Lemma 2.14 and Proposition 2.18 in the analysis from 2.2. The computation that gives Proposition 2.18 can still be carried out and will give the formula that we announced for the at trace of the weighted transfer operator. Thus we will only explain how we can

Ruelle resonances are intrinsic

As pointed out in the introduction, the Banach spaces that appear in Theorems 1, 3 and 5 are highly non-canonical. Hence, in order to see that the Ruelle spectrum (see Denitions 3, 5 and 7) is a well-dened object, we need a result to compare discrete spectrum of an operator acting on dierent spaces. For the discrete time case, we can for instance use the following lemma (which is slightly more general than [Bal18 Then, the intersection of the spectra of L 1 and L 2 with U coincide and so do the associated generalized eigenspaces (that are consequently included in

It follows from the fact that B is a Hausdor topological space that this formula indeed denes a norm on B 1 + B 2 . Moreover, B 1 + B 2 with this norm identies with the quotient of B 1 × B 2 by the closed subspace {{u, v} ∈ B 1 × B 2 : u = v}, so that this norm makes B 1 + B 2 a Banach space. We may consequently replace B by B 1 + B 2 and assume that

Since L 1 and L 2 coincide on the intersection B 1 ∩ B 2 , we may dene an operator L on B by setting Lu = L 1 v + L 2 w if u = v + w (this denition is independent of the choice of v and w). One easily checks that L is a bounded operator on B, and induces a bounded operator on B 1 ∩ B 2 that we endow with the complete norm

It follows from the discussion in [Kat66, Chapter III, 5] that for i = 1, 2 the resolvent (z -L i ) -1 dene a meromorphic family of bounded operators from B i to itself on U , with residues of nite rank. We dene then a meromorphic family on U of operators from B 1 ∩B 2 to B by

where we denote by the letter j inclusion maps. Now, if z ∈ U is such that |z| > max( L , L 1 , L 2 ), we can write

Since U is unbounded and connected, it follows then from the analytic continuation principle that the meromorphic map R 1 and R 2 coincides on U . Now, let λ ∈ U be an element of the spectrum of L 1 . By assumption, this is an isolated eigenvalue of nite multiplicity, so that the associated spectral projector is given by

where γ is a small enough circle around λ. We also know that P λ,1 has nite rank. Replacing 1 by 2 in (B.1), we dene similarly P λ,2 , which is the spectral projector associated to λ for L 2 if λ belongs to the spectrum of L 2 and 0 otherwise. Since R 1 = R 2 , we nd that

Now, for i = 1, 2, let E λ,i denotes the generalized eigenspace associated to λ for L i , that is the image of P λ,i . Since this is a nite-dimensional space, its topology is unambiguously dened. By assumption, B 1 ∩ B 2 is dense in B i , so that P λ,i (B 1 ∩ B 2 ) is dense in E λ,i . However, P λ,i (B 1 ∩ B 2 ) is a subspace of the nite-dimensional vector space E λ,i and is

Proof. The proof is basically the same as for Lemma B.1. The only dierence is that we replace the representation of the resolvent as a power series by the representation as a Laplace transform

which is valid for i = 1, 2 when Re z 1.

Factorization of the dynamical determinant

We establish here, under the hypotheses of Theorem 3.1, a Hadamard-like factorization (C.3) for the dynamical determinant d(z) dened by (12). We use the same notations as in 3.4 and 3.5. Let t 0 > 0 be shorter than any periodic orbit of (φ t ) t∈R . Then, working as in the proof of Proposition 3.3, we see that, for Re z 1, the essential spectral radius of

is zero. Here, as in 3.4 and 3.5, the operator P = X + V denotes the generator of the semi-group (L t ) t≥0 dened by (9). Then, applying holomorphic functional calculus in nite dimension as in the proof of Lemma 3.44, we see that the spectrum of (C.1) is made of the e λt 0 (z-λ) d+2 for λ in the spectrum of P . Then, for Re z 1, Proposition 3.32 implies that the right-hand side of (C.1) denes a trace class operator on H 0 . From Lemma B.1, we see that the spectrum of (C.1) is the same when acting on H or on h0 . Then, using Lidskii's Trace Theorem and Proposition 3.32, we see that 1 , λ∈σ R (P )

In order to discuss the condition υ < 2 in Theorem 3.

where β > 0 and α ∈ υ-1 υ , 1 (this is the same condition as in Proposition 3.17), endowed with the norm

Then the transfer operator for n ∈ Z. Using the fact that ln (1 + |n|)

we see that L acting on H α,β is trace class when α < 1 2 and is not trace class when α > 1 2 (in the case α = 1 2 it depends on the value of β, it corresponds to the case υ = 2 in 2.2.3). Thus, we need to chose α < 1 2 if we want L to be nuclear. For general maps, this choice is possible only when υ < 2 (see the condition in Proposition 3.17).

Consequently, using our method to prove the trace formula (TFF) for C κ,υ Anosov ows would require to construct Hilbert spaces in a totally dierent way, if υ > 2. The case υ = 2 also seems to be tricky, but the results from 2.2, and in particular 2.2.3, suggest that maybe something interesting can be proven in that case.

Appendix E

Heuristic argument

We detail here an argument that suggests that the dynamical determinant (dened by (4) and ( 5)) associated to a generic perturbation of the doubling map, Example 1, has innite order (hence supporting Conjecture 1). We denote by T the doubling map and by t → T t a perturbation of T in the C ∞ topology (such that T 0 = T ) . We also assume that this perturbation is C ∞ in t. We denote by L t the transfer operator (1) associated to T and by d t the associated dynamical determinant (4). Let us write for x ∈ S 1 Using the Implicit Function Theorem to follow periodic point of T t , we nd then that for n ∈ N * we have

where we introduced for n ∈ N * the linear form a n dened by .

(E.2)

We denote by E the space of C ∞ functions from S 1 to R with zero average. We endow E with the C ∞ topology. Notice that f belongs to E and that all functions in E may be realized as (E.1) by choosing the perturbation t → T t . We will prove that the derivative (E.2) is an entire function of the variable z that grows arbitrarily fast for generic f ∈ E.

Since d 0 (z) = 1 -z, this suggests that for t near 0 the dynamical determinant d t (z) grows arbitrarily fast for generic X. Indeed, we have

(1 -z) 1 + th f (z) + O(t 2 ) .

(E.3)

This asymptotic expansion a priori only holds 1 uniformly on all compact subsets of C, so that it cannot be used to investigate the growth of d t (z), but it can certainly guide our intuition.

It follows from EulerMacLaurin's formula [Kre98, Corollary 9.27] that for every f ∈ E the sequence (a n (f )) n∈N decreases faster than any geometric sequence, so that h f is an entire function. In order to see that any bound on the growth of h f fails generically, we only need to prove the following due to Cauchy's formula.

Lemma E.1. Let (b n ) n≥1 be a sequence of strictly positive real numbers that decays faster than any geometric sequence. Then there is a G δ dense subset A of E such that for every f ∈ A we have

for innitely many n's.

Proof. We start by constructing a particular element g ∈ E. Let P ⊆ N * denotes the set of prime integers. For every p ∈ P dene the function e p : x → exp (2iπ (2 p -1) x) .

Then, notice that for every p, q ∈ P, we have a 2 p -1 (e q ) = δ p,q .