Gabriel Rivi Ère
email: gabriel.riviere@math.univ-lille1.fr

LONG-TIME DYNAMICS OF THE PERTURBED SCHR ÖDINGER EQUATION ON NEGATIVELY CURVED SURFACES

We consider perturbations of the semiclassical Schrödinger equation on a compact Riemannian surface with constant negative curvature and without boundary. We show that, for scales of times which are logarithmic in the size of the perturbation, the solutions associated to initial data in a small spectral window become equidistributed in the semiclassical limit. As an application of our method, we also derive some properties of the quantum Loschmidt echo below and beyond the Ehrenfest time for initial data in a small spectral window.

Introduction

In this article, we consider (M, g) a smooth (C ∞ ), connected, orientable, boundaryless, compact and Riemannian manifold of dimension d. We want to study the long time dynamics of the following family of Schrödinger equations:

(1) ∀0 < ≤ 1, ı ∂u ∂t = P ( )u , with u ⌉ t=0 = ψ , where (ψ ) 0< ≤1 is a sequence of normalized initial data in L 2 (M) satisfying proper oscillatory assumptions, and

(2) P ( ) := -2 ∆ g 2 + ǫ V, with ∆ g the Laplace Beltrami operator induced by the Riemannian metric g, V ∈ C ∞ (M, R) and ǫ → 0 as → 0 + . We aim at describing the dynamics of these equations in the semiclassical limit → 0 + . Due to the semiclassical approximation, these properties will be related to the properties of the geodesic flow (G t 0 ) t∈R acting on T * M. For instance, if we denote by u (τ ) the solution of (1) at time τ , i.e. u (τ ) := e -iτ P ( ) ψ , then we can introduce its "Wigner distribution" 1 on T * M

(3) ∀a ∈ C ∞ c (T * M), µ (τ )(a) := u (τ ), Op (a)u (τ ) , where Op (a) is a -pseudodifferential operator with principal symbol a -see [START_REF] Zworski | Semiclassical analysis[END_REF]. This quantity describes the distribution of the solution of (1) in T * M. As an application of the Egorov theorem [START_REF] Zworski | Semiclassical analysis[END_REF], one finds that, for every fixed τ in R, [START_REF] Anantharaman | Rivière Dispersion and controllability for the Schrödinger equation on negatively curved manifolds[END_REF] ∀a ∈ C ∞ c (T * M), µ (τ )(a) = µ (0)(a • G τ 0 ) + o(1), as → 0 + . In other words, the "Wigner distribution" at time τ is related to the one at time 0 through the action of the geodesic flow. This simple relation illustrates the connection between the quantum evolution and the classical one in the semiclassical limit → 0 + .

Our goal is to study the long time dynamics of (1) through these distributions. For that purpose, we will study the properties of µ (τ ) where τ = τ will depend on > 0, and more specifically when it will tend to +∞ as → 0 + . We will focus on geometric situations where the geodesic flow enjoys some chaotic features, e.g. the Anosov property [START_REF] Katok | Hasselblatt Introduction to the modern theory of dynamical systems[END_REF][START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF] the main examples being negatively curved manifolds [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF]. These considerations are related to questions arising in the field of quantum chaos, where one wants to understand the influence of the chaotic properties of the classical system on its quantum counterpart.

In the definition of the Schrödinger operator (2), we added a self-adjoint perturbation ǫ V , where ǫ → 0. In fact, we will not look at the dynamics of the "free" Schrödinger equation (where V ≡ 0), and our goal is rather to understand the influence of this kind of self-adjoint perturbations on the long time dynamics. Looking at the influence of selfadjoint perturbations on the quantum dynamics of chaotic systems is related to the question of the quantum Loschmidt echo in the physics literature -see section 6 for a brief reminder or [START_REF] Gorin | Zdinaric Dynamics of Loschmidt echoes and fidelity decay[END_REF][START_REF] Jacquod | Entanglement and Irreversibility in Quantum Dynamical Systems with Few Degrees of Freedom[END_REF][START_REF] Goussev | Wisniacki Loschmidt Echo[END_REF] for recent surveys on these issues. In the mathematics literature, these kind of considerations have recently appeared in several places. In [START_REF] Bolte | Schwaibold Stability of wave packet dynamics under perturbations[END_REF][START_REF] Combescure | A phase-space study of the quantum Loschmidt Echo in the semiclassical limit[END_REF], the authors were interested by questions directly related to the quantum Loschmidt echo. In [START_REF] Eswarathasan | Average pointwise bounds for deformations of Schrodinger eigenfunctions[END_REF][START_REF] Canzani | On the distribution of perturbations of propagated Schrödinger eigenfunctions[END_REF], the authors looked at magnetic (or metric) perturbations of the Schrödinger operator on a general compact manifold, and they obtained some informations on the pointwise bounds of the solutions of the perturbed Schrödinger equation for a finite time, and for a typical choice of perturbations. In [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF], the questions were close to the ones considered in the present article, and we will compare more precisely below our results to those from this reference. The tools used here are in fact the continuation of the ones introduced in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF]. As another application of the methods from the present article, we will also deduce some properties on the quantum Loschmidt echo below and beyond the Ehrenfest time -see section 6. Finally, the long time dynamics of the "perturbed" Schrödinger equation for integrable systems is studied in [START_REF] Macià | [END_REF].

Remark 1.1. We underline that, even if the perturbations we will consider will be small in the semiclassical limit, they will be quite strong at the quantum level as we will require in our statements that ǫ ≥ ν for some 0 < ν < 1/2. In fact, according to the semiclassical Weyl law [START_REF] Zworski | Semiclassical analysis[END_REF], the "mean level spacing" for the eigenvalues is of order d where d is the dimension of M.

Statement of the main results

When the geodesic flow satisfies some chaotic properties, one of the classical results on the semiclassical distribution of the solutions of the "unperturbed" Schrödinger equation is the quantum ergodicity theorem [START_REF]Shnirelman Ergodic properties of eigenfunctions[END_REF][START_REF]Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces[END_REF][START_REF] De | Verdière Ergodicité et fonctions propres du Laplacien[END_REF][START_REF] Helffer | Robert Ergodicité et limite semi-classique[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF]. This property is usually formulated for stationary solutions of the Schrödinger equations but it can be generalized to more general solutions [START_REF] Anantharaman | Rivière Dispersion and controllability for the Schrödinger equation on negatively curved manifolds[END_REF]. In order to state this result, we fix some sequence (δ ) 0< ≤1 which satisfies δ → 0 as → 0 + and δ ≥ α for some fixed constant α > 0. We define then

H := 1 [1-δ ,1+δ ] -2 ∆ g L 2 (M).
If the Liouville measure L is ergodic for the geodesic flow on the unit cotangent bundle 2 S * M := (x, ξ) ∈ T * M : p 0 (x, ξ) = 1 2 , then it is known that this space is of dimension N( ) ∼ C M d /δ , for some fixed constant C M > 0 depending only on (M, g) [START_REF] Dimassi | Sjöstrand Spectral asymptotics in the semi-classical limit[END_REF]. The quantum ergodicity theorem can be stated as follows [START_REF]Shnirelman Ergodic properties of eigenfunctions[END_REF][START_REF]Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces[END_REF][START_REF] De | Verdière Ergodicité et fonctions propres du Laplacien[END_REF][START_REF] Helffer | Robert Ergodicité et limite semi-classique[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF][START_REF] Anantharaman | Rivière Dispersion and controllability for the Schrödinger equation on negatively curved manifolds[END_REF]: Theorem 2.1 (Quantum Ergodicity). Let (τ ) 0< ≤1 such that lim →0 + τ = +∞. Suppose that the Liouville measure L is ergodic for the geodesic flow on S * M. Then, for every orthonormal basis (ψ j ) j=1,...,N ( ) of H , one can find J( ) ⊂ {1, . . . , N( )} such that lim

→0 + ♯J( ) N( ) = 1,
and, for every a in C ∞ c (T * M), for every ϕ in L 1 (R), one has This theorem tells us that, under ergodicity of the Liouville measure, the solutions of the "free" Schrödinger equation become equidistributed for a "generic choice" of initial data microlocalized near S * M. We also observe that equidistribution occurs when we average over the time parameter t. This result is very robust and it holds for many quantum systems with an underlying chaotic classical system.

Again, this result holds for a typical choice of initial data, and it is a difficult problem to understand what can be said for a fixed sequence of normalized initial data (ψ ) 0< ≤1see [START_REF]Zelditch Recent developments in mathematical quantum chaos[END_REF][START_REF] Sarnak | Recent progress on the quantum unique ergodicity conjecture[END_REF][START_REF] Nonnenmacher | Anatomy of quantum chaotic eigenstates[END_REF] for recent reviews on these questions. We mention that, for short scales of times (which are at most logarithmic in > 0), one can also get a good description of the distributions (µ (τ )) 0< ≤1 for certain class of initial data, namely coherent states [START_REF] Combescure | Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow[END_REF][START_REF] Bonechi | Bièvre Exponential mixing and | log | time scales in quantized hyperbolic maps on the torus[END_REF][START_REF] Bouclet | Long time propagation and control on scarring for perturbed quantized hyperbolic toral automorphisms[END_REF], or Lagrangian states [START_REF] Schubert | Semiclassical behaviour of expectation values in time evolved Lagrangian states for large times[END_REF][START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Festival Yves Colin de Verdière[END_REF][START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF].

Instead of looking at particular families, or at generic families of initial data for the unperturbed equation, we will now try to understand, for general sequences of initial data, the quantum evolution under the perturbed Schrödinger equation (1), i.e. when V = 0. This question was already discussed in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF]. Precisely, we will now consider sequences of initial data (ψ ) 0< ≤1 which satisfy the following property: [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF] lim

R→+∞ lim sup →0 + 1 [1-R ,1+R ] -2 ∆ ψ -ψ L 2 (M ) -→ 0, and ∀0 < ≤ 1, ψ L 2 = 1.
2 Here p 0 (x, ξ) :=

g * x (ξ,ξ) 2
with g * the metric induced by g on the cotangent bundle.

Remark 2.2. Recall that in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF], it was only required that, for every δ 0 > 0,

(6) lim →0 + 1 [1-δ 0 ,1+δ 0 ] -2 ∆ g ψ -ψ L 2 (M ) = 0, and ∀0 < ≤ 1, ψ L 2 = 1.
which is more general than [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF]. In section 4, we will in fact consider slightly more general initial data than the ones satisfying ( 5); yet, the assumptions will still be more restricitive than [START_REF] Bambusi | Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time[END_REF].

Our goal is to understand the action for short logarithmic times of the "perturbed" Schrödinger propagator e -iτ P ( ) on initial data satisfying [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF], and to show that imposing (5) instead of ( 6) allows to improve substantially the results from [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] -see corollary 2.6 below. In section 6, we will also show the relevance of this approach in the study of the quantum Loschmidt echo.

2.1. Semiclassical measures. Let (τ 1 ( )) 0< ≤1 and (τ 2 ( )) 0< ≤1 be two sequences which satisfy τ 1 ( ) ≤ τ 2 ( ). We define then

M([τ 1 ( ), τ 2 ( )], → 0 + ),
as the set of accumulation points in D ′ (T * M) (as → 0 + ) of the sequences of distributions (µ (τ )) 0< ≤1 defined by [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Festival Yves Colin de Verdière[END_REF] where

• (ψ ) 0< ≤1 varies among sequences satisfying (5), • (τ ) 0< ≤1 varies among sequences satisfying τ 1 ( ) ≤ τ ≤ τ 2 ( ) for > 0 small enough. Any element in M([τ 1 ( ), τ 2 ( )], → 0 + )
is in fact a probability measure carried on S * M that is called a semiclassical (defect) measure [START_REF] Gérard | Mesures semi-classiques et ondes de Bloch[END_REF][START_REF]Burq Mesures semi-classiques et mesures de défaut[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF]. We underline that these measures have a priori no extra properties like invariance by the geodesic flow. Our goal is to describe the properties of the measures belonging to M([τ 1 ( ), τ 2 ( )], → 0 + ). Given a subsequence (ψ n ) n∈N satisfying [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF] with n → 0 + , we define the following subset of M([τ 1 ( ), τ 2 ( )], → 0 + ):

M(ψ n , [τ 1 ( n ), τ 2 ( n )], n → 0 + ),
where we restricted ourselves to the sequence of initial data (ψ n ) n∈N .

Remark 2.3. For a given sequence (ψ ) 0< ≤1 satisfying [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF], one can also extract a subsequence n → 0 + such that the subsequence (µ n (0)) n∈N converge in D ′ (T * M) to some accumulation point µ 0 which is a probability measure on S * M. We note that, if we make the stronger assumption that

(-2 ∆ g -1)ψ L 2 = o ( ) , ψ L 2 = 1,
then the accumulation point µ 0 is in fact invariant by the geodesic flow G t 0 acting on S * M [51] -Ch. 5. In the case of a compact congruence surface, it was proved that, if (ψ ) 0< ≤1 is also a sequence of o(1)-quasimodes for a certain given Hecke operator T p , then µ 0 = L [START_REF] Brooks | Lindenstrauss Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF].

Finally, we define another particular subset of M([τ 1 ( ), τ 2 ( )], → 0 + ), i.e.

M log ([τ 1 ( ), τ 2 ( )], → 0 + ),
where we impose the extra assumption that the initial data vary among sequences (ψ ) 0< ≤1 satisfying [START_REF] Barreira | Dimension and ergodic decompositions for hyperbolic flows[END_REF] (-

2 ∆ g -1)ψ L 2 = o | log | -1 , ψ L 2 = 1.
According to [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF], the semiclassical measures of such initial data have positive metric entropy.

Remark 2.4. We emphasize that all the sets of semiclassical measures we have introduced in this paragraph will depend implicitly on the choice of the sequence (ǫ ) 0< ≤1 , and on the potential V . They both play an important role in the statements below; however, in order to alleviate notations, they do not appear explicitely in the conventions we used for the sets of semiclassical measures M(. . .).

Main theorem.

In our different results, an important role will be played by the function

(8) ∀(x 0 , ξ 0 ) ∈ S * M, f V (x 0 , ξ 0 ) := g * x 0 (d x 0 V, ξ ⊥ 0 ), where ξ ⊥ 0 ∈ S *
x 0 M is the vector directly orthogonal to ξ 0 . In the case of negatively curved surfaces, it represents (up to a constant) the unstable component of the Hamiltonian vector field associated to V on T * M [24] -section 3. More precisely, our results will depend on the geometry of the set of "critical points" of f V :

(9) C V := +∞ j=0 (x 0 , ξ 0 ) ∈ S * M : (X j 0 .f V )(x 0 , ξ 0 ) = 0 ,
where X 0 is the geodesic vector field, i.e. X 0 (ρ) = d dt (G t 0 (ρ)) |t=0 for every ρ in S * M, and where X j 0 means that we differentiate j times in the direction of X 0 . Our main result is the following: Theorem 2.5. Suppose that dim(M) = 2 and that it has constant negative curvature K ≡ -1. Suppose that ǫ → 0 as → 0, and that there exists 0 < ν < 1/2 such that, for > 0 small enough, ν ≤ ǫ .

Then, for every sequence (ψ n ) n∈N which satisfies [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF] with n → 0 + , and which has an unique semiclassical measure µ 0 , for every

1 < c 1 ≤ c 2 < min{3/2, 1/(2ν)}, for every µ ∈ M(ψ n , [c 1 | log ǫ n |, c 2 | log ǫ n |], n → 0 + ),
and for every a ∈ C ∞ (S * M, R), one has

µ 0 (C V ) min{a} + (1 -µ 0 (C V )) S * M adL ≤ µ(a) ≤ µ 0 (C V ) max{a} + (1 -µ 0 (C V )) S * M adL.
This theorem is a consequence of proposition 4.4 below which is slightly more precise as it allows, for each choice of (ǫ ) 0< ≤1 to consider more general families of initial data -see remark 4.1.

In order to clarify our statement, we will give below two corollaries of this theorem. Before, we briefly observe that if we choose V satisfying µ 0 (C V ) < 1, then, for every nonempty open subset ω of S * M, one has µ(ω) > 0. In other words, provided the perturbation V satisfies some "generic" property with respect to the initial data, then the solutions of the perturbed Schrödinger equation put mass on every nonempty open subset of S * M for times of order | log(ǫ )|. In the case where µ(C V ) = 0, the limit measure is in fact the Liouville measure. Compared with the quantum ergodicity theorem, we emphasize that we do not need to average over the time parameter.

2.3. Some corollaries. The following corollary is a direct consequence of the main theorem:

Corollary 2.6. Suppose that dim(M) = 2 and that it has constant negative curvature K ≡ -1. Suppose that ǫ → 0 as → 0 + , and that there exists 0 < ν < 1/2 such that for > 0 small enough, ν ≤ ǫ .

Suppose also that

C V = ∅.
Then, for every

1 < c 1 ≤ c 2 < min{3/2, 1/(2ν)}, one has M ([c 1 | log(ǫ )|, c 2 | log(ǫ )|]) = {L}.
Remark 2.7. Observe that the set

C 0 V := {(x 0 , ξ 0 ) ∈ S * M : f V (x 0 , ξ 0 ) = 0}
can never be empty, as the set there always exists x 0 ∈ M such that d x 0 V = 0. In appendix C, it is shown that the assumption C V = ∅ is generic in the sense that it is satisfied on an open and dense subset of C ∞ (M, R).

In other words, the solutions of the perturbed Schrödinger become equidistributed for time scales of order | log(ǫ )| as soon as the potential satisfies some geometric admissibility condition -see appendix C for the construction of such potentials. This statement is very close to the main theorem of [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF]. In this reference, equidistribution was also shown to hold for times of order | log(ǫ )| under perturbation by a "multi-scaled" potential which satisfies some geometric admissibility condition. The main improvement compared with this reference are the following. First, the geometric assumptions on the potential are simpler, and we do not have to deal with a "multi-scaled perturbation". Moreover, we do not need to average over subintervals of [c 1 | log(ǫ )|, c 2 | log(ǫ )|] to obtain equidistribution of the solutions, i.e. equidistribution holds for any time in the above interval. Finally, the main result from [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] holds for a generic ǫ in the interval [0, ν ′ ], while the present result holds for any large enough ǫ -see proposition 4.4 for a more precise statement. Yet, it is important to underline that the restrictions on the energy localization of the initial data are much more restrictive here than in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] -see ( 5) and [START_REF] Bambusi | Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time[END_REF].

As was already mentioned, the main theorem shows that the solutions of the perturbed Schrödinger equation put some mass on every nonempty open subset as soon as we know that the initial data do not put all its mass on the set C V . The results from [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF] provide sufficient conditions to ensure this property, modulo the fact that the initial data satisfy the stronger assumption [START_REF] Barreira | Dimension and ergodic decompositions for hyperbolic flows[END_REF]. More precisely, we introduce the "maximal invariant" subset inside C V , i.e.

Λ V := t∈R G t 0 C V ,
and combining our main result to [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF], we obtain the following corollary:

Corollary 2.8. Suppose that dim(M) = 2 and that it has constant negative curvature K ≡ -1. Suppose that ǫ → 0 as → 0, and that there exists 0 < ν < 1/2 such ν ≤ ǫ .

Suppose that the topological entropy3 of Λ V satisfies

h top (Λ V ) < 1 2 .
Then, for every

1 < c 1 ≤ c 2 < min{3/2, 1/(2ν)}, for every µ ∈ M log ([c 1 | log(ǫ )|, c 2 | log(ǫ )|]),
and for every nonempty open subset ω in S * M, one has µ(ω) > 0.

Thanks to corollary 4 from [START_REF] Barreira | Dimension and ergodic decompositions for hyperbolic flows[END_REF], one knows that h top (Λ V ) < 1 2 is satisfied as soon as the Hausdorff dimension of Λ V is < 2. We also recall that,

Λ V ⊂ Λ 0 V := t∈R G t 0 C 0 V ⊂ C 0 V ⊂ S * M,
and we remark that, for a generic choice of V (say V has finitely many critical points), one has dim H (C 0 V ) = 2. 2.4. Organization of the article. In section 3, we briefly recall some properties of geodesic flows on negatively curved surfaces. In section 4, we use semiclassical tools to reduce the proof of theorem 2.5 to a question on hyperbolic dynamical systems. In section 5, we solve this dynamical systems question using strong structural stability and unique ergodicity of the horocycle flow. In section 6, we apply our method to the study of the quantum Loschmidt echo. In appendix A, we give a short toolbox on semiclassical analysis, and in appendix B, we recall the strong structural stability theorem and we give a brief account on the results from [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] that we will use in this article. Finally, appendix C provides a large class of potentials satisfying the assumptions of corollary 2. [START_REF] Bambusi | Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time[END_REF].

In all the article, M will denote a smooth (C ∞ ), connected, orientable, compact and Riemannian manifold without boundary.

Properties of geodesic flows on negatively curved surfaces

In all this article, we make the additional assumption that M is a surface with constant sectional curvature K ≡ -1. We will now draw some dynamical consequences of this geometric assumption. We refer to [START_REF] Katok | Hasselblatt Introduction to the modern theory of dynamical systems[END_REF][START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF] for a more detailed exposition.

In this geometric setting, the geodesic flow4 (G t 0 ) t∈R satisfies the Anosov property on S * M [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF]. Precisely, it means that, for every ρ = (x, ξ) in S * M, there exists a G t 0 -invariant splitting [START_REF] Bouclet | Long time propagation and control on scarring for perturbed quantized hyperbolic toral automorphisms[END_REF] T

ρ S * M = RX 0 (ρ) ⊕ E s (ρ) ⊕ E u (ρ),
where X 0 (ρ) is the Hamiltonian vector field associated to p 0 (x, ξ) = ξ 2 x 2 , E u (ρ) is the unstable direction and E s (ρ) is the stable direction. These three subspaces are preserved under the geodesic flow and there exist constants C 0 > 0 and γ 0 > 0 such that, for any t ≥ 0, for any v s ∈ E s (ρ) and any v u ∈ E u (ρ),

d ρ G t 0 v s G t 0 (ρ) ≤ C 0 e -γ 0 t v s ρ and d ρ G -t 0 v u G -t 0 (ρ) ≤ C 0 e -γ 0 t v u ρ ,
where . w is the norm associated to the Sasaki metric on S * M [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF]. Moreover, as dim(M) = 2, these three subspaces are 1-dimensional subspaces of T ρ S * M. As explained in section 3 of [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF], one can associate a direct orthonormal basis to this splitting that we denote by (X 0 (ρ), X s (ρ), X u (ρ)).

Remark 3.1. We emphasize that we make a slightly different choice of convention compared with [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF]. In this reference, the stable and unstable vector fields X s and X u were chosen to be of norm √ 2. Here, we will use the convention that they are unit vectors for the Sasaki metric on S * M.

Recall from Chapter 3 of [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF] (see also section 3 of [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF]) that there exists some C 0 > 0 such that, for every ρ in S * M and for every t in R, [START_REF] Bouzouina | Robert Uniform semiclassical estimates for the propagation of quantum observables[END_REF] d ρ G t 0 ≤ C 0 e t . In our proof, we will use two main properties of geodesic flows on negatively curved surfaces, namely

• strong structural stability -see appendix B for a brief reminder of this property [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF][START_REF] De La Llave | Moriyon Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation[END_REF]; • unique ergodicity of the horocycle flow [START_REF] Furstenberg | The unique ergodicity of the horocycle flow, Recent advances in topological dynamics[END_REF][START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF]. We conclude this preliminary section by a brief reminder on the ergodic properties of horocycle flows on negatively curved surfaces [START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF]. Thanks to the fact that we are considering negatively curved surfaces, one knows that X u defines a C 1 vector field on S * M [START_REF] Hirsch | Pugh Smoothness of horocycle foliations[END_REF]. The unstable horocycle flow is then defined as the

C 1 flow (H s u ) s∈R satisfying ∀ρ ∈ S * M, d ds (H s u (ρ)) = X u • H s u (ρ).
Recall from [START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF] that this parametrization of the horocycle flow is uniformly expanding in the sense that

∀ρ ∈ S * M, ∀(t, τ ) ∈ R 2 , G t 0 • H τ u (ρ) = H e t τ u • G t 0 (ρ).
Moreover, this flow is uniquely ergodic [START_REF] Furstenberg | The unique ergodicity of the horocycle flow, Recent advances in topological dynamics[END_REF][START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF]: Theorem 3.2. Suppose that dim(M) = 2, and that M has constant negative sectional curvature K ≡ -1. One has, for every a in C 0 (S * M),

lim T →+∞ sup 1 T T 0 a • H s u (ρ)ds - S * M adL : ρ ∈ S * M = 0,
where L is the desintegration of the Liouville measure on S * M.

As mentioned above, this property will be at the heart of our proof. In fact, it was shown in [START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF] (lemma 4.5) that this result implies that, for every a in C 0 (S * M) and every b > 0, [START_REF] Brooks | Lindenstrauss Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF] lim

s→+∞ 1 b b 0 a • H s u • G t 0 (ρ)dt = S * M adL,
uniformly in ρ. This property exactly says that small pieces of geodesics become equidistributed under the action of the horocycle flow, and it is the central step in the proof of strong mixing for horocycle flows given in [START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF]. After a reduction based on semiclassical techniques, we will have to understand some ergodic properties of perturbed geodesic flows to complete the proof of our main results. This will be done in section 5, and, even if the proof given there does not concern directly the horocycle flow, it will be modeled on a similar strategy as the proof of [START_REF] Brooks | Lindenstrauss Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF] given in [START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF]. The main difference is that we will replace H s u by a perturbation Gs ǫ of the geodesic flow and that we will need to prove that, in a certain regime of b, ǫ and s, it behaves like the unstable horocycle flow.

In order to clarify the proof given in section 5, we briefly recall from [START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF] how one can derive [START_REF] Brooks | Lindenstrauss Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF] from the unique ergodicity of the horocycle flow. Let η > 0. It is sufficient to prove that, for b > 0 small enough, there exists s 0 (b, η) such that, for every s ≥ s 0 (b, η), the average

1 b b 0 a • H s u • G t 0 (ρ) is within η of S * M adL. We write that 1 b b 0 a • H s u • G t 0 (ρ)dt = 1 b b 0 a • G t 0 • H se -t u (ρ)dt = 1 b b 0 a • H se -t u (ρ)dt + O(b).
Then, we make the change of variables τ = se -t and we get

1 b b 0 a • H s u • G t 0 (ρ)dt = - 1 b se -b s a • H τ u (ρ) dτ τ + O(b).
This quantity looks very much like a Birkhoff average, except that we have a Jacobian factor in the integral. In order to deal with this term, we apply the mean value Theorem and we get

1 b b 0 a • H s u • G t 0 (ρ)dt = - e -b -1 be -t 0 1 se -b -s se -b s a • H τ u (ρ)dτ + O(b), for some t 0 in [0, b]. We have that 1-e -b be -t 0 = 1 + O(b), which implies that 1 b b 0 a • H s u • G t 0 (ρ)dt = 1 se -b -s se -b s a • H τ u (ρ)dτ + O(b).
We can now apply unique ergodicity of the horocycle flow, and we find that there exists a nonincreasing function r(T ) → 0 as T → +∞ such that

1 b b 0 a • H s u • G t 0 (ρ)dt = S * M adL + O(b) + r(s(1 -e -b )) = S * M adL + O(b) + o((sb) -1 ),
which implies our result.

Reduction to classical dynamics

In this section, we consider a slightly more general setting than the one in the introduction in order to allow more general class of initial data. We fix a sequence (ǫ ) 0< ≤1 which satisfies ǫ → 0, as → 0 + and which represents the "strength" of our perturbation. We define then admissible sequences of initial data of order ν 0 > 0 as follows:

(13) lim →0 + 1 [1-ǫ -ν 0 ,1+ ǫ -ν 0 ] -2 ∆ ψ -ψ L 2 (M ) = 0, and, ∀ 0 < ≤ 1, ψ L 2 = 1,
where ν 0 > 0 is some positive constant.

Remark 4.1. For any choice of (ǫ ) 0< ≤1 , assumption (13) allows to consider sequences of normalized initial data satisfying ( 14) lim sup

→0 + 1 [1-R ,1+R ] -2 ∆ ψ -ψ L 2 (M ) -→ 0, as R → +∞.
We observe that this remark combined to proposition 4.4 below allows to deduce our main statement, i.e. theorem 2.5.

For every ǫ ∈ [0, ǫ ], we introduce the following self-adjoint deformation of the Schrödinger operator:

Pǫ ( ) := - 2 ∆ g 2 + ǫV,
where V belongs to C ∞ (M, R). We will denote by (G t ǫ ) t∈R the Hamiltonian flow associated to p ǫ (x, ξ) := p 0 (x, ξ) + ǫV (x).

Our goal is to study the long time dynamics of the corresponding quantum propagator U ,ǫ (t) := e -it Pǫ( ) on initial data satisfying [START_REF]Burq Mesures semi-classiques et mesures de défaut[END_REF], and for ǫ large enough belonging to [0, ǫ ]. For that purpose, we introduce a sequence of time scales (τ ) 0< ≤1 which satisfies τ → +∞ as → 0 + . Remark 4.2. In the following, our results will concern scales of times τ of order | log(ǫ )|, under the assumption that ǫ ≫ √ . In our geometric context, it exactly means that we consider a regime where the semiclassical approximation is valid, i.e. below the Ehrenfest time. In fact, in our setting, the Ehrenfest time in the Egorov theorem is known to be of order | log |/2 [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Festival Yves Colin de Verdière[END_REF][START_REF] Dyatlov | Guillarmou Microlocal limits of plane waves and Eisenstein functions[END_REF] -see appendix A for a brief reminder.

Given a sequence of normalized initial data (ψ ) 0< ≤1 satisfying (13), we define

u ǫ (τ ) := e -iτ Pǫ( ) ψ ,
and its "Wigner distribution" on T * M, i.e.

(

) ∀a ∈ C ∞ c (T * M), µ ǫ (τ )(a) := u ǫ (τ ), Op (a)u ǫ (τ ) L 2 15
The goal of this section is to describe the asymptotic properties of µ ǫ (τ ) as → 0 + , τ ∼ | log(ǫ )|, and ǫ ∈ [0, ǫ ] large enough.

Remark 4.3. The properties we will obtain will depend on the choice of V and on the properties of the semiclassical measures of the initial data.

In all of this section, we will choose ǫ and ν 0 in such a way that ǫ -ν 0 → 0 as → 0 + . Recall then that, one can extract a subsequence n → 0 such that the sequence of distributions (µ 0 n (0)) n∈N converges to a limit distribution which is in fact a probability measure µ 0 carried on the unit cotangent bundle S * M [51] -Chapter 5. We underline that µ 0 does not have a priori extra properties like invariance by the geodesic flow, even under the stronger assumption [START_REF] Canzani | On the distribution of perturbations of propagated Schrödinger eigenfunctions[END_REF].

We will only consider sequences of initial data with an unique semiclassical measure µ 0 and, in order to alleviate the notations, we denote the extraction n → 0 by → 0 + . Finally, the properties we will obtain are related to the set of critical points of order J ≥ 0, i.e. ( 16)

C J V := J j=0 (x 0 , ξ 0 ) ∈ S * M : (X j 0 .f V )(x 0 , ξ 0 ) = 0 .
4.1. Statement of the main result. Before stating our main result, we start with a simple observation which follows directly from the results described in paragraph A.3. Without any assumptions on (τ ) 0< ≤1 , on (ǫ ) 0< ≤1 (except that ǫ → 0) and on the geometry of the manifold, we always have [START_REF] Combescure | Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow[END_REF] min

S * M {a} ≤ lim inf →0 + ,ǫ∈[0,ǫ ] µ ǫ (τ )(a) ≤ lim sup →0 + ,ǫ∈[0,ǫ ] µ ǫ (τ )(a) ≤ max S * M {a}.
The proposition below will show that, under proper assumptions on the geometry of the manifold and on V , and for strong enough perturbations (ǫ ) 0< ≤1 , one can find a scale of times (τ ) 0< ≤1 for which (µ ǫ (τ )(a)) →0 + asymptotically belong to a smaller interval. More precisely, we will show Proposition 4.4. Suppose that dim(M) = 2, and that M has constant negative sectional curvature K ≡ -1. Suppose that lim →0 ǫ = 0, and that there exists 0 < ν < 1 2 such that, for > 0 small enough, one has ǫ ≥ ν . Let J be a nonnegative integer. Let ν 0 > 0 and ν 1 ≥ 0 satisfying

1 + ν 1 + (3J + 1)ν 0 < min 3 2 , 1 2ν
.

Suppose that there exists

1 + ν 1 + (3J + 1)ν 0 < c 1 ≤ c 2 < min 3 2 , 1 2ν such that, for > 0 small enough, c 1 | log(ǫ )| ≤ τ ≤ c 2 | log(ǫ )|.
Then, for any normalized sequence (ψ ) →0 + in L 2 (M) which satisfies (13) with ν 0 , and which has an unique semiclassical measure µ 0 , one has, for every a in C ∞ c (T * M, R),

µ 0 (C J V ) min S * M {a} + (1 -µ 0 (C J V )) S * M adL ≤ lim inf →0 + ,ǫ∈[ǫ 1+ν 1 ,ǫ ] µ ǫ (τ )(a) ≤ lim sup →0 + ,ǫ∈[ǫ 1+ν 1 ,ǫ ] µ ǫ (τ )(a) ≤ µ 0 (C J V ) max S * M {a} + (1 -µ 0 (C J V )) S * M
adL.

Remark 4.5. This result implies Theorem 2.5 when we restrict ourselves to sequences of initial data satisfying ( 14) -see remark 4.1.

Remark 4.6. Our assumptions on the different parameters impose that 0

< ν 0 < 1 2(3J+1) ≤ 1 2
and ǫ ≥ √ asymptotically. In particular, one has ǫ -ν 0 → 0 when → 0 + , as in the context of remark 4.3.

The statement of the proposition is a little bit technical as it involves many parameters, and it should be understood as follows. If we suppose that ǫ ≫ √ , then one can find admissibility conditions on frequencies of the initial data, and a scale of times for which the sequence (µ ǫ (τ )(a)) ,ǫ belongs asymptotically to the interval

I µ 0 ,V,J (a) := µ 0 (C J V ) min S * M {a} + (1 -µ 0 (C J V )) S * M adL, µ 0 (C J V ) max S * M {a} + (1 -µ 0 (C J V )) S * M adL ,
which is a subinterval of [min a, max a] that appeared in [START_REF] Combescure | Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow[END_REF]. The interval is smaller as soon as µ 0 , V and J satisfy µ 0 (C J V ) < 1. In the case where they satisfy µ 0 (C J V ) = 0, then the matrix elements converge in fact to S * M adL. Remark 4.7. We underline that, if we fix a normalized sequence (ψ ) →0 + with an unique semiclassical measure µ 0 , then one has µ 0 (C J+1 V ) ≤ µ 0 (C J V ) and thus I µ 0 ,V,J+1 (a) ⊂ I µ 0 ,V,J (a). Yet, one has to be careful as the condition on (ψ ) →0 + becomes more restrictive as we increase the parameter J. For instance, given a normalized sequence (ψ ) →0 + , it could happen that the proposition could be applied for some J but not necessarly for J + 1.

This result is very close to theorem 4.1 from [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF], and the main improvement compared with this reference is that we do not need to average over the perturbation parameter ǫ ∈ [0, ǫ ] in order to get an equidistribution property. However, we have to make a much stronger restriction on the family of initial data as it was only required in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] that

∀δ 0 > 0, lim →0 + 1 [1-δ 0 ,1+δ 0 ] -2 ∆ ψ -ψ L 2 (M )
= 0, which is obviously a much weaker assumption than (13). 4.2. Preliminary observations. Thanks to the frequency assumption [START_REF]Burq Mesures semi-classiques et mesures de défaut[END_REF] and to Remark 4.6, we can suppose without loss of generality that a is homogeneous in a neighborhood of S * M, i.e. there exists 0 < δ 0 < 1/2 such that [START_REF] Combescure | A phase-space study of the quantum Loschmidt Echo in the semiclassical limit[END_REF] ∀(x, ξ)

satisfying p 0 (x, ξ) ∈ [1/2 -δ 0 , 1/2 + δ 0 ], a(x, ξ) = a x, ξ ξ .
Without loss of generality, we can also suppose that ( 19)

1 [1-ǫ -ν 0 ,1+ǫ -ν 0 ] -2 ∆ ψ = ψ .
As mentionned in remark 4.3, the semiclassical measure µ 0 has a priori no invariance properties under such general assumptions. Still, we can observe invariance for very short scales of times in . In fact, according to [START_REF] De La Llave | Moriyon Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation[END_REF], one has (20) e -it(-2 ∆g -1)

2 ψ = ψ + O(|t|ǫ -ν 0 ).
In particular, for every ν 2 > ν 0 , one has

(21) ∀|t| ≤ ǫ ν 2 , e it ∆g 2 ψ = e -it 2 ψ + o(1),
where the remainder is uniform for t in this interval.

Remark 4.8. Equality ( 21) is crucial in our proof of proposition 4.4. In fact, our argument will make use of an equidistribution result, and we will need to average over some parameter in order to use this equidistribution property. In [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF], the averaging was performed over the perturbation parameter ǫ ∈ [0, ǫ ]. Here, we will take advantage of the fact that the initial data satisfies a stronger spectral localization, and we will use [START_REF] Dyatlov | Guillarmou Microlocal limits of plane waves and Eisenstein functions[END_REF] in order to average over a time parameter t ∈ [0, ǫ ν 2 ].

4.3.

Proof of proposition 4.4. The proof of this proposition can be divided in two main steps: (1) we apply semiclassical rules in order to reduce ourselves to a question on ergodic properties of geodesic flows; (2) we use tools from hyperbolic dynamical systems in order to answer this "dynamical systems" question. This paragraph is devoted to the first step, and the proof of the second step is postponed to section 5. Let J, ν, ν 0 , ν 1 , c 1 and c 2 be as in the statement of proposition 4.4. Recall that ǫ ≥ ν and that

c 1 | log(ǫ )| ≤ τ ≤ c 2 | log(ǫ )|,
for > 0 small enough. Let a be an element in C ∞ c (T * M, R) as in paragraph 4.2, i.e. which is 0-homogeneous in a neighborhood of size δ 0 of S * M. We underline that it is sufficient to prove the lower bound as the upper bound can then be obtained by replacing a by -a.

4.3.1.

Truncation in phase space. Let 0 < δ < δ 0 /4. We introduce 0 ≤ χ δ ≤ 1 a smooth function on R which is equal to 1 on the interval [(1δ)/2, (1 + δ)/2] and 0 outside some interval [1/2δ, 1/2 + δ]. Thanks to [START_REF] De La Llave | Moriyon Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation[END_REF], we can write

ψ = χ δ ( P0 ( ))ψ + o(1).
Recall that the operator χ δ ( Pǫ ( )) is a -pseudodifferential operator in Ψ -∞,0 (M) with principal symbol χ δ • p ǫ (x, ξ) [START_REF] Zworski | Semiclassical analysis[END_REF] (Ch. 14) and with the semi-norms which are uniformly bounded for ǫ ∈ [0, ǫ ]. Then, thanks to the Calderón-Vaillancourt theorem [START_REF] Zworski | Semiclassical analysis[END_REF], we find that, uniformly for ǫ ∈ [0, ǫ ],

ψ = χ δ ( Pǫ ( ))ψ + o(1).
This implies that, one has

µ ǫ (τ )(a) = ψ , e + iτ Pǫ( ) Op (a)χ δ ( Pǫ ( ))e -iτ Pǫ( ) ψ + o(1),
where the remainder is uniform for ǫ ∈ [0, ǫ ]. We now apply the composition formula for pseudodifferential operators and we find that

(22) µ ǫ (τ )(a) = ψ , e + itτ Pǫ( ) Op (a × χ δ • p ǫ )e -itτ Pǫ( ) ψ + o(1),
where the remainder is still uniform for ǫ ∈ [0, ǫ ].

4.3.2.

Long time Egorov property. Observe now that, for > 0 small enough and ǫ

∈ [0, ǫ ], the function a × χ δ • p ǫ is compactly supported in the energy layer {(x, ξ) : 1/2 -2δ ≤ |ξ| 2 /2 ≤ 1/2 + 2δ}. Recall also that 0 ≤ τ ≤ c 2 | log(ǫ )| ≤ c 2 ν| log |,
with c 2 ν < 1/2. Thus, we can choose 0 < δ < δ 0 /4 small enough in a way that depends only on c 2 ν ∈ [0, 1/2) and such that we can apply Egorov Theorem up to the time τ (uniformly for ǫ ∈ [0, ǫ ] and τ ≤ c 2 | log(ǫ )|) -see [START_REF] Smale | An infinite dimensional version of Sard's theorem[END_REF] in the appendix. In other words, we have that, for 0 < δ < δ 0 /4 small enough,

µ ǫ (τ )(a) = ψ , Op (a • G τ ǫ × χ δ • p ǫ )ψ + o(1)
, where the remainder is uniform for ǫ ∈ [0, ǫ ] and 0 ≤ τ ≤ c 2 | log(ǫ )|. 4.3.3. Invariance for short times. We now use [START_REF] Dyatlov | Guillarmou Microlocal limits of plane waves and Eisenstein functions[END_REF], i.e. invariance by the free Schrödinger equation over short intervals of times. More precisely, we write that one has

(23) µ ǫ (τ )(a) = 1 ǫ ν 2 ǫ ν 2 0 ψ , e -is ∆g 2 Op (a • G τ ǫ × χ δ • p ǫ )e is ∆g 2 ψ ds + o(1)
,

where ν 2 > ν 0 . As discussed in appendix A, the symbol a • G τ ǫ × χ δ • p ǫ remains in a class of symbols S -∞,0 ν ′ (T * M) with 0 ≤ c 2 ν < ν ′ < 1/2 (with the semi-norms which are uniformly bounded for τ ≤ c 2 | log(ǫ )| and ǫ ∈ [0, ǫ ]).
In particular, we can apply Egorov theorem for finite time (here ǫ ν 2 → 0), and we find that

µ ǫ (τ )(a) = ψ , Op 1 ǫ ν 2 ǫ ν 2 0 a • G τ ǫ • G s 0 × χ δ • p ǫ • G s 0 ds ψ + o(1).
Using Calderón-Vaillancourt Theorem one more time, one gets

µ ǫ (τ )(a) = ψ , Op χ δ • p 0 × 1 ǫ ν 2 ǫ ν 2 0 a • G τ ǫ • G s 0 ds ψ + o(1). 4.3.4.
Using ergodic properties of the classical flow. As all the symbols are compactly supported and as they belong to an admissible class of symbols S -∞,0

ν ′ (T * M) with 0 ≤ c 2 ν < ν ′ < 1/2,
we can use the results from paragraph A. [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Festival Yves Colin de Verdière[END_REF]. It means that wa can replace Op by a positive quantization Op + (see [START_REF]Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces[END_REF] for instance), i.e.

µ ǫ (τ )(a) = ψ , Op + χ δ • p 0 × 1 ǫ ν 2 ǫ ν 2 0 a • G τ ǫ • G s 0 ds ψ + o(1)
.

Fix now η 0 > 0 and introduce the following subset of S * M:

K J V (η 0 ) := (x, ξ) ∈ S * M : ∃0 ≤ j ≤ J, (X 0 -Id) j .f V (x, ξ) ≥ η 0 , where we recall that f V (x, ξ) := g * x (d x V, ξ ⊥ ).
We also define the following compact subset of

T * M KJ V (η 0 , δ) := (x, ξ) ∈ T * M : x, ξ ξ ∈ K J V (η 0 ) and p 0 (x, ξ) ∈ [1/2 -δ, 1/2 + δ] .
Remark 4.9. The points in the set KJ V (η 0 , δ) corresponds to the points (x, ξ) in T * M for which we are able to control uniformly the convergence of the integral

1 ǫ ν 2 ǫ ν 2 0 a • G τ ǫ • G s 0 (x, ξ)ds.
We refer to proposition 5.1 for a precise statement.

We introduce a smooth cutoff function χ η 0 V which is identically equal to 1 on KJ V (η 0 , δ) and which vanishes outside KJ V (η 0 /2, δ). We define

(24) A ,η 0 ,δ (ǫ) := inf 1 ǫ ν 2 ǫ ν 2 0 a • G τ ǫ • G s 0 (x, ξ)ds : (x, ξ) ∈ KJ V (η 0 /2, δ) ,
Using these notations and the positivity of Op + , we derive that min

S * M {a}µ 0 (0)(χ δ • p 0 (1 -χ η 0 V )) + A ,η 0 ,δ (ǫ)µ 0 (0)(χ δ • p 0 χ η 0 V ) ≤ µ ǫ (τ )(a).
Recall that a is homogeneous in a neighborhood of size δ 0 > 0 of S * M. Thus, as G τ ǫ • G s 0 (x, ξ) remains in this neighborhood for all (t, s) when 1/2δ ≤ p 0 (x, ξ) ≤ 1/2 + δ (provided we choose ǫ ≥ 0 small enough), we can replace a by ã in the definition of A ,η 0 ,δ , where ã(x, ξ) := a(x, ξ/ ξ ) for every (x, ξ) in T * M -M. In particular, provided we pick 0 < δ < δ 0 /4 small enough and ǫ ∈ [ǫ 1+ν 1 , ǫ ], we can apply proposition 5.1 which implies that A ,η 0 ,δ (ǫ) converges to S * M adL.

We now take the limit → 0 + , and we deduce that

µ 0 (K J V (η 0 /2) c ) min S * M {a} + µ 0 (K J V (η 0 )) S * M adL ≤ lim inf →0,ǫ∈[ǫ 1+ν 1 ,ǫ ] µ ǫ (τ )(a).
This property holds for any η 0 > 0. Thus, we finally derive

µ 0 (C J V ) min S * M {a} + µ 0 ((C J V ) c ) S * M adL ≤ lim inf →0,ǫ∈[ǫ 1+ν 1 ,ǫ ] µ ǫ (τ )(a),
which concludes the proof of proposition 4.4.

Perturbations of the geodesic flow

Thanks to the results of section 4, the proof of our main result is now reduced to a purely dynamical systems question as it only remains to estimate the quantity A ,η 0 ,δ (ǫ) defined by [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF]. Precisely, for a fixed 0-homogeneous C 1 function ã on T * M -M, we need to understand the asymptotic behaviour of

I x 0 ,ξ 0 (ǫ, b, T ) := 1 b b 0 ã • G T ǫ • G s 0 (x 0 , ξ 0 )ds,
as b, ǫ → 0 and T → +∞. Recall that (G t ǫ ) t∈R is the Hamiltonian flow associated to the function p ǫ (x, ξ) := ξ 2 x 2 + ǫV (x). This integral looks very much like the integral involved in [START_REF] Brooks | Lindenstrauss Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF] except that the unstable horocycle has been replaced by a perturbed geodesic flow. The way we will deal with the convergence of I x 0 ,ξ 0 (ǫ, b, T ) will in fact be very similar to the proof of [START_REF] Brooks | Lindenstrauss Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF] given in section 3. The additional arguments we will need will be:

• the strong structural stability theorem [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF][START_REF] De La Llave | Moriyon Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation[END_REF] which will allow us to "replace" the perturbed geodesic flow by a "reparametrization" of the horocycle flow; • a theorem due to Cartan [START_REF] Cartan | Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications[END_REF] on polynomials which helps us to estimate the size of the Jacobian factor. Fix now J ≥ 0 and η 0 > 0. In order to give our main result on the convergence of I x 0 ,ξ 0 (ǫ, b, T ), recall that we defined the following subset of S * M:

(25) K J V (η 0 ) := (x, ξ) ∈ S * M : ∃0 ≤ j ≤ J, (X 0 -Id) j .f V (x, ξ) ≥ η 0 , where we set f V (x, ξ) := g * x (d x V, ξ ⊥ ).
Using this convention, we have the following statement: Proposition 5.1. Suppose that dim(M) = 2, and that M has constant negative sectional curvature K ≡ -1. Let J ≥ 0, and let η 0 > 0. Let ν 1 + (3J + 1)ν 2 < 1/2 with ν 1 ≥ 0 and ν 2 > 0. Let

1 + ν 1 + (3J + 1)ν 2 < c 1 ≤ c 2 < 3 2 .
Then, there exists δ 1 > 0 such that, for every C 1 function ã on T * M -M which is 0homogeneous, one has

lim ǫ 0 →0 sup ( * ) 1 ǫ ν 2 0 ǫ ν 2 0 0 ã • G c| log ǫ 0 | ǫ • G s 0 (x 0 , ξ 0 )ds - S * M ãdL = 0,
where ( * ) means that we take the supremum over the set

(x 0 , ξ 0 , c, ǫ) ∈ T * [1/2-δ 1 ,1/2+δ 1 ] M × [c 1 , c 2 ] × [ǫ 1+ν 1 0 , ǫ 0 ] : x 0 , ξ 0 ξ 0 ∈ K J V (η 0 ) , with T * [1/2-δ 1 ,1/2+δ 1 ] M := {(x 0 , ξ 0 ) ∈ T * M : 1/2 -δ 1 ≤ p 0 (x 0 , ξ 0 ) ≤ 1/2 + δ 1 }.
This proposition tells us that small pieces of geodesics become equidistributed under the action of a perturbed geodesic flow provided that the perturbation is nontrivial on the small piece we consider. The fact that the perturbation is nontrivial is exactly guaranteed by the fact that we require (x 0 , ξ 0 / ξ 0 ) to be on the subset K J V (η 0 ). We emphasize that this statement looks very much like the results5 of section 6 in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF], more precisely corollary 6.4. In this reference, instead of averaging over the time parameter s, the average was performed over the perturbation parameter ǫ. Then, the result was established as an equidistribution property for similar scales of times, and the admissibility condition on the perturbation involved the nonvanishing of the following integral transform:

∀ρ 0 = (x 0 , ξ 0 ) ∈ S * M, β u V (x 0 , ξ 0 ) := 1 √ 2 +∞ 0 g * x(τ ) (d x(τ ) V, ξ ⊥ (τ ))e -τ dτ,
where (x(τ ), ξ(τ )) := G τ 0 (x 0 , ξ 0 ). We refer to appendix B to see how this transform appears naturally when we apply the strong structural stability theorem. This function will also play an important role in our proof and our admissibility condition will in fact be related to it. According to [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF], this function is Hölder continuous for every6 γ < 1/2. Yet, we have more regularity if we look at the direction of the geodesic flow. In fact, one has

e -s β u V • G s 0 (x 0 , ξ 0 ) = β u V (x 0 , ξ 0 ) - s 0 e -τ f V • G τ 0 (x 0 , ξ 0 ) dτ,
and we can then observe that the map s → e -s β u V • G s 0 (x 0 , ξ 0 ) is of class C ∞ (for a fixed choice of (x 0 , ξ 0 )). Moreover, the quantities appearing in the definition of K J V (η 0 ) are exactly the derivatives of this map at s = 0.

We describe now more precisely the main stages of the proof:

(1) in paragraph 5.1, using homogeneity properties of our problem, we "project" everything on S * M; (2) in paragraph 5.2, we use the strong structural stability theorem to replace the perturbed geodesic flow by a "reparametrized" horocycle flow involving the derivatives of the map s → e -s β u V • G s 0 (x 0 , ξ 0 ); (3) in paragraph 5.3, we make use of the unique ergodicity of the horocycle flow to conclude. The main lines of the proof are very close to the arguments given in section 6 of [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF]; yet, some aspects need a slightly different treatment, especially in steps ( 2) and (3). 5.1. Reduction to S * M. As was already explained, we will first "project" on S * M all the quantities involved in the definition of I x 0 ,ξ 0 (ǫ, b, T ). We follow the same procedure as in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] (paragraph 5.1) and we refer to it for the details.

Let (x 1 , ξ 1 ) be an element in a small neighborhood of S * M and define Σ ǫ x 1 ,ξ 1 := {(x, ξ) ∈ T * M : p ǫ (x, ξ) = p ǫ (x 1 , ξ 1 )}, which is an energy layer for the Hamiltonian p ǫ . Introduce also the two following diffeomorphisms:

θ ǫ x 1 ,ξ 1 : Σ ǫ x 1 ,ξ 1 → S * M, (x, ξ) → (x, ξ/ ξ
), and its inverse

θ ǫ x 1 ,ξ 1 -1 : S * M → Σ ǫ x 1 ,ξ 1 , (x, ξ) → x, 2(p ǫ (x 1 , ξ 1 ) -ǫV (x))ξ .
Thanks to these two maps, we can define a new flow on S * M, i.e.

ϕ t ǫ,x 1 ,ξ 1 = θ ǫ x 1 ,ξ 1 • G t/ ξ 1 ǫ • θ ǫ x 1 ,ξ 1 -1 .
Recall that we can compute explicitely the vector field

(26) Y ǫ x 1 ,ξ 1 (ρ) := d dt ϕ t ǫ,x 1 ,ξ 1 (ρ) t=0 .
associated to this new flow. More precisely, one has Lemma 5.2. One has, for every ρ = (x, ξ) in S * M,

(27) Y ǫ x 1 ,ξ 1 (ρ) = c ǫ,x 1 ,ξ 1 (x)X 0 (ρ) + ǫ √ 2 ξ 1 c ǫ,x 1 ,ξ 1 (x) g * x d x V, ξ ⊥ (X s (ρ) -X u (ρ)) ,
where c ǫ,x 1 ,ξ 1 (x) := pǫ(x 1 ,ξ 1 )-ǫV (x)

p 0 (x 1 ,ξ 1 )
.

The proof of this lemma follows from a direct calculation and it was given in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] (lemma 5.2).

Remark 5.3. We can reestablish the dependence in (x 1 , ξ 1 ) more clearly and write:

c ǫ,x 1 ,ξ 1 (x) = 1 + 2 ξ 1 2 ǫ(V (x 1 ) -V (x)) = 1 + O x,x 1 ,ξ 1 (ǫ).
We can rewrite I x 0 ,ξ 0 (ǫ, b, T ) using this new flow, and we get (28)

I x 0 ,ξ 0 (ǫ, b, T ) = 1 b b 0 ã • ϕ T ξ 0 ǫ,x(s),ξ(s) • G s ξ 0 0 x 0 , ξ 0 ξ 0 ds,
where (x(s), ξ(s)) := G s 0 (x 0 , ξ 0 ).

5.2.

Applying strong structural stability. We will now use the strong structural stability theorem in order to transform the integral I x 0 ,ξ 0 (ǫ, b, T ) into an integral involving the horocyle flow. Precisely, we start by proving the following lemma:

Lemma 5.4. Suppose that dim(M) = 2, and that M has constant negative sectional curvature K ≡ -1. Let ã be a C 1 function on T * M -M which is 0-homogeneous, let N ≥ 1 and let 0 < γ < 1/2. There exist δ 1 , ǫ 1 , s 1 , T 1 > 0 and C 1 > 0 such that, for every ǫ ∈ [0, ǫ 1 ], for every s ∈ [0, s 1 ], for every T ≥ T 1 and for every (x 0 , ξ 0 ) satisfying p 0 (x 0 , ξ 0

) ∈ [1/2 -δ 1 , 1/2 + δ 1 ], one has ã • ϕ T ξ 0 ǫ,x(s),ξ(s) • G s ξ 0 0 x 0 , ξ 0 ξ 0 -ã • H -ǫP N x 0 ,ξ 0 (s)e T ξ 0 u (ρ(ǫ, T, x 0 , ξ 0 )) ≤ C 1 ǫT + ǫ 1+γ e T ξ 0 + s + s N +1 ǫe T ξ 0 , where ρ(ǫ, T, x 0 , ξ 0 ) := G T ξ 0 0 • H ǫβ u V x 0 , ξ 0 ξ 0 u x 0 , ξ 0 ξ 0 , and (29)
P N x 0 ,ξ 0 (s) := N -1 p=0 ( ξ 0 s) p+1 (p + 1)! ((X 0 -1) p .f V ) x 0 , ξ 0 ξ 0 .
In the end, we will need to average over the time parameter s. Thanks to this lemma, this average will now correspond to an average along "reparametrized" trajectories of the horocycle flow which is known to be uniquely ergodic [START_REF] Furstenberg | The unique ergodicity of the horocycle flow, Recent advances in topological dynamics[END_REF][START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF]. Remark 5.5. We emphasize that our assumption on the set K J V (η 0 ) involves some nonvanishing conditions on the coefficients of the polynom appearing in the "reparametrization" of the horocycle flow.

Remark 5.6. Before giving the proof of this lemma, we start with a simple observation which explains why we need to consider time scales larger than | log(ǫ 0 )| in the statement of proposition 5.1. In fact, if one has ǫ ≪ e -T ξ 0 and s ≪ 1, then the previous lemma implies that

ã • ϕ T ξ 0 ǫ,x(s),ξ(s) • G s ξ 0 0 x 0 , ξ 0 ξ 0 -ã • G T 0 (x 0 , ξ 0 ) = o(1).
In particular, averaging over the time parameter s for such scales would not provide any equidistribution.

Proof. Let ã be a smooth function on T * M -M which is 0-homogeneous. We will use the conventions of appendix B. We fix 0 < γ < 1/2. Let (x 0 , ξ 0 ) be an element in a small neighborhood of S * M. First, we write the strong structural stability equation. More precisely, thanks to (51), we have, for every ρ in S * M and for every s in R,

ã • ϕ T ξ 0 ǫ,x(s),ξ(s) (ρ) = ã • h ǫ x(s),ξ(s) • G τ ǫ x(s),ξ(s) (T ξ 0 ,ρ) 0 • h ǫ x(s),ξ(s) -1 (ρ).
Using the "smoothness" of the maps withr espect to ǫ -see (52), we can write that ã • ϕ

T ξ 0 ǫ,x(s),ξ(s) (ρ) = ã • G T ξ 0 0 • h ǫ x(s),ξ(s) -1 (ρ) + O(ǫT ),
where the constant in the remainder is uniform for ρ in S * M, s ∈ R and (x 0 , ξ 0 ) in a small neighborhood of S * M.

We now replace h ǫ x(s),ξ(s)

-1
by its approximate expression given by (55). Then, according to [START_REF] Bouzouina | Robert Uniform semiclassical estimates for the propagation of quantum observables[END_REF], we get

ã • ϕ T ξ 0 ǫ,x(s),ξ(s) (ρ) = ã • G T ξ 0 0 • exp -ṽ ǫ x(s),ξ(s) (ρ) + O(ǫT ) + O(ǫ 1+γ e T ξ 0 ),
where ṽǫ x(s),ξ(s) is defined by (53) and is C 1 in ǫ. Again, the constants in the remainders are still uniform for ρ in S * M, s ∈ R and (x 0 , ξ 0 ) in a small neighborhood of S * M.

Thanks to lemma 1 in [START_REF] Moser | On a theorem of Anosov[END_REF] -see also remark B.3 from the appendix, we have

d C 0 exp -ṽ ǫ x(s),ξ(s) , exp - ǫ ξ 0 β s V X s • exp - ǫ ξ 0 β u V X u = O(ǫ 1+γ ),
where the constant is uniform for s ∈ R and (x 0 , ξ 0 ) in a small neighborhood of S * M. In particular, one has ã•ϕ

T ξ 0 ǫ,x(s),ξ(s) (ρ) = ã•G T ξ 0 0 •exp - ǫ ξ 0 β s V X s •exp - ǫ ξ 0 β u V X u (ρ)+O(ǫT )+O(ǫ 1+γ e T ξ 0 ).
We will now approximate these two maps by the unstable and stable horocycle flows. For that purpose, we fix ρ in S * Mand we observe that the maps

ǫ → exp ρ - ǫ ξ 0 β u V X u , and ǫ → H -ǫ ξ 0 β u V (ρ) u (ρ)
are of class C 1 . Moreover, their derivatives coincides up to an error of order O(ǫ) where the constant in the remainder is uniform for ρ in S * M and ξ 0 close to 1. In particular,

d S * M exp ρ - ǫ ξ 0 β u V X u , H -ǫ ξ 0 β u V (ρ) u (ρ) = O(ǫ 2 ),
where the constant in the remainder is still uniform for ρ in S * M and ξ 0 close to 1. The same holds for the maps generated by the stable vector field X s . Thus, up to other error terms of the same order, we can replace the maps exp -ǫ ξ 0 β s V X s and exp -ǫ ξ 0 β u V X u by the stable and unstable horocycle flows, i.e.

ã • ϕ T ξ 0 ǫ,x(s),ξ(s) (ρ) = ã • G T ξ 0 0 • H -ǫ ξ 0 β s V s • H -ǫ ξ 0 β u V u (ρ) + O(ǫT ) + O(ǫ 1+γ e T ξ 0 ),
where the constant in the remainders are still uniform for ρ in S * M, s ∈ R and (x 0 , ξ 0 ) in a small neighborhood of S * M. This implies that ã • ϕ

T ξ 0 ǫ,x(s),ξ(s) (ρ) = ã • G T ξ 0 0 • H -ǫ ξ 0 β u V (ρ) u (ρ) + O(ǫT ) + O(ǫ 1+γ e T ξ 0 ), as G t 0 • H τ s = H e -t τ s
• G t 0 for every t and τ in R [START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF]. Up to this point, the proof is exactly the same as in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF], and we will now analyse more precisely the reparametrization constant β u (ρ).

We use the fact that G t 0 • H τ u = H e t τ u • G t 0 , and we find

(30) ã•ϕ T ξ 0 ǫ,x(s),ξ(s) (ρ) = ã•G s ξ 0 0 •H -e (T -s) ξ 0 ǫβ u V (ρ) ξ 0 u •G (T -s) ξ 0 0
(ρ)+O(ǫT )+O(ǫ 1+γ e T ξ 0 ).

We now write that

e -s ξ 0 β u V • G s ξ 0 0 x 0 , ξ 0 ξ 0 = β u V x 0 , ξ 0 ξ 0 - s ξ 0 0 e -τ f V • G τ 0 x 0 , ξ 0 ξ 0 dτ, where f V (x, ξ) = g * x (d x V, ξ ⊥ ).
Then, we find that, for every p ≥ 0,

d p+1 ds p+1 e -s ξ 0 β u V • G s ξ 0 0 x 0 , ξ 0 ξ 0 s=0 = ξ 0 p+1 (X 0 -1) p .f V x 0 , ξ 0 ξ 0 .
In particular, we can write the order N expansion as s → 0, i.e.

(31)

e -s ξ 0 β u V • G s ξ 0 0 x 0 , ξ 0 ξ 0 = β u V x 0 , ξ 0 ξ 0 -P N x 0 ,ξ 0 (s) + O(s N +1 ),
where P N x 0 ,ξ 0 (s) is defined by [START_REF] Goussev | Wisniacki Loschmidt Echo[END_REF]. Finally, combining ( 30) and ( 31), we find that

ã • ϕ T ξ 0 ǫ,x(s),ξ(s) • G s ξ 0 0 x 0 , ξ 0 ξ 0 = ã • H ǫe T ξ 0 β u V x 0 , ξ 0 ξ 0 -P N x 0 ,ξ 0 (s) u • G T ξ 0 0 x 0 , ξ 0 ξ 0 + O(ǫT ) + O(ǫ 1+γ e T ξ 0 ) + O(s) + O(s N +1 ǫe T ξ 0 ),
where the constant in the remainders are uniform for (x 0 , ξ 0 ) in a small neighborhood of S * M. This concludes the proof of the lemma.

5.3.

Using unique ergodicity of the horocycle flow. Thanks to lemma 5.4, we can write, for every N ≥ 1 (32)

I x 0 ,ξ 0 (ǫ, b, T ) = 1 b b 0 ã••H -ǫP N x 0 ,ξ 0 (s)e T ξ 0 u (ρ(ǫ, T, x 0 , ξ 0 ))ds+O(ǫT )+O(ǫ 1+γ e T ξ 0 )+O(b)+O(b N +1 ǫe T ξ 0 )
Regarding the previous formula, we need to understand the asymptotic behaviour of

Ĩx 0 ,ξ 0 (ǫ, b, T ) := 1 b b 0 ã • •H -ǫP N
x 0 ,ξ 0 (s)e T ξ 0 u (ρ(ǫ, T, x 0 , ξ 0 ))ds as ǫ, b → 0, and T → +∞. We are now in a situation which looks very much like [START_REF] Brooks | Lindenstrauss Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF]. The main difference is the polynomial term in the time reparametrization.

It would be natural to make the change of variables s ′ = P N x 0 ,ξ 0 (s). However, the polynom (P N x 0 ,ξ 0 ) ′ (s) may vanish on certain points of the interval. This is the reason why we require the point (x 0 , ξ 0 ) to belong to the subset K J V (η 0 ). This hypothesis means that, for such points, at least one the first J + 1 coefficients of the polynom (P N x 0 ,ξ 0 ) ′ (s) does not vanish.

In particular, a first observation we can make is that the Jacobian factor (P N x 0 ,ξ 0 ) ′ (s) can only vanish at finitely many places.

Our first step will be to understand precisely the subsets where the Jacobian of the change of variables is very small (paragraph 5.3.1). Then, it will allow us to make the change of variables on proper subintervals of [0, b] (paragraph 5.3.2) and to apply unique ergodicity of the horocycle flow (paragraph 5.3.3).

5.3.1.

Estimates on the Jacobian of the change of variables. In order to study the size of the Jacobian in our change of variables, we will proceed as in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF], i.e. make use of the following theorem due to Cartan [START_REF] Cartan | Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications[END_REF]: Theorem 5.7. Given any number H > 0 and any complex numbers z 1 , . . . , z n , there is a system of p disks in the complex plane, with p ≤ n and with the sum of the radii equal to 2H, such that for each point z lying outside these disks, one has the inequality

|z -z 1 |.|z -z 2 |. . . . .|z -z n | > H e n .
From this theorem, one can in fact deduce the following property Lemma 5.8. Let θ be some positive parameter satisfying 0 < θ < 1. Let η 0 > 0, J ≥ 0 and N ≥ J + 1.

Then, there exists δ 1 > 0, 0 < s 1 < 1 and C 0 > 0 such that, for every 0 < s 0 < s 1 , and for every

(x 0 , ξ 0 ) in K J V (η 0 ) satisfying 1/2 -δ 1 ≤ p 0 (x 0 , ξ 0 ) ≤ 1/2 + δ 1
, one has a system of subintervals, some of which can be empty, L 1 (x 0 , ξ 0 ), . . . , L J (x 0 , ξ 0 ) of [0, s 0 ] with the sum of their length bounded by C 0 s 1+ θ 2(J +1) 0 and satisfying

A x 0 ,ξ 0 (s 0 ) := s ∈ [0, s 0 ] : |(P N x 0 ,ξ 0 ) ′ (s)| ≤ s J+θ 0 ⊂ J p=1 L p (x 0 , ξ 0 ).
Proof. Modulo minor modifications, the proof follows the same lines as the proof of proposition 5.17 in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF]. For the sake of completeness, we briefly recall how one can deduce this lemma from Cartan's Theorem. We fix θ > 0. First, we write

(P N x 0 ,ξ 0 ) ′ (s) = N -1 p=0 ξ 0 p+1 p! (X 0 -1) p .f V x 0 , ξ 0 ξ 0 s p .
If we choose δ 1 > 0 small enough, one knows that, for every (x 0 , ξ 0 ) in

K J V (η 0 ) satisfying 1/2 -δ 1 ≤ p 0 (x 0 , ξ 0 ) ≤ 1/2 + δ 1 , there exists 0 ≤ p 1 ≤ J such that ξ 0 p 1 +1 p 1 ! (X 0 -1) p 1 .f V x 0 , ξ 0 ξ 0 ≥ η 0 2J! .
Let (x 0 , ξ 0 ) be such a point and let p 1 be the first integer for which the coefficient of the polynom is ≥ η 0 2J! . The case p 1 = 0 is straightforward as we can choose all the intervals to be empty. Suppose now p 1 = 0.

In this case, we introduce

q x 0 ,ξ 0 (s) := p 1 p=0 ξ 0 p+1 p! (X 0 -1) p .f V x 0 , ξ 0 ξ 0 s p ,
which will be the "dominant part" of the Jacobian factor (P N x 0 ,ξ 0 ) ′ (s). In fact, one can define

B x 0 ,ξ 0 (s 0 ) := s ∈ [0, s 0 ] : |(P N x 0 ,ξ 0 ) ′ (s)| ≤ s J+θ 0 , and
|q x 0 ,ξ 0 (s)| ≥ s p 1 + θ 2 0 .
Then, there exists an uniform constant C J,N,δ 1 > 0 such that the following holds

B x 0 ,ξ 0 (s 0 ) ⊂ s ∈ [0, s 0 ] : C J,N,δ 1 s p 1 +1 ≥ s p 1 + θ 2 0 (1 -s J-p 1 + θ 2 0
) .

As Jp 0 + θ 2 > 0 and as θ < 1, this set is empty for s 0 > 0 small enough (depending only on C J,N,δ 1 , J, θ and θ ′ ). This shows that the dominant part of the Jacobian is encoded by the polynom q x 0 ,ξ 0 (s). In other words, for s 0 > 0 small enough, one has

A x 0 ,ξ 0 (s 0 ) ⊂    s ∈ [0, s 0 ] : p 1 !|q x 0 ,ξ 0 (s)| ( ξ 0 ) p 1 +1 (X 0 -1) p 1 .f V x 0 , ξ 0 ξ 0 ≤ 2J!s p 1 + θ 2 0 η 0    .
We are now in a situation where we can apply Cartan's Theorem on polynoms. Thus, there exists a system of subintervals, some of which can be empty, L 1 (x 0 , ξ 0 ), . . . , L J (x 0 , ξ 0 ) of [0, s 0 ] with the sum of their length bounded by 2J!e η 0 s

1+ θ 2p 1 0
and satisfying

A x 0 ,ξ 0 (s 0 ) ⊂ J p=1
L p (x 0 , ξ 0 ), which concludes the proof of the lemma.

Change of variables.

We will now perform a change of variables in the integral defining Ĩx 0 ,ξ 0 (ǫ, b, T ). For that purpose, we will split the interval [0, b] in small subintervals where the Jacobian of the change of variables is large enough. Let 0 < θ < 1. Let (x 0 , ξ 0 ) be an element in K J V (η 0 ) satisfying 1/2δ 1 ≤ p 0 (x 0 , ξ 0 ) ≤ 1/2 + δ 1 , where δ 1 is given by lemma 5.8. For b > 0 small enough, the subset A x 0 ,ξ 0 (b) ∩ [0, b] is included in the union of at most J subintervals of [0, b] whose total length is bounded by Cb 1+ θ 2(J +1) . Outside of these "bad" subintervals, the Jacobian of the change of variables does not vanish and is bounded from below by b J+θ . We will now split the complementary of these "bad" subintervals into a family of L disjoint subintervals (B k ) k=1,...L (depending on x 0 , ξ 0 ) of individual length b 1+2J+2θ and the union of at most J + 1 intervals whose total length is bounded by (J + 1)b 1+2J+2θ . More precisely, we write [START_REF] Jacquod | Entanglement and Irreversibility in Quantum Dynamical Systems with Few Degrees of Freedom[END_REF] Ĩx

0 ,ξ 0 (ǫ, b, T ) = 1 b L k=1 B k ã • H -ǫP N x 0 ,ξ 0 (s)e T ξ 0 u (ρ(ǫ, T, x 0 , ξ 0 ))ds + O(b θ 2(J +1) ),
where one has

• for every 1 ≤ k ′ = k ≤ L, B k ′ ∩ B k = ∅; • for every 1 ≤ k ≤ L, B k is an interval of length b 1+2J+2θ ; • for every 1 ≤ k ≤ L and for every s in B k , one has |(P N x 0 ,ξ 0 ) ′ (s)| ≥ b J+θ ; • 1 b L k=1 |B k | = 1 + O(b θ 2(J +1) ).
We will now consider each of the subintegrals independently, i.e. for every 1

≤ k ≤ L Ĩk x 0 ,ξ 0 (ǫ, b, T ) := 1 |B k | B k ã • H -ǫP N
x 0 ,ξ 0 (s)e T ξ 0 u (ρ(ǫ, T, x 0 , ξ 0 ))ds, and we will verify that this converges to S * M ãdL for a proper range of ǫ, b → 0, and T → +∞. We fix 1 ≤ k ≤ L and we make the change of variables s ′ = P N x 0 ,ξ 0 (s) where s ∈ B k . We obtain

Ĩk x 0 ,ξ 0 (ǫ, b, T ) = 1 |B k | P N x 0 ,ξ 0 (B k ) ã • H -ǫs ′ e T ξ 0 u (ρ(ǫ, T, x 0 , ξ 0 )) ds ′ |(P N x 0 ,ξ 0 ) ′ • (P N x 0 ,ξ 0 ⌉ B k ) -1 (s ′ )| .
Remark 5.9. We observe that P N x 0 ,ξ 0 (B k ) is an interval whose length is bounded from below by b 1+3J+3θ and from above by O(b 1+2J+2θ ). In the following, we will denote by Bk this interval.

We now write

1 (P J x 0 ,ξ 0 ) ′ • (P J x 0 ,ξ 0 ⌉ B k ) -1 (s ′ ) - 1 (P J x 0 ,ξ 0 ) ′ (s k )) ≤ sup s∈B k |(P J x 0 ,ξ 0 ) ′′ (s) (P J x 0 ,ξ 0 ) ′ s 2 ×|(P J x 0 ,ξ 0 ⌉ B k ) -1 (s ′ )-s k |,
where s k is the left end point of B k . In particular, we find that

Ĩk x 0 ,ξ 0 (ǫ, b, T ) = 1 |B k | (P J x 0 ,ξ 0 ) ′ (s k )) Bk ã • H -ǫs ′ e T ξ 0 u (ρ(ǫ, T, x 0 , ξ 0 ))ds ′ + O(b),
where the constant in the remainder is uniform for 1 ≤ k ≤ L (and for (x 0 , ξ 0 ) in the allowed energy layers).

Remark 5.10. Taking the particular case ã = 1, we also observe that

| Bk | = |B k | (P J x 0 ,ξ 0 ) ′ (s k )) (1 + O(b)
). In the end, we have obtained that

(34) Ĩk x 0 ,ξ 0 (ǫ, b, T ) = 1 | Bk | Bk ã • H -ǫs ′ e T ξ 0 u (ρ(ǫ, T, x 0 , ξ 0 ))ds ′ + O(b),
where Bk is an interval whose length is bounded from below by b 1+3J+3θ .

Remark 5.11. Let (x 0 , ξ 0 ) be an element in K J V (η 0 ) satisfying 1/2-δ 1 ≤ p 0 (x 0 , ξ 0 ) ≤ 1/2+δ 1 , where δ 1 > 0 was given by lemma 5.8. Combining [START_REF] Jacquod | Entanglement and Irreversibility in Quantum Dynamical Systems with Few Degrees of Freedom[END_REF] and ( 34), we have that, for every 0

< θ < 1, Ĩx 0 ,ξ 0 (ǫ, b, T ) = 1 L L k=1 1 | Bk | Bk ã • H -ǫse T ξ 0 u (ρ(ǫ, T, x 0 , ξ 0 ))ds + O b θ 2(J +1)
, where Bk is an interval whose length is bounded from below by b 1+3J+3θ and which depends on (x 0 , ξ 0 ) (but not on ǫ and T ).

Unique ergodicity.

We can now conclude using unique ergodicity of the horocycle flow [START_REF] Furstenberg | The unique ergodicity of the horocycle flow, Recent advances in topological dynamics[END_REF][START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF], which implies

r(T 0 ) := sup |τ |≥T 0 sup 1 τ τ 0 ã • H s ′ u (ρ)ds ′ - S * M ãdL : ρ ∈ S * M
is a nonincreasing function which tends to 0 as T 0 → +∞. Then, one has, for every

1 ≤ k ≤ L, 1 | Bk | Bk ã • H -ǫse T ξ 0 u (ρ(ǫ, T, x 0 , ξ 0 ))ds = S * M ãdL + r b 1+3J+3θ ǫe T ξ 0 .
Thanks to remark 5.11 and to [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators III[END_REF], it implies that

I x 0 ,ξ 0 (ǫ, b, T ) = S * M ãdL+r b 1+3J+3θ ǫe T ξ 0 +O b θ 2(J +1) +O(ǫT )+O(ǫ 1+γ e T ξ 0 )+O(b N +1 ǫe T ξ 0 ).
As N can be chosen arbitrarly large, θ > 0 arbitrarly small and γ arbitrarly close to 1/2, this concludes the proof of proposition 5.1, where we took

ǫ 0 → 0 + with b = ǫ ν 2 0 , T = c| log ǫ 0 |, and ǫ ∈ [ǫ 1+ν 1 0 , ǫ 0 ].

Decay of the quantum Loschmidt echo

Motivated by the fact that the unitarity of the quantum propagator e -iτ P0 ( ) (with P0 ( ) := -2 ∆g 2 ) does not allow one to observe any sensitivity to perturbations of the initial conditions, Peres argued in [START_REF] Peres | Stability of quantum motion in chaotic and regular systems[END_REF] that both the classical and the quantum system should be sensitive to pertubations of the Hamiltonian. For that reason, he suggested that one should look at perturbations of the Hamiltonian for fixed sequences of initial data in order to study the influence of perturbations both in the classical and in the quantum setting. For the quantum counterpart, he proposed to look at the overlap between the solutions of the unperturbed and the perturbed Schrödinger equation for fixed initial data. Precisely, given a normalized sequence of initial data (ψ ) 0< ≤1 and V ∈ C ∞ (M, R), one should study the following quantity:

(35) F V ,ǫ (ψ , τ ) := e -iτ P0 ( ) ψ , e -iτ ( P0 ( )+ǫV ) ψ 2 .
Peres expected that this overlap should typically decay for any quantum system, and that it should decay to a much lower value for chaotic sytems than for regular ones. One of the main difficutly one encounters when studying this overlap is that we want to understand the limit as τ → +∞ but also as → 0 and ǫ → 0. In [START_REF] Jalabert | Pastawski Environment-independent decoherence rate in classically chaotic systems[END_REF], motivated by experiments in nuclear magnetic resonance, Jalabert and Pastawski were also interested 7 in studying properties of F V ,ǫ (ψ , τ ) for chaotic systems. They considered the situation where the initial data are given by a sequence of coherent states which are microlocalized at some point (x 0 , ξ 0 ) in phase space, and where the potential is given by V (x) = u 1 V 1 (x) + . . . + u J V J (x), where the (u i ) i=1,...J are independent random variables. They observed that, on average and for a certain range of parameters, the quantum Loschmidt echo is exponentially decaying with a rate which is asymptotically given by the mean of the Lyapunov exponents of the classical system. This regime can be observed for times of order the Ehrenfest time, and for strong enough perturbations (meaning that ǫ → 0 is large compared with the mean level spacing of P0 ( )). This regime is known as the Lyapunov regime. In [START_REF] Jacquod | Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo[END_REF], it was emphasized that the situation becomes slightly more complicated for smaller perturbations, and that one can observe other kind of regimes like the so-called Fermi golden rule regime (exponential decay with a rate depending on ǫ). Besides the above works, many progresses have been made recently in the physics literature concerning the asymptotic behaviour of F V ,ǫ (ψ , τ ), and we refer to [START_REF] Gorin | Zdinaric Dynamics of Loschmidt echoes and fidelity decay[END_REF][START_REF] Jacquod | Entanglement and Irreversibility in Quantum Dynamical Systems with Few Degrees of Freedom[END_REF][START_REF] Goussev | Wisniacki Loschmidt Echo[END_REF] for detailed reviews on these questions. It is important to note that, when studying this problem, the decay rates depend in a subtle way on various quantities like ǫ, τ and , but also the shape (or the statistical properties) of the perturbation, and the choice of initial data.

The aim of this last section is to use the tools developed in the previous section for the study of the asymptotic properties of the quantum Loschmidt echo on surfaces with constant negative curvature. We will look at strong perturbations, namely ǫ ≫ √ and at scales of times ≥ | log ǫ|. For simplicity of exposition 8 , we will only consider sequences of normalized initial data satisfying (5), i.e. lim R→+∞ lim sup

→0 + 1 [1-R ,1+R ] -2 ∆ ψ -ψ L 2 (M ) = 0, and ∀ 0 < ≤ 1, ψ L 2 (M ) = 1.
Our main results on these questions are propositions 6.2 and 6.4 which state that the quantum Loschmidt echo becomes asymptotically strictly less than 1. Compared with the results described above, it does not provide any decay rate but is valid for any sequence of initial data (with a proper localization in frequencies) and for any V satisfying a certain explicit admissibility condition. In particular, we do not have to average over a family of perturbations.

6.1. Preliminary lemma. As a first step in the study of the properties of the quantum Loschmidt echo, we study the restriction of the perturbed propagator e -it ( P0 ( )+ǫ V ) on the eigenspaces of the unperturbed Schrödinger operator P0 ( ). The following lemma is the key result of this section: Lemma 6.1. Suppose that dim(M) = 2, and that M has constant negative sectional curvature K ≡ -1. Suppose that lim →0 ǫ = 0, and that there exists 0 < ν < 1 2 such that, for > 0 small enough, one has ǫ ≥ ν . 8 As in the statements of section 4, our arguments could be generalized to deal with slightly more general normalized initial data satisfying [START_REF]Burq Mesures semi-classiques et mesures de défaut[END_REF] for some small enough ν 0 > 0.

Let J be a nonnegative integer such that

C J V := J j=0 (x 0 , ξ 0 ) ∈ S * M : (X j 0 .f V )(x 0 , ξ 0 ) = 0 = ∅. Let c 1 , c 2 , ν 0 > 0 satisfying 1 + (3J + 1)ν 0 < c 1 ≤ c 2 < min {3/2, 1/(2ν)} .
Then, there exists 0 < c 0 < 1 such that, for every sequence (τ ) 0< ≤1 satisfying

c 1 | log(ǫ )| ≤ τ ≤ c 2 | log(ǫ )|,
one has lim sup

→0 + Π(1, ǫ -ν 0 )e -iτ ( P0 ( )+ǫ V ) Π(1, ǫ -ν 0 ) L 2 (M )→L 2 (M ) ≤ c 0 , where Π(1, ǫ -ν 0 ) := 1 [1-ǫ -ν 0 ,1+ǫ -ν 0 ] (-2 ∆ g ).
This lemma shows that, under some geometric assumptions on the perturbation, the norm of the perturbed propogator restricted to the eigenspaces of P0 ( ) is uniformly strictly less than 1. In appendix C, it is shown that the assumption on V is "generic".

Proof. We proceed by contradiction, i.e. we suppose that, for every integer n ≥ 1, one can find a sequence (τ n ) 0< ≤1 satisfying

c 1 | log(ǫ )| ≤ τ n ≤ c 2 | log(ǫ )|,
and such that lim sup

→0 + Π(1, ǫ -ν 0 )e -iτ n ( P0 ( )+ǫ V ) Π(1, ǫ -ν 0 ) L 2 (M )→L 2 (M ) ≥ 1 - 1 n .
Thus, for any integer n ≥ 1, one can find 9 0 < n ≤ 1 n and ψ n such that

ψ n L 2 = 1, c 1 | log ǫ n | ≤ τ n n ≤ c 2 | log ǫ n |, and (36)
Π(1, n ǫ -ν 0 n )e - iτ n n n ( P0 ( n)+ǫ n V ) Π(1, n ǫ -ν 0 n )ψ n L 2 (M )→L 2 (M ) ≥ 1 - 2 n .
We define ψ n := Π(1, n ǫ -ν 0 n )ψ n , which satisfies, thanks to the previous inequality, lim n→+∞ ψ n = 1. We will now use two different procedures to compute the limit of the following quantity:

A n := M (V * ) 2 e - iτ n n n ( P0 ( n)+ǫ n V ) ψ n 2 dvol g , where V * = V -S * M V • πdL (with π(x, ξ) = x)
, and vol g is the Riemannian volume on M. Using proposition 4.4 and the fact that C J V is empty, we first obtain that

lim n→+∞ A n = S * M V • π - S * M V • πdL 2 dL.
9 Without loss of generality, we can suppose 0 < n+1 < n .

verifying, for > 0 small enough, ǫ ≥ ν .

Let 1 < c 1 ≤ c 2 < min{3/2, 1/(2ν)}.
Then, there exists 0 ≤ c 0 < 1 such that, for every sequence (ψ ) 0< ≤1 satisfying (5), and for every sequence (τ ) 0< ≤1 satisfying for > 0 small enough

c 1 | log ǫ | ≤ τ ≤ c 2 | log ǫ |, one has 0 ≤ lim sup →0 F V ,ǫ (ψ , τ ) ≤ c 0 < 1.
Remark 6.3. In order to deduce this proposition from lemma 6.1, one should observe that every sequence (ψ ) 0< ≤1 satisfying (5) verify, for every ν 0 > 0,

lim →0 Id -1 [1-ǫ -ν 0 ,1+ǫ -ν 0 ] (-2 ∆ g ) ψ L 2 = 0.
Moreover, by a compactness argument, C V = ∅ implies that there exists J ≥ 0 such that

C J V = ∅.
Our result shows that the quantum Loschmidt echo is strictly less than 1 for times of order | log ǫ |. Recall that, under our assumptions on ǫ , we are looking at a scale times for which the standard semiclassical rules apply. In other words, we are below the so-called Ehrenfest time which is | log | 2 in this geometric context. We also observe that our result holds for any sequence of initial data satisfying proper energy localization, namely [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF]. Compared with the results from the physics literature mentioned in the introduction of this section, we emphasize that our proposition does not provide a priori a decay rate for the quantum Loschmidt echo. According to the physics literature, it should typically be a nonincreasing function of time. Thus, one expects that it will at least remain strictly smaller than 1 for larger times. The following proposition provides some informations in this direction. Proposition 6.4. Suppose that dim(M) = 2, and that M has constant negative sectional curvature K ≡ -1. Suppose also that

C V = +∞ j=0 (x 0 , ξ 0 ) ∈ S * M : (X j 0 .f V )(x 0 , ξ 0 ) = 0 = ∅.
Let (ǫ ) 0< ≤1 be a sequence such that lim →0 ǫ = 0, and such that there exists 0 < ν < 1 2 verifying, for > 0 small enough, ǫ ≥ ν . Then, there exists 0 ≤ c 0 < 1 such that, for every sequence (ψ ) 0< ≤1 satisfying (5), and for every sequence (τ ) 0< ≤1 satisfying lim

→0 + τ | log ǫ | = +∞, one has 0 ≤ lim sup →0 1 0 F V ,ǫ (ψ , tτ )dt ≤ c 0 < 1.
Compared with proposition 6.2 which was for instance valid for any time of order c| log ǫ | (with c > 0 fixed in a convenient interval), this result holds on average over an interval of times of order τ . The advantage is that we can consider much larger times, e.g. we can choose τ ≫ | log |. A natural scale of times is τ = 1/ , which is called the Heisenberg time in the physics literature. In this case, the result reads as follows: Corollary 6.5. Suppose that dim(M) = 2, that M has constant negative sectional curvature K ≡ -1, and that C V = ∅. Let 3/2 < α < 2.

Then, there exists 0 ≤ c 0 < 1 such that for every sequence (ψ ) 0< ≤1 satisfying (5), one has

∀τ 0 > 0, lim sup →0 1 τ 0 τ 0 0 e it∆g 2 ψ , e it(∆g+ -α V ) 2 ψ 2 dt ≤ c 0 < 1.
σ 0 : Ψ m,k (M) → S m,k /S m-1,k-1 (T * M) is intrinsically defined. Most of the rules (for example the composition of operators, the Egorov and Calderón-Vaillancourt Theorems) that hold on R 2d still hold in the case of Ψ m,k (M). Because our study concerns the behavior of quantum evolution for logarithmic times in , a larger class of symbols should be introduced as in [START_REF] Zworski | Semiclassical analysis[END_REF], for 0

≤ ν < 1/2, ( 46
) S m,k ν (T * M) := (a ) ∈(0,1] ∈ C ∞ (T * M) : |∂ α x ∂ β ξ a | ≤ C α,β -k-ν|α+β| ξ m-|β| .
Results of [START_REF] Zworski | Semiclassical analysis[END_REF] (as Calderón-Vaillancourt and Egorov theorems) can also be applied to this new class of symbols.

A.2. Egorov theorem. In this paragraph, we briefly recall the Egorov theorem for short logarithmic times on constant negatively curved surfaces. The proof of this result on compact manifold was given in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Festival Yves Colin de Verdière[END_REF][START_REF] Dyatlov | Guillarmou Microlocal limits of plane waves and Eisenstein functions[END_REF] building on earlier proofs on R d [START_REF] Bambusi | Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time[END_REF][START_REF] Bouzouina | Robert Uniform semiclassical estimates for the propagation of quantum observables[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF]. Here, we will in fact be interested in generalizations of the results from [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Festival Yves Colin de Verdière[END_REF][START_REF] Dyatlov | Guillarmou Microlocal limits of plane waves and Eisenstein functions[END_REF] in the context of "perturbed" Schrödinger operators Pǫ ( ) := P0 ( ) + ǫV . It was explained in appendix B of [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] how the Egorov Theorem for short logarithmic times can be extended to Pǫ ( ) by slightly adapting the arguments from [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Festival Yves Colin de Verdière[END_REF][START_REF] Dyatlov | Guillarmou Microlocal limits of plane waves and Eisenstein functions[END_REF]. We will just recall the result we need, and we refer the reader to the above references for more details. Let δ > 0 and let a be a smooth function which is compactly supported in the following neighborhood of S * M of size δ, i.e.:

T * [1/2-δ,1/2+δ] M := {(x, ξ) ∈ T * M : p 0 (x, ξ) ∈ [1/2 -δ, 1/2 + δ]} ,
where p 0 (x, ξ) := ξ 2 2 . According to [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] (appendix B), there exists ǫ 0 > 0 such that, for every ǫ ∈ [0, ǫ 0 ] and for every smooth function a compactly supported in T * [1/2-δ,1/2+δ] M, one has that, for every 0 ≤ t ≤ belongs to Ψ -∞,0 ν (M) for some 0 < ν < 1/2 (which depends on δ and ǫ 0 ). Moreover, its principal symbol is equal to a • G t ǫ and all the involved semi-norms can be uniformly bounded in terms of ǫ ∈ [0, ǫ 0 ] and of t in the above range. Also, thanks to the Calderón-Vaillancourt Theorem, we find that, uniformly for ǫ ∈ [0, ǫ 0 ] and 0

≤ t ≤ (1-δ)| log | 2 √ 1+6δ , one has (47) e it Pǫ( ) Op (a)e -it Pǫ( ) -Op (a • G t ǫ ) L 2 →L 2 = o(1), as → 0.
A.3. Positive quantization. Even if the Weyl procedure is a natural choice to quantize an observable a on R 2d , it is sometimes preferrable to use a quantization procedure Op + that satisfies the following property : Op + (a) ≥ 0 if a ≥ 0. This can be achieved thanks to the anti-Wick procedure Op AW , see [START_REF] Helffer | Robert Ergodicité et limite semi-classique[END_REF] for instance. For a in S 0,0 ν (R 2d ), that coincides with a function on R d outside a compact subset of T

* R d = R 2d , one has (48) Op w (a) -Op AW (a) L 2 ≤ C 1≤|α|≤D |α| 2 ∂ α a ,
where C and D are some positive constants that depend only on the dimension d. To get a positive procedure of quantization on a manifold, one can replace the Weyl quantization by the anti-Wick one in definition [START_REF] Schubert | Semiclassical behaviour of expectation values in time evolved Lagrangian states for large times[END_REF]. This new choice of quantization (that we will denote by Op + ) is positive and it is well defined for every element a in S 0,0 ν (T * M) of the form c 0 (x) + c(x, ξ) where c 0 belongs to S 0,0 ν (T * M) and c belongs to10 C ∞ c (T * M) ∩ S 0,0 ν (T * M). We can also require that Op + (1) = Id L 2 (M ) . The main observation is that, for such symbols, one has [START_REF]Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces[END_REF] Op

+ (a) -Op (a) L 2 ≤ C ′ 1≤|α|≤D ′ |α| 2 ∂ α a ,
where C ′ and D ′ are some positive constants that depend only on the manifold M and on the choice of coordinate charts.

Appendix B. Strong structural stability

In our proof, we needed to use the strong structural stability property for Anosov flows [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF], and more precisely, we needed to use the fact that the conjugating homeomorphism and the reparametrization function depend in a "smooth" way on the perturbation parameter ǫ. This regularity was observed by De La Llave, Marco and Moriyon [START_REF] De La Llave | Moriyon Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation[END_REF] based on "analytic" proofs of structural stability for Anosov diffeomorphisms due to Moser [START_REF] Moser | On a theorem of Anosov[END_REF] and Mather [START_REF] Smale | Differentiable dynamical systems[END_REF]. In this appendix, we recall a few facts on the geometric properties of the conjugating homeomorphism that were proved in section 5 of [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] based on the arguments of [START_REF] De La Llave | Moriyon Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation[END_REF].

First, we briefly recall strong structural stability property for Anosov flows in the same way as it was stated in [START_REF] De La Llave | Moriyon Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation[END_REF]. For that purpose, we introduce some manifolds of mappings that will be involved in this theorem -see [START_REF] Jr | On the geometry of function spaces[END_REF][START_REF] Abraham | Lectures of Smale on differential topology[END_REF][START_REF]Eels Jr A setting for global analysis[END_REF] or the appendix of [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] for a brief reminder on their differential structure. Define

C X 0 (S * M) := h ∈ C 0 (S * M, S * M) : ∀ρ ∈ S * M, d dt h • G t 0 (ρ) t=0 = D X 0 h exists ,
which can be endowed with a smooth differential structure modeled on the Banach spaces of continuous sections s : S * M → h * T S * M which are differentiable along the geodesic flow. This manifold contains an "adapted" submanifold M which contains Id S * M in its interior and for which the elements are in some sense "transversal" to the geodesic vector field X 0 [START_REF] De La Llave | Moriyon Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation[END_REF] -appendix A (see also appendix of [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF]). The structural stability theorem can be then stated as follows (theorem A.2 in [START_REF] De La Llave | Moriyon Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation[END_REF]):

Theorem B.1. [Strong structural stability] Assume X 0 is an Anosov vector field. There exists an open neighborhood U 0 (X 0 ) of X 0 in V 2 (S * M) and a unique C 2 map S 0 : U 0 (X 0 ) → M × C 0 (S * M, R) such that S 0 (X 0 ) = (Id S * M , 1) and if S 0 (X) = (h, τ ), then

D X 0 h -τ X • h = 0 S * M (h), (50)
where 0 S * M is the zero section.

Remark B.2. In fact, the neighborhood U 0 (X 0 ) can be chosen small enough to ensure that h is an homeomorphism -appendix A of [START_REF] De La Llave | Moriyon Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation[END_REF] or remark 5.5 of [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF]. Using that h is an homeomorphism, we can also write the following formula connecting the two flows:

(51) ∀ t ∈ R, h • G τ (t,ρ) 0 • h -1 (ρ) = G t X (ρ), where τ (t, ρ) := t 0 ds τ • h -1 • G s X (ρ)
.

We now describe some geometric properties of the conjugating homeomorphism when we apply the strong structural stability theorem to the perturbations Y ǫ x 1 ,ξ 1 . Recall that they define small C 1 perturbation of the geodesic vector field X 0 . We refer to section 5 of [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] for the details of the proofs.

Observe that there exists ǫ 0 > 0 such that, for every (x 1 , ξ 1 ) in a small neighborhood of S * M and for every ǫ ∈ [0, ǫ 0 ], Y ǫ x 1 ,ξ 1 belongs to the neighborhood of the previous theorem. We write

S 0 (Y ǫ x 1 ,ξ 1 ) = (h ǫ x 1 ,ξ 1 , τ ǫ x 1 ,ξ 1 ). As the map S 0 is of class C 1 , we can write that (52) sup ρ∈S * M d(h ǫ x 1 ,ξ 1 (ρ), ρ), |τ ǫ x 1 ,ξ 1 (ρ) -1| = O(ǫ),
where the constant in the remainder is uniform for (x 1 , ξ 1 ) in a small neighborhood of S * M.

In our proof, we also need to understand precisely the properties of the map (h ǫ x 1 ,ξ 1 ) -1 . We observe that the map ǫ → (h ǫ x 1 ,ξ 1 ) -1 ∈ C 0 (S * M, S * M) has a priori no reason to be of class C 1 -see remark 5.7 in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF]. In order to solve this problem, we will write h ǫ x 1 ,ξ 1 := exp(v ǫ x 1 ,ξ 1 ), where v ǫ x 1 ,ξ 1 is a continuous vector field and exp is the exponential map induced by the Riemannian structure on S * M. We also introduce the following vector field on S * M:

(53) ṽǫ x 1 ,ξ 1 := ǫ ξ 1 (β s V X s + β u V X u ) , where
β s V (x, ξ) := 1 √ 2 +∞ 0 g * x(-t) d x(-t) V, ξ ⊥ (-t) e -t dt, and
β u V (x, ξ) := 1 √ 2 +∞ 0 g * x(t) d x(t) V, ξ ⊥ (t) e -t dt,
with G t 0 (x, ξ) := (x(t), ξ(t)). We observe that these two functions do not depend on (x 1 , ξ 1 ). It was proved in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF] that β u V and β s V are C γ -Hölder for every γ < 1/2 -lemma 5.13 from this reference. According to lemma 1 in [START_REF] Moser | On a theorem of Anosov[END_REF], we can write that (54) exp(-ṽ ǫ x 1 ,ξ 1 ) • exp v ǫ x 1 ,ξ 1 = exp(-ṽ ǫ x 1 ,ξ 1 + v ǫ x 1 ,ξ 1 + r(ṽ ǫ x 1 ,ξ 1 , v ǫ x 1 ,ξ 1 )), where r(ṽ ǫ x 1 ,ξ 1 , v ǫ x 1 ,ξ 1 ) C 0 ≤ C ṽǫ x 1 ,ξ 1 C γ v ǫ

x 1 ,ξ 1 γ C 0 , for some uniform constant C > 0 (depending on the manifold and on γ). In particular, we have that, in our setting, r(ṽ ǫ x 1 ,ξ 1 , v ǫ x 1 ,ξ 1 ) C 0 = O(ǫ 1+γ ) with the constant involved in the remainder which is uniform for (x 1 , ξ 1 ) in a small neighborhood of S * M.

Remark B.3. The proof of this fact was given in the appendix of [START_REF] Moser | On a theorem of Anosov[END_REF] for the general case of vector fields on a Riemannian manifold. The only difference is that the proof given in this reference is for γ = 1. Yet, the proof can be directly adapted to get the above estimate involving Hölder norms.

Finally, according to paragraphs 5.3.1 and 5.3.2 in [START_REF] Eswarathasan | Rivière Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces[END_REF], one has that ṽǫ

x 1 ,ξ 1 is equal to v ǫ

x 1 ,ξ 1 up to an error of order O(ǫ 2 ) in the C 0 -topology. This property followed from the differentiation of the implicit equation [START_REF]Zelditch Recent developments in mathematical quantum chaos[END_REF]. In particular, thanks to (54), we can derive that exp(-ṽ ǫ x 1 ,ξ 1 ) • h ǫ x 1 ,ξ 1 is close to identity up to an error of order O(ǫ 1+γ ), where the constant involved is uniform for (x 1 , ξ 1 ) in a small neighborhood of S * M. We underline that this property holds for every 0 < γ < 1/2. To summarize, for every 0 < γ < 1/2, we have that (55) sup

ρ∈S * M d h ǫ x 1 ,ξ 1 -1 (ρ), exp(-ṽ ǫ x 1 ,ξ 1 )(ρ) = O(ǫ 1+γ ),
where the constant in the remainder is uniform for (x 1 , ξ 1 ) in a small neighborhood of S * M. Thus, even if the map ǫ → (h ǫ x 1 ,ξ 1 ) -1 ∈ C 0 (S * M, S * M) is not "smooth", it can be approximated in a precise way by a smooth map which has a very explicit expression.

Proof. Recall that the (natural) Fréchet topology on C ∞ (M, R) is in fact equivalent to the topology induced by the following metric:

∀V, W ∈ C ∞ (M, R), D(V, W ) := j≥0 1 2 j+1 min {1, V -W C j } ,
where, for every i ≥ 0, . C i is the usual norm on C i (M, R).

We will first prove that the set U is open. We fix V in U. By compactness, we observe that there exists some J 0 > 0 such that, for every ρ in S * M, there exists 0 ≤ j ≤ J 0 such that X j 0 .f V (ρ) = 0. We introduce fV (ρ) := max |X j 0 .f V (ρ)| : 0 ≤ j ≤ J 0 , which is continuous on S * M. From our assumption, there exists 0 < δ 0 < 1 such that fV ≥ δ 0 on S * M. We now observe that, for every j ≥ 0, there exists a constant c j ≥ 1 (depending only on (M, g) and on j) such that, for every ρ ∈ S * M and for every W in C ∞ (M, R), one has

|X j 0 .(f V -f W )(ρ)| ≤ c j V -W C j+1 . If we take W in C ∞ (M, R) such that D(V, W ) ≤ δ 0
2 J 0 +2 max 0≤j≤J 0 c j , then, for every 0 ≤ j ≤ J 0 , one has V -W C j ≤ δ 0 2 max 0≤j≤J 0 c j . Then, we deduce that W belongs to U. It remains to show that the set U is dense. This will follow from the Sard-Smale's theorem [START_REF] Smale | An infinite dimensional version of Sard's theorem[END_REF]. Before applying this theorem, we make a simple observation. We fix V in C ∞ (M, R) and δ 0 > 0. We observe that there exists J 0 ≥ 5 such that, for every W in C ∞ (M, R),

D(V, W ) ≤ J 0 j=0 1 2 j+1 V -W C j + δ 0 .
Suppose now that we are able to find W in C J 0 (M, R) which is δ 0 close to V in the C J 0 topology and such that, for every ρ ∈ S * M, there exists 0 ≤ j ≤ 3 verifying X j 0 .f W (ρ) = 0. Then, we can use the fact that C ∞ (M, R) is dense in C J 0 (M, R) for the C J 0 topology and conclude. In fact, by density, we can find, for every 0 < δ < δ 0 , W in C ∞ (M, R) such that W -W C J 0 ≤ δ max 0≤j≤J 0 c j . In particular, for such a W , one has D(V, W ) ≤ 3δ 0 , and

∀ρ ∈ S * M, max 0≤j≤3 {|X j 0 .f W (ρ)|} ≥ max 0≤j≤3 {|X j 0 .f W (ρ)|} -δ.
Taking δ > 0 small enough to ensure that the above quantity is positive for every ρ in S * M, we have found W in C ∞ (M, R) which is δ 0 close to V in the C ∞ topology and such that, for every ρ in S * M, there exists 0 ≤ j ≤ 3 satisfying X j 0 .f W (ρ) = 0. It now remains to prove that, for every J 0 ≥ 5, the set U J 0 := V ∈ C J 0 (M, R) : ∀ρ ∈ S * M, ∃0 ≤ j ≤ 3 s.t. X j 0 .f V (ρ) = 0 contains a dense subset of C J 0 (M, R). For that purpose, we define

L : (V, ρ) ∈ C J 0 (M, R) × S * M → (f V (ρ), X 0 .f V (ρ), X 2 0 .f V (ρ), X 3 0 .f V (ρ)) ∈ R 4 .
This defines a C 1 map on the Banach manifold C J 0 (M, R) × S * M, and we say that 0 is a regular value of L if, for every (V, ρ) satisfying L(V, ρ) = 0, the tangent map D (V,ρ) L is a continuous surjective linear map whose kernel has a closed complement. One can verify that it is in fact continuous, and that the kernel has a closed complement (the kernel has finite codimension). In order to verify the surjectivity, it is sufficient to show that (56) ∀z ∈ R 4 , ∃W ∈ C J 0 (M, R) such that D (V,ρ) L.(W, 0) = z.

We note that D (V,ρ) L.(W, 0) = L(W, ρ), and we introduce the following linear maps, for every ρ 0 = (x 0 , ξ 0 ) in S * M: L (0) ρ 0 : W ∈ B 0 := C J 0 (M, R) → f W (ρ 0 ) ∈ R, and, for 1 ≤ j ≤ 3,

L (j)
ρ 0 : W ∈ B j := ∩ j-1 l=0 Ker L (l) ρ 0 → X j 0 .f W (ρ 0 ) ∈ R. For every 0 ≤ j ≤ 3, one can verify that these maps are nonvanishing linear forms on the infinite dimensional Banach space B j -see remark C.2 below. In particular, taking W j which does not belong to the kernel of L (j) ρ 0 for every 0 ≤ j ≤ 3, we can write that, for every 0 ≤ j ≤ 3, L(W j , ρ 0 ) = (a j 0 , a j 1 , a j 2 , a j 3 ), where, by construction, a j j = 0, and a j p = 0 for every p < j. In particular, R 4 = span{D (V,ρ) L.(W j , 0) : 0 ≤ j ≤ 3}, and the map D (V,ρ) L is surjective.

Remark C.2. Above, we claimed that L (j) ρ 0 is a nonvanishing linear form when it acts on the Banach space B j ⊂ C J 0 (M, R). This can be proved as follows. Let 0 ≤ j ≤ 3. Consider κ : U → V ⊂ R 2 a chart centered at x 0 in M. This chart can be lifted to a chart centered at (x 0 , 0) in T * M as follows κ : T * U → T * V ⊂ R 4 , (x, ξ) → (u k , v l ) := (κ(x), (dκ(x) T ) -1 ξ).

For a fixed smooth function W on M, we define in a small neighborhood of 0 the function W (u k ) = W • κ -1 (u k ). In these local coordinates, the map f W can rewritten:

f W (x, ξ) := k l gk,l (u)v l ∂ k W (u).
where (g k,l ) k,l is a non-degenerate 2-form. For ξ not equal to 0, we note that at least one of the two coefficients ṽk := l gk,l (u)v l does not vanish. In particular, at the point (x 0 , ξ 0 ), one has f W (x 0 , ξ 0 ) := d 0 W (ṽ), with ṽ = 0 independent of W (as ξ 0 = 0). More generally, for 1 ≤ j ≤ 3, one can write in local coordinates, X j 0 .f W (x 0 , ξ 0 ) := d j+1 W (ṽ, v, . . . , v) + α∈N 2 :|α|<j+1 a j α ∂ α W (0, 0),

  lim →0,j∈J( ) R ϕ(t) ψ j , e -itτ ∆g Op (a)e itτ ∆g ψ j dt = R ϕ(t)dt × S * M adL.

6. 2 . 2 .

 22 Beyond the | log ǫ |-scale. The previous proposition holds for times of order | log ǫ |, and it is natural to ask what are the properties of the quantum Loschmidt echo beyond this time scale.

  )e -it Pǫ( )

We refer the reader to[START_REF] Katok | Hasselblatt Introduction to the modern theory of dynamical systems[END_REF] for a precise definition.

In this case, (G t 0 ) t∈R is the Hamiltonian flow associated to p 0 .

We emphasize that the results are not equivalent, and that they cannot a priori be deduced one from the other.

The constant 1/2 appearing here is the main reason for the factor 3/2(= 1 + 1/2) involved in the statement of the proposition.

It seems that the terminology "quantum Loschmidt echo" was introduced in this article.

Here we mean that there exists a compact subset K ⊂ T * M such that supp(a ) ⊂ K for every 0 < ≤ 1.

Recall that a Fredholm map is a C 1 map whose tangent map defines a Fredholm operator. The index of a Fredholm map is then the index of its tangent map.

Acknowledgements

The author is partially supported by the Agence Nationale de la Recherche through the Labex CEMPI (ANR-11-LABX-0007-01) and the ANR project GeRaSic (ANR-13-BS01-0007-01). We warmly thank Suresh Eswarathasan for his comments on a preliminary version of this work, and Jean-Yves Welschinger for indicating us the proof of proposition C.1.

We will now compute the limit of A n in a slightly different manner, and then get the contradiction. For that purpose, we will admit that the following property holds: [START_REF] Lang | Introduction to differentiable manifolds[END_REF] V e - iτ n n n

. We postpone the proof of this equality to the end, and we first show how it allows us to conclude. As in paragraph 4.3, we introduce a smooth cutoff function χ δ to microlocalize the symbols near S * M. Combining this to relation [START_REF] Lang | Introduction to differentiable manifolds[END_REF], we obtain:

where we set

As in paragraph 4.3, we can use the invariance of the state ψ n on short intervals of time, precisely [START_REF] Dyatlov | Guillarmou Microlocal limits of plane waves and Eisenstein functions[END_REF]. We find that

for some fixed ν 2 > ν 0 . Using Egorov and Calderón-Vaillancourt theorems [START_REF] Zworski | Semiclassical analysis[END_REF] (Chapters 4 and 11), we have

, and thus

So, it remains to analyse the operator 1

2

V n e is n∆g 2 ds whose complete expression is

ds.

The proof of proposition 4.4 was in fact reduced to studying the convergence of this kind of operator -see equations [START_REF] Jr | On the geometry of function spaces[END_REF] and [START_REF]Eels Jr A setting for global analysis[END_REF]. In particular, we proved that, modulo small error terms, this operator is a n -pseudodifferential operator with principal symbol

Combining the facts that C J V is empty and that S * M V * dL = 0 to proposition 5.1, it can be shown that the norm of this operator is in fact o(1) as n → +∞. Thus, one has

which provides the contradiction as C J V is empty. It remains now to verify that (37) holds. We observe that, up to this point, we did not use all the informations contained in [START_REF] Katok | Hasselblatt Introduction to the modern theory of dynamical systems[END_REF]. In particular, by construction of ψ n , one knows that, as n → +∞,

Then, we fix a bounded sequence (δ n ) n≥1 such that ǫ n δ -1 n → 0, and δ n ǫ -1 n r n → 0 as n → +∞. We let 0 ≤ χ 1 ≤ 1 be a smooth cutoff function which is equal to 1 in a small neighborhood of 0 and 0 outside a slightly larger interval, say [-1/2, 1/2]. As 1 + ν 0 < 1 ν by assumption, and as

Using functional calculus for pseudodifferential operators (Ch. 14 in [START_REF] Zworski | Semiclassical analysis[END_REF]) and the fact that δ n ≥ ν for some 0 < ν < 1/2, we know that the operators

Then, using the Calderón-Vaillancourt theorem [START_REF] Zworski | Semiclassical analysis[END_REF] (Ch. 5), we find that (39)

Using [START_REF] Macià | [END_REF] and twice this equality, we find that

Thanks to the composition properties of -pseudodifferential operators, this can also be rewritten as

.

We now remark that

Implementing this property in [START_REF] Moser | On a theorem of Anosov[END_REF], we get

Applying [START_REF] Macià | [END_REF], [START_REF] Marcus | Ergodic properties of horocycle flows for surfaces of negative curvature[END_REF] and the composition rule for pseudodifferential operators in the other way, we finally obtain

, which is exactly equality (37). 6.2. Properties of the quantum Loschmidt echo.

6.2.1. Times of order | log ǫ |. As a direct application of lemma 6.1, we obtain the following property of the quantum Loschmidt echo: Proposition 6.2. Suppose that dim(M) = 2, and that M has constant negative sectional curvature K ≡ -1. Suppose also that

Let (ǫ ) 0< ≤1 be a sequence such that lim →0 ǫ = 0, and such that there exists 0 < ν < 1

We will now give the proof of proposition 6.4 which also follows from lemma 6.1.

Proof. As C V = ∅ and as S * M is compact, there exists J ≥ 0 such that C J V = ∅. We fix ν 0 > 0 small enough to ensure 1 + (3J + 1)ν 0 < min{3/2, 1/(2ν)} as in the statement of lemma 6.1.

Using remark 6.3, we first observe that lim sup

We will in fact show something slightly stronger. Precisely, we will prove that there exists 0 ≤ c 0 < 1 such that for every sequence (ψ ) 0< ≤1 satisfying (5), and for every sequence (τ ) 0< ≤1 satisfying lim

We proceed by contradiction. We suppose that there exist a sequence (ψ n ) n≥1 satisfying ( 5), and a sequence (

In particular, as ψ n = 1, one has (42) lim

and also, from the Jensen's inequality, [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF] lim

We fix 1 + (3J + 1)ν 0 < c < min{3/2, 1/(2ν)}. By changing the variables in equation ( 41), we deduce that

Then, from ( 41), ( 42) and ( 43), one gets

In particular, one has

which is ≤ c 2 0 < 1 from lemma 6.1, and thus provides the contradiction.

Appendix A. Semiclassical analysis on manifolds

In this appendix, we review some basic facts on semiclassical analysis that can be found for instance in [START_REF] Zworski | Semiclassical analysis[END_REF] -chapter 14.

A.1. General facts. Recall that we define on R 2d the following class of admissible symbols:

Let M be a smooth Riemannian d-manifold without boundary. Consider a smooth atlas

Consider now a smooth locally finite partition of identity (φ l ) adapted to the previous atlas (f l , V l ). That means l φ l = 1 and φ l ∈ C ∞ (V l ). Then, any observable a in C ∞ (T * M) can be decomposed as follows: a = l a l , where a l = aφ l . Each a l belongs to C ∞ (T * V l ) and can be pushed to a function ãl = ( f -1 l ) * a l ∈ C ∞ (T * W l ). As in [START_REF] Zworski | Semiclassical analysis[END_REF], define the class of symbols of order m and index k

Then, for a ∈ S m,k (T * M) and for each l, one can associate to the symbol ãl ∈ S m,k (R 2d ) the standard Weyl quantization

where u ∈ S(R d ), the Schwartz class. Consider now a smooth cutoff ψ l ∈ C ∞ c (V l ) such that ψ l = 1 close to the support of φ l . A quantization of a ∈ S m,k (T * M) is then defined in the following way [START_REF] Zworski | Semiclassical analysis[END_REF]: [START_REF] Schubert | Semiclassical behaviour of expectation values in time evolved Lagrangian states for large times[END_REF] Op (a)(u) :

where u ∈ C ∞ (M). This quantization procedure Op sends (modulo O( ∞ )) S m,k (T * M) onto the space of pseudodifferential operators of order m and of index k, denoted Ψ m,k (M) [START_REF] Zworski | Semiclassical analysis[END_REF].

It can be shown that the dependence in the cutoffs φ l and ψ l only appears at order 1 in (Theorem 18.1.17 in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators III[END_REF] or Theorem 9.10 in [START_REF] Zworski | Semiclassical analysis[END_REF]) and the principal symbol map

In this appendix, we will prove the following proposition which shows that the assumptions appearing in section 6 and corollary 2.6 are in some sense "generic": Proposition C.1. Let M be a smooth compact oriented Riemannian boundaryless surface. Then, the set

is open and dense in C ∞ (M, R) endowed with its natural topology of Fréchet space.

We note that we do not require M to be negatively curved in this statement. Recall that we have defined f V (x, ξ) := g * x (d x V, ξ ⊥ ), and C V := ρ ∈ S * M : ∀ j ≥ 0, X j 0 .f V (ρ) = 0 . The proof below was indicated to us by Jean-Yves Welschinger.

where a j α are real numbers which are independent of W . If we choose W such that d p x 0 W = 0 for every p < j, then W belongs to the space B j , and we have, for such a function W , L (j) ρ 0 (W ) := X j 0 .f W (x 0 , ξ 0 ) := d j+1 W (ṽ, v, . . . , v). As the vectors v and ṽ are both nonzero, we can find a function W whose derivatives up to order j vanish at (0, 0) and such that the previous quantity does not vanish.

From the previous discussion, we can conclude that L -1 (0) defines a C 1 submanifold of codimension 4 [START_REF] Lang | Introduction to differentiable manifolds[END_REF] -Ch. 1 and 2. We are now in position to conclude. For that purpose, we define the projection map Λ : (V, ρ) ∈ C J 0 (M, R) × S * M → V ∈ C J 0 (M, R).

In the terminology 11 of [START_REF] Smale | An infinite dimensional version of Sard's theorem[END_REF], this defines a Fredholm map of index 3. One can also consider the restriction Λ⌉ L -1 (0) of this map to the codimension 4 submanifold L -1 (0). This map can also be written Λ⌉ L -1 (0) = Λ • I, where I is the inclusion map from L -1 (0) to C J 0 (M, R) × S * M. Thanks to the above observation, I is a Fredholm map of index -4. By the composition rules for Fredholm operators, one has that Λ⌉ L -1 (0) is a Fredholm map of index -1. Thanks to the Sard-Smale's theorem -for instance corollary 1.5 in [START_REF] Smale | An infinite dimensional version of Sard's theorem[END_REF], there exists a dense subset D of C J 0 (M, R) such that, for every V in D, (Λ⌉ L -1 (0) ) -1 (V ) is empty. In particular, every V in D belongs to U J 0 ; thus, U J 0 is a dense subset.