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

A N Appendix

S T Éphane

Gabriel Rivi Ère

EIGENMODES OF THE DAMPED WAVE EQUATION AND SMALL HYPERBOLIC SUBSETS

We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of β-damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of β-damped trajectories of the geodesic flow.

The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip without eigenvalues below the real axis, under a pressure condition on the set of undamped trajectories.

Introduction

Let M be a smooth, connected, compact Riemannian manifold of dimension d ≥ 2 and without boundary. We will be interested in the high frequency analysis of the damped wave equation,

∂ 2 t -∆ + 2a(x)∂ t v(x, t) = 0, where ∆ is the Laplace-Beltrami operator on M and a ∈ C ∞ (M, R) is the damping function. The case of damping corresponds actually to a ≥ 0 but our results will be valid for any real valued function a. Our main concern in this article is to study asymptotic properties of solutions of the form v(t, x) = e -ıtτ u τ (x),

where τ belongs to C and u τ (x) is a non trivial element in L 2 (M ). Such a mode is a solution of (1) if one has

(2) (-∆τ 2 -2ıτ a)u τ = 0.

From the spectral analysis of (1), there exist countably many (τ n ) solving this nonselfadjoint eigenvalue problem. One can also verify that their imaginary parts remain in a bounded strip parallel to the real axis and they satisfy lim n→+∞ Re τ n = ±∞ [START_REF] Sjöstrand | Asymptotic distributions of eigenfrequencies for damped wave equations[END_REF][START_REF]Hitrik Eigenfrequencies and Expansions for Damped Wave Equations[END_REF][START_REF] Nonnenmacher | Spectral theory of damped quantum chaotic systems[END_REF]. We also recall that (τ, u τ ) solves the eigenvalue problem [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF] if and only if (-τ , u τ ) solves it [START_REF] Nonnenmacher | Spectral theory of damped quantum chaotic systems[END_REF]. Our main concern in the following will be to describe some asymptotic properties of sequences (τ n , u n ) n solving (2) with Re τ n → +∞ and Im τ n → β, where β ∈ R. Very general results on the asymptotic distribution of the τ n and its links with the properties of (1) have been obtained by various authors. For instance, in a very general context, Lebeau related the geometry of the undamped geodesics, the spectral asymptotics of the τ n and the energy decay of the damped wave equation [START_REF] Lebeau | Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli[END_REF]. Related results were also proved in several geometric contexts where the family of undamped geodesics was in some sense not too big: closed elliptic geodesic [START_REF]Hitrik Eigenfrequencies and Expansions for Damped Wave Equations[END_REF], closed hyperbolic geodesic [START_REF] Christianson | Semiclassical nonconcentration near hyperbolic orbits[END_REF][START_REF] Burq | Christianson Imperfect control for the damped wave equation[END_REF], subsets satisfying a condition of negative pressure [START_REF] Schenck | Energy decay for the damped wave equation under a pressure condition[END_REF][START_REF] Schenck | Exponential stabilization without geometric control[END_REF][START_REF] Nonnenmacher | Spectral theory of damped quantum chaotic systems[END_REF]. Concerning the distribution of the τ n , Sjöstrand gave a precise asymptotic description of the τ n on a general compact manifold [START_REF] Sjöstrand | Asymptotic distributions of eigenfrequencies for damped wave equations[END_REF]. We also refer the reader to [START_REF]Hitrik Eigenfrequencies for Damped Wave Equations on Zoll manifolds[END_REF] in the case of Zoll manifolds and to [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF] in the case of negatively curved manifolds.

1.1. Semiclassical reduction. We will mention more precisely some of these results related to ours but before that we would like to proceed to a semiclassical reformulation of our problem as it was performed in [START_REF] Sjöstrand | Asymptotic distributions of eigenfrequencies for damped wave equations[END_REF]. Thanks to the different symmetries of our problem, we can restrict ourselves to the limit Re τ → +∞. We will look at eigenfrequencies τ of order -1 (where 0 < ≪ 1 will be the semiclassical parameter of our problem) and we will set

τ = √ 2z
, where z( ) = 1 2 + O( ).

In the following, we will often omit the dependence of z( ) = z in in order to simplify the notations. Thanks to this change of asymptotic parameters, studying the high frequency modes of the problem [START_REF] Anantharaman | Spectral deviations for the damped wave equation[END_REF] corresponds to look at sequences (z( ) = 1 2 + O( )) 0< ≪1 and (ψ ) 0< ≪1 in L 2 (M ) satisfying 1 (3) (P( , z)z( ))ψ = 0, where P( , z

) := - 2 ∆ 2 -ı 2z( )a(x).
Recall that, for every t in R, the quantum propagator associated to P( , z) is given by (4) U t := exp -ıtP( , z) .

It was proved by Markus-Matsaev and Sjöstrand that the "horizontal" distribution of the eigenvalues of P( , z) satisfies a Weyl law in the semiclassical limit → 0 -see Theorem 5.2 in [START_REF] Sjöstrand | Asymptotic distributions of eigenfrequencies for damped wave equations[END_REF] for the precise statement. Translated in this semiclassical setting, our goal is to describe asymptotic properties of a sequence of normalized eigenmodes (ψ ) →0 + satisfying (3) with z( ) = 1 2 + O( ) and Im z( ) = β + o(1), as → 0. A way to study these eigenmodes is to look at the following distributions on T * M [START_REF]Burq Mesures semi-classiques et mesures de défaut[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF]:

(5) ∀b ∈ C ∞ o (T * M ), µ ψ (b) := ψ , Op (b)ψ L 2 (M)
, where Op (b) is a -pseudodifferential operator (see section 5 for a brief reminder). Under our assumptions, one can prove that, as tends to 0, µ ψ converges (up to an extraction) to a probability measure µ on the unit cotangent bundle S * M = {(x, ξ) ∈ T * M : ξ x = 1}. Moreover, this probability measure satisfies the following invariance relation: [START_REF] Bouzouina | Robert Uniform semiclassical estimates for the propagation of quantum observables[END_REF] ∀b ∈ C 0 (S * M ), µ(b) = µ b • g t e -2βt-2 t 0 a•g s ds ,

where g t is the geodesic flow on S * M . Such a probability measure is called a semiclassical measure of the sequence (ψ ) →0 + [START_REF]Burq Mesures semi-classiques et mesures de défaut[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF] and one can verify that the support of such a measure is invariant under the geodesic flow. Following [START_REF] Lebeau | Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli[END_REF][START_REF] Sjöstrand | Asymptotic distributions of eigenfrequencies for damped wave equations[END_REF][START_REF] Asch | The spectrum of the damped wave operator for a bounded domain in R 2[END_REF], one can introduce the following dynamical quantities:

A + = lim T →+∞ 1 T sup ρ∈S * M - T 0 a • g s (ρ)ds,
and

A -= lim T →+∞ 1 T inf ρ∈S * M - T 0 a • g s (ρ)ds.
Then, β ∈ [A -, A + ]. As in the selfadjoint case, one can try to understand properties of these semiclassical measures -see [START_REF] Asch | The spectrum of the damped wave operator for a bounded domain in R 2[END_REF] for some general results. For instance, if {γ} is a periodic orbit on which the Birkhoff average of -a,

lim T →+∞ - 1 T T 0 a • g s (ρ)ds, ρ ∈ {γ},
is not equal to β, then one has µ({γ}) = 0. However, if the Birkhoff average along γ is equal to β, this can be no longer true. When specified in the case of hyperbolic periodic orbits, our main result will give informations on this kind of issues.

1.2. Results in the selfadjoint case. Before stating our result, we would like to recall related results in the selfadjoint case a ≡ 0 -see also [START_REF]Zelditch Recent developments in mathematical quantum chaos[END_REF], section 5 for a more detailed account on the results we will mention. In this case, it means that we look at eigenfunctions of the Laplacian on M in the large eigenvalue limit.

In [START_REF] De Verdière | Parisse Équilibre instable en régime semi-classique. I. Concentration microlocale[END_REF], Colin de Verdière and Parisse have exhibited geometric situations where one can find a sequence of eigenmodes (ψ ) >0 whose semiclassical measure is an invariant probability measure carried by an hyperbolic periodic orbit γ. Yet, they show that if such a concentration occurs, it must at happen at a slow rate. Precisely, they prove that if U is a fixed small neighborhood of their geodesic γ, then there exists a positive constant C such that

M\U |ψ (x)| 2 dvol M (x) ≥ C | log | , as → 0.
This result has been generalized2 to more general Hamiltonian flows involving a hyperbolic closed geodesic by Burq-Zworski [START_REF] Burq | Zworski Geometric control in the presence of a black box[END_REF] and Christianson [START_REF] Christianson | Semiclassical nonconcentration near hyperbolic orbits[END_REF]. In [START_REF] Toth | L p norms of eigenfunctions in the completely integrable case[END_REF], Toth and Zelditch also consider a related question and they look at the concentration of eigenmodes in shrinking tubes in S * M of size ν around a closed hyperbolic geodesic (where 0 < ν < 1 2 ) -see also paragraph 5.1 of [START_REF]Zelditch Recent developments in mathematical quantum chaos[END_REF]. Roughly speaking, they prove that, in their specific geometric situation (completely integrable flow), not all the mass of the eigenmodes can be localized on such shrinking tubes. In this article, we will consider similar questions for more general hyperbolic subsets and for stationary modes of the damped wave equation.

Finally, under a global assumption on the geodesic flow (namely it should be Anosov), Anantharaman proved that semiclassical measures associated to eigenmodes of ∆ cannot be completely carried by closed orbit of the geodesic flow (which are hyperbolic in this case) [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF]. In our main statement, we will not make any global assumption on the dynamical properties of the geodesic flow and it would be interesting to understand how Anantharaman's statement could be extended to the damped wave equation -see [START_REF] Rivière | Delocalization of slowly damped eigenmodes on Anosov manifolds[END_REF] for results in this sense.

1.3. Statement of the main result. We now turn back to eigenmodes of the damped wave equation. We underline that, to the knowledge of the author, even if there is an important literature concerning eigenfunctions of the Laplacian on M , much less seems to be known on the asymptotic description of eigenmodes for the damped wave equation. Our results concerning these questions will be here of two types:

• we extend the study of concentration in shrinking tubes of size ν to more general hyperbolic subsets satisfying a condition of negative topological pressure; • we consider the situation where a is a general smooth and real valued function on M (and not only the case a ≡ 0). As it will be involved in the statement of our main result, we recall now what is the topological pressure. Let Λ be a compact and hyperbolic subset of S * M invariant under the geodesic flow g t . For any ǫ > 0 and T > 0, we say that the subset F in Λ is (ǫ, T )-separated if, for any ρ and ρ

′ in F , ∀0 ≤ t ≤ T, d(g t ρ, g t ρ ′ ) ≤ ǫ =⇒ ρ = ρ ′ .
Then, we can define the topological pressure of the subset Λ with respect to 1 2 log J u where J u is the unstable Jacobian -see paragraph 2.1 below. It is defined as [START_REF] Pesin | Dimension Theory in Dynamical Systems: Contemporary Views and Applications[END_REF]

P top Λ, g t , 1 2 log J u := lim ǫ→0 lim sup T →+∞ 1 T log sup F    ρ∈F exp 1 2 T 0 log J u • g s (ρ)ds    ,
where the supremum is taken over all (ǫ, T )-separated subsets F . In this definition, we have two phenomena. On the one hand, the Birkhoff average of 1 2 log J u leads to exponentially small terms when T → ∞; on the other hand, depending on the complexity of the dynamics on Λ, the cardinal of F could grow exponentially when T → ∞. Thus, saying that the topological pressure is negative means that the contribution of the first quantity is more important. If Λ is a (or a collection of) closed hyperbolic geodesics, then P top Λ, g t , 1 2 log J u is negative.

We say that a function is (Λ, , ν)-localized if it is a smooth cutoff function in a ν -neighborhood of Λ -see § 3.1.1 for a precise definition. We can now state our main result.

Theorem 1.1. Suppose Λ is a compact, invariant, hyperbolic subset satisfying

P top Λ, g t , 1 2 log J u < 0,
and such that

(7) sup ρ∈Λ - T 0 a • g s (ρ)ds ≤ βT + O(1) when T → +∞ .
Fix 0 < ν < 1 2 and a (Λ, , ν)-localized function Θ Λ, ,ν . Then, there exists a constant c Λ,a,ν < 1 such that, for any sequence (ψ ) →0 + of eigenmodes satisfying (3) with

z( ) = 1 2 + O( ) and Im z( ) ≥ β + o | log | -1 , as → 0 + , one has lim sup →0 Op (Θ Λ, ,ν ) ψ , ψ ≤ c Λ,a,ν < 1.
We underline that we allow the imaginary parts of z( ) to go a little bit below the horizontal axis {Im z = β}. Precisely, we authorize an error of order o( | log | -1 ), that will be crucial for the results proven in the appendix. A more comfortable statement is given by the following corollary which can be deduced from Theorem 1.1:

Corollary 1.2. Suppose Λ is a compact, (g t ) t -invariant hyperbolic satisfying P top Λ, g t , 1 2 log J u < 0.
Suppose also that there exists a positive constant C such that

∀T > 0, ∀ρ ∈ Λ, -C + βT ≤ - T 0 a • g s (ρ)ds ≤ βT + C.
Fix 0 < ν < 1 2 and a (Λ, , ν)-localized function Θ Λ, ,ν . Then, there exists a constant c Λ,a,ν < 1 such that, for any sequence (ψ ) →0 + of eigenmodes satisfying (3) with z( ) = 1 2 + O( ) as → 0 + , one has lim sup

→0 Op (Θ Λ, ,ν ) ψ , ψ ≤ C Λ,a,ν < 1.
Proof. Let us briefly explain how Corollary 1.2 can be obtained from Theorem 1.1. One can proceed by contradiction and suppose that there exists a sequence ( l ց 0) l∈N and a sequence (ψ l ) l of normalized eigenmodes satisfying (3) with z( l ) = 1 2 + O( l ) and lim

l→+∞ Op l (Θ Λ, l ,ν ) ψ l , ψ l = 1.
This implies that, for any semiclassical measure µ associated to this sequence, one must have µ(Λ) = 1. In particular, thanks to relation [START_REF] Bouzouina | Robert Uniform semiclassical estimates for the propagation of quantum observables[END_REF], this implies that Im z( l ) l tends to β as l tends to infinity. Thanks to Theorem 1.1, one also obtains that Im z( l ′ ) l ′ ≤ β for an infinite subsequence of integers l ′ . On the other hand, our assumption also implies that [START_REF]Burq Mesures semi-classiques et mesures de défaut[END_REF] lim

l ′ →+∞ Op l ′ ΘΛ, l ′ ,ν ψ l ′ , ψ l ′ = 1,
where ΘΛ, l ′ ,ν (x, ξ) = Θ Λ, l ′ ,ν (x, -ξ). The function ΘΛ, l ′ ,ν satisfies the assumption of Theorem 1.1 with the set Λ replaced by Λ ′ := {(x, ξ) : (x, -ξ) ∈ Λ}. Moreover, the sequence (ψ l ′ ) l ′ solves (3) if we replace a by -a and z( l ′ ) by z( l ′ ). In particular, since Im z( l ′ ) l ′ ≥ -β and since Λ ′ satisfies the assumption of Theorem 1.1 w.r.t. the pair (-a, -β), we can apply the Theorem to this new sequence: the conclusion of the Theorem contradicts the limit [START_REF]Burq Mesures semi-classiques et mesures de défaut[END_REF].

In the selfadjoint case a ≡ 0, this corollary slightly improves Toth-Zelditch's result as we only impose the hyperbolic subsets to satisfy a condition of negative topological pressure. A default of our approach is yet that the upper bound c Λ,a,ν is not very explicit compared to the constant appearing in [START_REF]Zelditch Recent developments in mathematical quantum chaos[END_REF] -section 5. Our interest in proving this result was also to show that this property remains true in the nonselfadjoint case where a is non constant. As was already mentioned, nothing forbids a priori that eigenmodes with damping parameter β concentrate on a β-damped closed geodesic3 : corollary 1.2 prevents fast concentration on such orbits if they are hyperbolic.

If the geodesic flow is ergodic for the Liouville measure on S * M (manifolds of negative curvature are the main example), Sjöstrand showed that most of the imaginary parts converge to the spatial average of -a [START_REF] Sjöstrand | Asymptotic distributions of eigenfrequencies for damped wave equations[END_REF]. Thus, in this case, our result says that if there is a hyperbolic closed geodesic with such a Birkhoff average, then eigenmodes cannot concentrate on it too fast. As was already pointed out, it would be interesting to understand what can be said under the additional assumption that the geodesic flow is Anosov on S * M (e.g. if M is of negative curvature). For instance, can one prove in the Anosov case that semiclassical measures cannot be completely carried by a β-damped closed orbit?

Finally, we would like to say a few words about the proof. Our argument relies crucially on hyperbolic dispersive estimates as they were obtained by Anantharaman and Nonnenmacher in the Anosov case [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF][START_REF] Anantharaman | Half delocalization of eigenfunctions of the Laplacian on an Anosov manifold[END_REF] and by Nonnenmacher and Zworski in the context of chaotic scattering [START_REF] Nonnenmacher | Zworski Quantum decay rates in chaotic scattering[END_REF]. More precisely, we will use a generalization of these properties in a nonselfadjoint setting similar to the results obtained by Schenck in [START_REF] Schenck | Energy decay for the damped wave equation under a pressure condition[END_REF].

These hyperbolic estimates give an upper bound for the growth of "quantum cylinders" associated to ψ and localized near the hyperbolic set Λ. These cylinders are a kind of analogues in a quantum setting of the Bowen balls used in the theory of dynamical systems [START_REF] Katok | Hasselblatt Introduction to the Modern Theory of Dynamical Systems[END_REF][START_REF] Pesin | Dimension Theory in Dynamical Systems: Contemporary Views and Applications[END_REF]. Under our dynamical assumption on Λ, one can show that the mass of "quantum cylinders" near the set Λ is exponentially small for cylinders of length K| log | (with K > 0 very large but independent of ) -paragraph 3.2.2. Then, the main difficulty is that it is hard to connect these estimates for long cylinders to estimates which are valid for shorter cylinders to which we could apply the semiclassical approximation, e.g. of length less than the Ehrenfest time κ 0 | log | [START_REF] Bouzouina | Robert Uniform semiclassical estimates for the propagation of quantum observables[END_REF] (with κ 0 > 0 small independent of ). It turns out that if we restrict ourselves to cylinders that remain in a νneighborhood of Λ, the mass on the quantum cylinders (far from this neighborhood) is positive and it satisfies a "subadditive structure" -paragraph 3.2.3. A similar property was already observed and used by Anantharaman in a selfadjoint context [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF]. In our case, it implies that if the mass on the cylinders of length K| log | far from the ν -neighborhood is positive, then this property remains true for cylinders of shorter length κ 0 | log |. This observation is crucial in our proof and it allows to get the conclusion using standard semiclassical rules-paragraph 3.2.4

Organization of the article. In section 2, we introduce the dynamical setting of the article. We also build an open cover of S * M that will be used to define quantum cylinders in the subsequent section. Then, in section 3, we give the proof of Theorem 1.1 and postpone the proof of several semiclassical results to section 4. In section 5, we give a short toolbox on pseudodifferential calculus on a manifold.

Finally, in an appendix in collaboration with Stéphane Nonnenmacher, we explain how these methods can be used to derive inverse logarithmic spectral gaps for the damped wave equationsee [START_REF] Christianson | Semiclassical nonconcentration near hyperbolic orbits[END_REF][START_REF] Burq | Christianson Imperfect control for the damped wave equation[END_REF][START_REF] Christianson | From resolvent estimates to damped waves[END_REF] for related results.

Dynamical setting

The Hamiltonian function associated to the geodesic flow on S * M will be denoted p 0 (x, ξ) = ξ 2 x 2 in the following of this article. Under proper assumptions (see remark 2.3), we underline that our proof should also work for more general Hamiltonian flows as in [START_REF] Nonnenmacher | Zworski Quantum decay rates in chaotic scattering[END_REF][START_REF] Sjöstrand | Asymptotic distributions of eigenfrequencies for damped wave equations[END_REF]; yet, for simplicity of exposition, we restrict ourselves to the case of geodesic flows.

2.1. Hyperbolic sets. From this point, we make the assumption that the set Λ is a compact, invariant and hyperbolic subset of S * M . The hyperbolicity hypothesis means that one has the following decomposition [START_REF] Katok | Hasselblatt Introduction to the Modern Theory of Dynamical Systems[END_REF]

∀ρ ∈ Λ, T ρ S * M = RX p0 (ρ) ⊕ E u (ρ) ⊕ E s (ρ),
where RX p0 (ρ) is the direction of the Hamiltonian vector field, E u (ρ) is the unstable space and E s (ρ) is the stable space. In particular, there exist a constant C > 0 and 0 < λ < 1 such that for every t ≥ 0, one has

∀v u ∈ E u (ρ), d ρ g -t v u ≤ Cλ t v u and ∀v s ∈ E s (ρ), d ρ g t v s ≤ Cλ t v s .
Due to the specific structure of our Hamiltonian, the above properties remain true for any energy layer 4 associated to E > 0

E E := p -1 0 ({E}) = {(x, ξ) ∈ T * M : p 0 (x, ξ) = E} . Define now the unstable Jacobian at point ρ ∈ S * M and time t ≥ 0 J u t (ρ) := det d g t ρ g -t |E u (g t ρ)
,

where the unstable spaces at ρ and g t ρ are equipped with the induced Riemannian metric. It defines a Hölder continuous function on S * M [START_REF] Katok | Hasselblatt Introduction to the Modern Theory of Dynamical Systems[END_REF] (that can be extended to any energy layer E E ). We underline that this quantity tends to 0 with an exponential rate as t tends to infinity. Moreover, it satisfies the following multiplicative property

J u t+t ′ (ρ) = J u t (g t ′ ρ)J u t ′ (ρ).
In the following, we will use the notation J u (ρ) = J u 1 (ρ) on S * M . 2.2. Topological pressure. In the statement of Theorem 1.1, we made an assumption on the topological pressure of the subset Λ. Let us explain what informations are provided by this hypothesis following the observations of paragraph 5.2 in [START_REF] Nonnenmacher | Zworski Quantum decay rates in chaotic scattering[END_REF] -see also [START_REF] Pesin | Dimension Theory in Dynamical Systems: Contemporary Views and Applications[END_REF], chapter 4 for general definitions of topological pressure.

Fix a small δ > 0. Then, for every E ∈ [ 1-δ 2 , 1+δ 2 ], the set

Λ E = (x, ξ) ∈ E E : x, ξ √ 2E ∈ Λ is hyperbolic. We fix a finite open cover V = (V a ) a∈A of (9) Λ δ := 1-δ 2 ≤E≤ 1+δ 2 Λ E
of diameter less than some small ǫ > 0 and such that, for every a in A, one has

V a ⊂ E δ := p -1 0 (1/2 -δ, 1/2 + δ) .
For every integer n 0 , the refined cover V (n0) is the collection of the open sets

V α = n0-1 j=0 g -j V αj , where α = (α 0 , α 1 , . . . , α n0-1 ) ∈ A n0 .
Equivalently, V α contains the points ρ, the trajectory of which sits in V α0 at time 0, in V α1 at time 1, etc, and in V αn-1 at time n -1.

The fact that P top (Λ, g t , 1 2 log J u ) < 0 implies the existence of a positive constant P 0 such that for δ small enough, for any cover of small enough diameter (say ǫ ≤ ǫ 0 ) and for any n 0 ∈ N large enough (depending on ǫ), one can extract a subcover W (n0) ⊂ V (n0) of Λ δ such that (10)

Vα∈W (n 0 ) sup ρ∈Vα∩Λ δ exp 1 2 n0 0 log J u • g t (ρ)dt ≤ e -2n0P0
4 For more general Hamiltonian, it would remain true in a small vicinity of the energy layer due to the stability of the hyperbolic structure [START_REF] Katok | Hasselblatt Introduction to the Modern Theory of Dynamical Systems[END_REF].

(we may assume that any V α ∈ W (n0) intersects Λ δ ). Thanks to assumption [START_REF] Bowen | Ruelle The ergodic theory of Axiom A flows[END_REF] on Λ, we can also verify that for n 0 large enough, one also has ( 11)

Vα∈W (n 0 ) sup ρ∈Vα∩Λ δ exp n0 0 1 2 log J u -a • g t (ρ)dt ≤ e n0(β-P0) .
Remark 2.1. In our proof, we will fix an open cover of small diameter ǫ ≤ ǫ 0 in order to get a subcover W (n0) satisfying [START_REF] Christianson | Semiclassical nonconcentration near hyperbolic orbits[END_REF]. Such a choice can be made for every ǫ ≤ ǫ 0 . Moreover, we choose such an epsilon in order to have ǫ ≤ ǫ0 /2, where ǫ0 is the constant appearing in lemma 2.2. We also take ǫ small enough to have the factor 1 + O(ǫ) in estimate [START_REF] Lebeau | Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli[END_REF] smaller than e

P 0 2 .
Once V is chosen with the above requirements, we also select n 0 and W (n0) such that (11) holds. All these parameters will remain fixed for the rest of the proof.

We will call W the family of words α = (α 0 , α 1 , . . . , α n0-1 ) corresponding to the elements V α ∈ W (n0) . We also complete the cover, by selecting an open set V ∞ such that V ∞ ∩ Λ δ = ∅, and such that

V ∞ ∪ α∈W V α = E δ .
Finally, we denote W = W ∪ {∞}.

2.3.

A lemma from dynamical systems. Before entering the details of our proof, we mention the following lemma which is taken from the appendix of [START_REF] Bowen | Ruelle The ergodic theory of Axiom A flows[END_REF] (lemma A.2): Lemma 2.2. Let Λ be a hyperbolic set in S * M satisfying assumption [START_REF] Bowen | Ruelle The ergodic theory of Axiom A flows[END_REF]. There exists ǫ0 > 0 (depending on M , δ and a(x)) such that, for any

E ∈ [ 1 2 -δ, 1 2 + δ],
for any p > 0 and any

ρ 2 ∈ E δ satisfying ∃ρ 1 ∈ Λ δ such that ∀0 ≤ k ≤ p -1, d T * M (g k ρ 1 , g k ρ 2 ) ≤ ǫ0 , one has - p 0 a • g s (ρ 2 )ds ≤ βp + O(1),
where the constant involved in O(1) is independent of ρ 2 and p.

In particular, this lemma will allow us to extend the inequality [START_REF] Bowen | Ruelle The ergodic theory of Axiom A flows[END_REF] to a small (dynamical) neighborhood of Λ δ . The proof of this lemma was given in [START_REF] Bowen | Ruelle The ergodic theory of Axiom A flows[END_REF] where the authors treated the case of a single energy layer (δ = 0). Yet, their proof can be adapted to get a uniform ǫ0 on the energy interval E δ . We verify below that their argument can be extended to a small neighborhood of S * M .

Proof. The proof of this lemma relies on two observations:

• if the trajectory of ρ 2 remains close to the one of ρ 1 ∈ Λ δ in the future, then ρ 2 must have an "exponentially small unstable component"; • the Birkhoff averages -p 0 a • g s ds on Λ δ are uniformly bounded by βp + O(1). We closely follow the presentation of [START_REF] Bowen | Ruelle The ergodic theory of Axiom A flows[END_REF] and refer the reader to it for more details. We start by giving a precise meaning to the first observation. For that purpose, we write the following decomposition of the tangent space, for any ρ = (x, ξ) ∈ Λ δ ,

T ρ E δ = E 0 (ρ) ⊕ E s (ρ) ⊕ E u (ρ),
where E 0 (ρ) is the vector space generated by X p0 (ρ) and the energy direction ρ(t) = (x, tξ) and E u/s are still the unstable/stable directions. For v in T ρ E δ , we denote v = v 0 + v s + v u the decomposition adapted to these subspaces. For ǫ ′ > 0 small enough and any ρ ∈ Λ δ , one can construct a smooth chart 5 φ ρ : 5 Here, H(ǫ ′ ) means that we consider a ball of radius ǫ ′ around 0 in the subspace H. with

T ρ E δ (ǫ ′ ) → E δ satisfying φ ρ (E 0 (ρ) + E s (ρ))(ǫ ′ ) ⊂ W cs (ρ), and φ ρ (E 0 (ρ) + E u (ρ))(ǫ ′ ) ⊂ W cu (ρ),
W cs/cu (x, ξ) = t∈R 1-δ 2 ≤E≤ 1+δ 2 W s/u g t x, √ 2E ξ ξ ,
where W s/u (ρ ′ ) denote the stable/unstable manifold at point ρ ′ . Moroeover, one can choose φ ρ such that d 0 φ ρ is given by the identity. The construction is a straightforward adaptation of property A.1 in [START_REF] Bowen | Ruelle The ergodic theory of Axiom A flows[END_REF] to a small neighborhood of S * M . For ǫ ′ > 0 small enough, introduce now

F ρ = φ -1 g 1 ρ • g 1 • φ ρ : T ρ E δ (ǫ ′ ) → T g 1 ρ E δ , which is tangent to d ρ g 1 at the origin. Define also D(ǫ ′ , p) := {v ∈ T ρ E δ : ∀0 ≤ k ≤ p -1, F g k ρ • . . . F ρ v g k ρ ≤ ǫ ′ }. Let v = v 0 + v s + v u be an element in D(ǫ ′ , p).
One can mimick again the proof of [START_REF] Bowen | Ruelle The ergodic theory of Axiom A flows[END_REF] (precisely the proof of inequality A.5 in this reference) and verify that there exist uniform constants C > 0 and 0 < λ < 1 such that

(12) ∀0 ≤ k ≤ p -1, F g k ρ • . . . F ρ (v 0 + v s + v u ) -F g k ρ • . . . F ρ (v 0 + v s ) g k ρ ≤ Cǫ ′ λ p-1-k .
This upper bound is obtained thanks to the hyperbolicity assumption (combined to a Taylor formula near the origin). This result expresses the first property mentionned at the beginning of our proof. Precisely, it shows that a point which remains close to ρ 1 ∈ Λ δ during a time p has an exponentially small unstable component (in our system of charts).

We will now use this family of charts to prove lemma 2.2. First, we observe that there exists a constant C > 0 such that

∀ρ 1 , ρ 2 ∈ E δ , 1 0 a • g s (ρ 1 )ds - 1 0 a • g s (ρ 2 )ds ≤ Cd(ρ 1 , ρ 2 ). Fix now ρ 1 = (x 1 , ξ 1 ) in Λ δ and ρ 2 in E δ satisfying ∀0 ≤ k ≤ p -1, d T * M (g k ρ 1 , g k ρ 2 ) ≤ ǫ0 ,
where ǫ0 is some small positive parameter. In particular, we choose it small enough to have v = φ -1 ρ1 (ρ 2 ) belongs to T ρ E δ (ǫ ′ /2) for every ρ 1 ∈ Λ δ and any ρ 2 ∈ E δ satisfying d(ρ 1 , ρ 2 ) ≤ ǫ0 . Define then w = v 0 + v s and introduce ρ 3 = φ ρ1 (w). Thanks to our construction, one has

ρ 3 ∈ W s (g τ ρ1 ) for some |τ | ≤ C 0 ǫ ′ and some ρ1 = (x 1 , E ′ ξ 1 ) ∈ Λ E ′ with |E ′ | ≤ C 0 ǫ ′ .
Thanks to the fact that a does not depend on ξ and that ρ1 belongs to Λ E ′ , the assumption (7) directly implies that

- p 0 a • g s (ρ 1 )ds ≤ βp + O(1),
where the constant involved in the remainder is uniform for |E ′ | ≤ C 0 ǫ ′ . To extend the assumption [START_REF] Bowen | Ruelle The ergodic theory of Axiom A flows[END_REF] to every energy layer Λ E ′ , we have crucially used the fact that a is independent of ξ, and the homogeneity of the geodesic flow -see remark 2.3 below for generalizations of this fact.

We will now compare the average along the trajectory of ρ 2 with the average along the trajectory of ρ1 . Thanks to the upper bound [START_REF] Christianson | Applications of Cutoff Resolvent Estimates to the Wave Equation[END_REF] and to the construction of ρ 3 , one has that, for any 0

≤ k ≤ p -1, d(g k ρ 3 , g k ρ 2 ) ≤ C 1 ǫ ′ λ p-1-k and d(g k+τ ρ1 , g k ρ 3 ) ≤ C ′ 1 ǫ ′ (λ ′ ) k , for some uniform C ′ 1 > 0 and 0 < λ ′ < 1.
We now use these properties to bound -p 0 a • g s (ρ 2 )ds. We write

- p 0 a • g s (ρ 2 )ds ≤ - p 0 a • g s (ρ 1 )ds + p 0 a • g s (ρ 1 )ds - p 0 a • g s+τ (ρ 1 )ds + p-1 k=0 1 0 a • g s+k+τ (ρ 1 )ds - 1 0 a • g s+k (ρ 3 )ds + 1 0 a • g s+k (ρ 3 )ds - 1 0 a • g s+k (ρ 2 )ds .
Using the different properties mentioned above, one gets

- p 0 a • g s (ρ 2 )ds ≤ βp + O(1) + 2C 0 ǫ ′ a ∞ + CC ′ 1 ǫ ′ 1 -λ ′ + CC 1 ǫ ′ 1 -λ ,
which is the expected conclusion.

Remark 2.3. At this point, we would like to mention something on the generalization of Theorem 1.1 to more general nonselfadjoint operators as in [START_REF] Sjöstrand | Asymptotic distributions of eigenfrequencies for damped wave equations[END_REF]. In order to adapt the previous lemma (which will be crucial in our proof) for more general Hamiltonian flows, one has to make the assumption that the Birkhoff averages of the corresponding damping function are bounded by βp + O(1) for every trajectory in a small neighborhood Λ δ of the hyperbolic subset Λ. Here this property was satisfied due to the specific structure of the "damping function" a and of the geodesic flow.

Proof of the main Theorem

We fix β a spectral parameter. Let (ψ ) 0< ≤ 0 be a sequence of normalized vector in L 2 (M ) such that

P( , z)ψ = z( )ψ , where z( ) satisfies (13) ∀0 < ≤ 0 , z( ) = 1 2 + O( ) and Im z( ) ≥ β + o | log | -1 .
Remark 3.1. Such a family may be defined by a discrete sequence n → 0 as n tends to infinity. Yet, in order to avoid heavy notations and to fit semiclassical notations [START_REF] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF], we will use the standard convention → 0 to denote the limit.

3.1. Concentration properties and discretization of the energy layer. In this paragraph, we describe the setting we will use to prove Theorem 1.1. We introduce Λ a compact, hyperbolic and invariant subset of S * M satisfying the assumption [START_REF] Bowen | Ruelle The ergodic theory of Axiom A flows[END_REF]. As in paragraph 2.2, we fix a small neighborhood of size δ > 0 around S * M (thanks to our assumption on Re z( ), the eigenmodes are microlocalized on S * M when tends to 0). We make the assumption that P top Λ, g t , 1 2 log J u < 0 and we will use the open covers introduced in §2.2.

3.1.1.

Cutoff functions near Λ. We fix 0 < ν < 1/2 a positive parameter and we introduce a cutoff function 0 ≤ Θ Λ, ,ν ≤ 1 around the set Λ. This function belongs to C ∞ c (T * M ) and satifies the following assumptions:

• Θ Λ, ,ν (x, ξ) = 0 for ξ 2 / ∈ [1/4, 2]; • Θ Λ, ,ν (x, ξ) = Θ Λ, ,ν (x, ξ/ ξ ) for ξ 2 ∈ [1/2, 3/2]; • for every ρ in S * M satisfying d(ρ, Λ) ≤ ν /2, Θ Λ, ,ν (ρ) = 1; • for every ρ in S * M satisfying d(ρ, Λ) ≥ 2 ν , Θ Λ, ,ν (ρ) = 0;
• the growth of the derivatives of Θ Λ, ,ν is controlled by powers of -ν and so the functions are amenable to -pseudodifferential calculus [START_REF] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF] (see also appendix 5 for a brief reminder);

We say that such a function is (Λ, , ν)-localized. Our goal is to prove that

(14) lim inf →0 Op (1 -Θ Λ, ,ν ) ψ , ψ ≥ c Λ,a,ν > 0,
for some positive constant c Λ,a,ν that depends only on Λ, a and ν (and, in particular, not on the sequence (ψ ) →0 ).

3.1.2. Smooth discretization of the energy layer. We now introduce a smooth partition of unity associated to our open cover (V α ) α∈W , namely a family of smooth functions

P α ∈ C ∞ c (V α , [0, 1]) which satisfy α∈W P α (ρ) = 1 near E δ/2 .
This smooth partition can be quantized into a family of pseudodifferential operators (π α ∈ Ψ -∞,0 (M )) α∈W such that for each α ∈ W , P α is the principal symbol of π α , and

W F (π α ) ⊂ V α , π * α = π α and α π α = Id microlocally near E δ/2 ,
We also introduce the following "refined" operators:

∀γ = (γ 0 , γ 1 , . . . , γ n-1 ) ∈ W n , Π γ := U n0 π γ n-1 U n0 . . . π γ 1 U n0 π γ 0 , Π γ := Π γ U -nn0 .
This new family of operators satisfies [START_REF] De Verdière | Parisse Équilibre instable en régime semi-classique. I. Concentration microlocale[END_REF] |γ|=n

Π γ = U nn0 microlocally near E δ/2 , equivalently |γ|=n Π γ = Id microlocally near E δ/2 ,
uniformly for times 0 ≤ n ≤ C| log |, for any fixed C > 0.

We notice that for n = |γ| finite, each operator Π γ admits for principal symbol ( 16)

P γ := P γ n-1 • g -n0 . . . P γ 1 • g (1-n)n0 P γ 0 • g -nn0 ,
which is supported in the "backward refined set" 6

V γ := g n0 V γ n-1 ∩ g 2n0 V γ n-2 ∩ • • • ∩ g nn0 V γ 0 .
In subsection 4.2 we will see that this connection between Π γ and P γ extends to times n ≤ κ 0 | log |, for κ 0 > 0 small enough. We already have two families of n-cylinders: the full set of n-cylinders

W n = (γ 0 , γ 1 , . . . , γ n-1 ) : ∀0 ≤ j ≤ n -1, γ j ∈ W ,
covering the whole energy slab E δ , and the set of n-cylinders

W n = (γ 0 , γ 1 , . . . , γ n-1 ) : ∀0 ≤ j ≤ n -1, γ j ∈ W ,
corresponding to trajectories remaining ǫ-close to Λ δ during a time nn 0 . We will distinguish a subfamily of n-cylinders, corresponding to points very close to Λ. Namely, we define Λ n ⊂ W n to be the set of n-cylinders satisfying supp Θ Λ, ,ν × P γ = ∅.

3.1.3. Preliminary lemmas. We will now make two simple (but crucial) observations that will be at the heart of our proof.

Lemma 3.2. There exists κ 0 > 0 small enough (depending on ν, δ, Λ and V ∞ ) such that, for small enough, for any point ρ ∈ supp Θ Λ, ,ν × P γ and any |t| ≤ κ 0 | log |, one has

d(g t (ρ), Λ δ ) ≤ ν/2 .
In particular, Λ n ⊂ W n .

6 Vγ contains the points ρ which were sitting in V γ n-1 at time -n 0 , in V γ n-2 at time -2n 0 ,..., in V γ 0 at time -nn 0 . The word γ thus describes the backward trajectory of ρ.

The proof of this lemma derives from the following observation. Any point in ρ ∈ supp Θ Λ, ,ν × P γ is at distance ≤ 2 ν from Λ δ . Due to the hyperbolicity assumption, the distance from Λ δ can grow at most exponentially with time: there is a uniform 0 < λ < 1 such that ( 17)

d(g t (ρ), Λ δ ) ≤ C ν λ |t| , ∀t ∈ R .
This is an important property as it will allow us to apply hyperbolic dispersive estimates to cylinders in Λ n -see paragraph 3.2.2. If we had chosen a larger "tube" around Λ, our argument would a priori not work as we will need to work with logarithmic times in -see paragraph 3.2.3. We will also need the following feature of cylinders in Λ n . Lemma 3.3. There exists κ 0 > 0 small enough (depending on ν, δ, Λ and V ∞ ) such that, for small enough, any n ≤ [κ 0 | log |], any γ ∈ Λ n and any ρ ∈ supp( P γ ), one has

- nn0-1 0 a • g s-nn0 (ρ)ds ≤ (nn 0 -1)β + O(1).
Proof. The proof relies on lemma 2.2. Choose ρ ∈ supp( P γ ). By definition of Λ n , there exists ρ γ ∈ supp(Θ Λ, ,ν × P γ ). The diameter of the open cover has been selected to be smaller than ǫ0 /2, where ǫ0 is the parameter of lemma 2.2. Hence, since g -k (ρ) and g -k (ρ γ ) belong to the same open sets V a k for all times k = 1, . . . , nn 0 , we have

∀1 ≤ k ≤ n 0 n, d g -k (ρ), g -k (ρ γ ) ≤ ǫ0 2 .
Since ρ γ is at distance ≤ 2 ν from Λ δ , we can choose a point ργ ∈ Λ δ such that d(ρ γ , ργ ) ≤ 2 ν . For κ 0 small enough, one gets d(g -t (ρ γ ), g -t (ρ γ )) ≤ C ν/2 for all 0 ≤ t ≤ κ 0 | log | -see property [START_REF] Helffer | Robert Ergodicité et limite semi-classique[END_REF]. As a consequence, for small enough,

∀1 ≤ k ≤ nn 0 , d g -k (ρ), g -k (ρ γ ) ≤ ǫ0 .
Using lemma 2.2, we deduce that

(18) - nn0-1 0 a • g s-nn0 (ρ)ds ≤ β(nn 0 -1) + O(1).
As in the previous lemma, if we want to work with logarithmic times in , we need to have a tube of size ν around Λ in order to obtain a remainder uniform w.r.t. . We underline that, in both lemmas, our choice of κ 0 > 0 depends on M , on Λ and on our choice of open cover, of n 0 and of ν.

3.2.

Proof of Theorem 1.1. We are now in the position to give the proof of our main result. Our strategy is to prove a positive lower bound for the norm

γ∈Λ c n Π γ ψ ,
where Λ c n is the complementary of Λ n in W n and n is a "short logarithmic time". It will roughly say that a positive part of the mass of ψ is far from Λ. We will first use a hyperbolic dispersive estimate [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF][START_REF] Nonnenmacher | Zworski Quantum decay rates in chaotic scattering[END_REF][START_REF] Schenck | Energy decay for the damped wave equation under a pressure condition[END_REF] in order to obtain a lower bound for a similar quantity corresponding to cylinders of length kn -see paragraph 3.2.2, with k ≫ 1 fixed (kn is a "large logarithmic time"). Then, by a subadditive argument (paragraph 3.2.3), we will derive the desired lower bound for cylinders of length n. Finally, we show in paragraph 3.2.4 how to derive Theorem 1.1 from this lower bound.

3.2.1.

Different scales of times. First, we select open covers V and W (n0) as in paragraph 2.2, in particular the diameter of V is small enough to get the requirements of remark 2.1.

We will then fix some κ 0 > 0 small enough, so that the bound of lemma 3.2 applies, and also such that the quantum evolution of observables supported in the energy slab E δ is under control for times |t| ≤ κ 0 n 0 | log | (see subsection 4.1 on this matter). We then introduce a "short" logarithmic time [START_REF]Hitrik Eigenfrequencies and Expansions for Damped Wave Equations[END_REF] n(

) := [κ 0 | log |] .
In particular, the arguments of lemma 3.3 and of paragraphs 3.2.3 and 3.2.4 will be valid for 0 ≤ n ≤ n( ). The choice of κ 0 depends on the open cover V, on the damping function a, on n 0 , on δ (the size of the energy slab we work on) and on the exponent ν used to define Θ ,Λ,ν . We fix k ≥ 2 a large positive integer, satisfying kκ 0 > d n0P0 -see paragraph 3.2.2. We will then define a second ("large") logarithmic time kn( ).

We will omit the dependence n( ) = n in to avoid heavy notations.

Remark 3.4. We underline that the different parameters we have introduced so far (namely n 0 , δ, κ 0 , k, P 0 and the open cover) are chosen in a way that depends only on Λ, a and ν. They will not depend on our choice of sequence ψ .

3.2.2.

Using hyperbolic dispersive estimates. The first step of our proof is to use the property (15) (still valid for "large" logarithmic times) and the fact that ψ is an eigenmode of U , in order to write (20)

Γ∈W kn Π Γ ψ , ψ = exp - ıknn 0 z( ) + O( ∞ ).
Here we have implicitly used the fact that the eigenstate ψ is microlocalized on the energy layer E 1/2 = S * M . Then, we split the above sum using the decomposition of W kn as

W kn = Λ k n ⊔ Λ k n c , where Λ k n = {Γ 0 Γ 1 . . . Γ k-1 : ∀0 ≤ j ≤ k -1, Γ j ∈ Λ n } and Λ k n c
is the complementary of Λ k n in W kn . We find then ( 21)

Γ∈Λ k n Π Γ ψ , ψ + Γ∈(Λ k n ) c Π Γ ψ , ψ = exp - ıknn 0 z( ) + O( ∞ ).
We will now use a hyperbolic dispersion estimate to bound the sum over Λ k n which is a subset of W nk -see lemma 3.2. We are almost in the situation of [24, §7.2], except that our generator P( , z) is nonselfadjoint. Still, like in [START_REF] Schenck | Energy decay for the damped wave equation under a pressure condition[END_REF], we can use the strategy of [START_REF] Nonnenmacher | Zworski Quantum decay rates in chaotic scattering[END_REF]Sec.4] by taking into account the nonselfadjoint contribution in the WKB Ansatz. The output is that, for every k ≥ 2, there exist constants C k > 0 and k > 0 (depending on k, on a, on the choice of the partition and on Λ) such that, for any ≤ k and any cylinder Γ = α 0 • • • α nk-1 ∈ W nk , the following hyperbolic dispersive estimate holds:

(22) Π α0•••α nk-1 L 2 →L 2 ≤ C k -d 2 (1 + O(ǫ)) knn0 nk-1 j=0 sup ρ∈Vα j ∩Λ δ exp n0 0 1/2 log J u -a) • g t (ρ) dt ,
where the constant involved in O(ǫ) depends only on the manifold and on a. Recall ǫ is an upper bound on the diameter of the partition V. Summing over all cylinders Γ ∈ W nk and using the assumption [START_REF] Christianson | Semiclassical nonconcentration near hyperbolic orbits[END_REF], we obtain, for small enough, ( 23)

Γ∈W nk Π Γ ψ ≤ C k (1 + O(ǫ)) knn0 e knn0(β-P0) -d 2 + O( ∞ ) ,
which is the adaptation of the last upper upper bound in [START_REF] Nonnenmacher | Zworski Quantum decay rates in chaotic scattering[END_REF]Sec.7] to our nonselfadjoint setting. Lemma 3.2 shows that Λ n ⊂ W n , so the above sum can be restricted to Λ k n :

(24)

Γ∈Λ k n Π Γ ψ ≤ C k (1 + O(ǫ)) knn0 e knn0(β-P0) -d 2 + O( ∞ ) , Remark 3.5.
Let us say a few words on the proof of the crucial hyperbolic dispersive estimate [START_REF] Lebeau | Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli[END_REF]. First, we observe that any normalized state v microlocalized near the energy layer can be locally decomposed into Lagrangian states. Precisely, in a local chart f l :

V l ⊂ M → B(0, ǫ) ⊂ R d , one can represent (modulo O L 2 ( ∞ )) v as an integral of the form (2π ) -d 2 B(0,2) ṽ ,l (η)e ı y,η dη,
where, for each "momentum" η ∈ B(0, 2), the function ṽ ,l (η) is smooth and compactly supported in the variable y ∈ B(0, ǫ) -see for instance [24, §7]. Translating back to the manifold, it gives us a representation of v as a superposition of Lagrangian states. The prefactor -d 2 in this decomposition is responsible for the appearance of -d 2 in the upper bound [START_REF] Lebeau | Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli[END_REF]. Thus, in order to prove our estimate, it "remains" to find uniform upper bounds for the norms of

Π α0•••α nk-1 (a e ıS ),
where a e ıS →0 is a sequence of Lagrangian states microlocalized near S * M (given by the Fourier decomposition described above). This uniform upper bounds can be obtained thanks to a careful WKB procedure. The difficulty comes from the fact that we have to deal with quantum evolution up to order knn 0 ≍ K| log | with K > 0 arbitrarly large. In particular, it could be delicate to represent the evolved state in a simple formula, because the involved Lagrangian leaves will spread over the manifold under the evolution. Here, the operator Π α0•••α nk-1 does not only evolve the state up to large logarithmic times but it also cuts the phase space into small pieces, thanks to the cutoff operators π αj that we have inserted every time n 0 of the evolution. Due to this localization, it turns out that one can obtain a "simpler" description (through the WKB procedure) of the Lagrangian state evolved by Π α0•••α nk-1 . This can be done up to large logarithmic times provided we choose a good family of Lagrangian states. This property was first observed in [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF] and then used in several other situations [START_REF] Anantharaman | Half delocalization of eigenfunctions of the Laplacian on an Anosov manifold[END_REF][START_REF] Nonnenmacher | Zworski Quantum decay rates in chaotic scattering[END_REF][START_REF] Schenck | Energy decay for the damped wave equation under a pressure condition[END_REF].

There is a natural choice of Lagrangian states which is associated to the vertical bundle of the energy layer. These particular states were used by Anantharaman and Nonnenmacher in a selfadjoint setting [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF][START_REF] Anantharaman | Half delocalization of eigenfunctions of the Laplacian on an Anosov manifold[END_REF] and also by Schenck in [START_REF] Schenck | Energy decay for the damped wave equation under a pressure condition[END_REF] in the context of the damped wave equation. In these references, these Lagrangian states remain under control up to large logarithmic times, due to the global structure of the geodesic flow (it was supposed to be Anosov). Indeed, the Anosov hypothesis implied that the associated Lagrangian submanifolds become uniformly close to the unstable foliation and that they do not develop caustics under the evolution (thanks to the absence of conjugate points) -see [28, §4] for details.

Even if we consider the same equation, our situation differs from the one considered by Schenck in [START_REF] Schenck | Energy decay for the damped wave equation under a pressure condition[END_REF], because we do not make any global assumption on the geodesic flow: we only assume it to be hyperbolic on Λ. Hence, we cannot a priori use the same decomposition, because our dynamical assumptions do not forbid the existence of conjugate points or caustics. Instead, we may consider the more flexible Fourier decomposition introduced by Nonnenmacher and Zworski in [START_REF] Nonnenmacher | Zworski Quantum decay rates in chaotic scattering[END_REF]. The Lagrangian leaves involved in this decomposition are transversal to the stable manifolds, and therefore remain under control up to large logarithmic times -see [24, §5.1 and 7.1] for details.

Thus, we use the Fourier decomposition of [START_REF] Nonnenmacher | Zworski Quantum decay rates in chaotic scattering[END_REF] and we follow carefully their proof in order to prove the hyperbolic estimate [START_REF] Lebeau | Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli[END_REF]. The main difference with this reference is that we have to take into account the damping function in the WKB procedure (like in [START_REF] Schenck | Energy decay for the damped wave equation under a pressure condition[END_REF]). This implies that the term in the upper bound is in our setting of the form [24, §7]. Following this strategy, we obtain hyperbolic dispersion estimates for cylinders that always remain in a small vicinity of the invariant hyperbolic set 7 Λ, meaning the cylinders in W kn .

nk-1 j=0 sup ρ∈Vα j ∩Λ δ exp n0 0 1/2 log J u -a) • g t (ρ) dt and not nk-1 j=0 sup ρ∈Vα j ∩Λ δ exp n0 0 1/2 log J u ) • g t (ρ) dt as in
Remark 3.6. The constant C k and k involved in the hyperbolic dispersive estimate above can be chosen independently of the sequence ψ .

As was mentionned in remark 2.1, the diameter ǫ of our initial cover was chosen small enough to have the factor (1 + O(ǫ)) ≤ e P 0 2 . As mentioned in §3.2.1, we choose kκ 0 > d n0P0 , so that the factor -d 2 e -knn0 P 0 2 = o(1). Using the assumption (13) on z( ) and the fact that the time knn 0 = O(| log |), we derive

Γ∈Λ k n Π Γ ψ ≤ Γ∈Λ k n Π Γ ψ = o e knn0 Im z( ) when → 0 .
Comparing this with the estimate ( 21), we get the following lower bound when → 0: ( 25)

Γ∈(Λ k n ) c Π Γ ψ ≥ e knn0 Im z( ) 1 + o(1) .
This lower bound concerns the large logarithmic time knn 0 , for which the operators Π Γ or Π Γ cannot be analyzed in terms of pseudodifferential calculus.

Subadditivity property.

We will now show that the left hand side of ( 25) satisfies a kind of "subadditive" property 8 for logarithmic times -see Eq. ( 28). For that purpose, we decompose

(Λ k n ) c into (Λ k n ) c = k-1 j=0 Γ = Γ 0 . . . Γ j . . . Γ k-1 : ∀i < j, Γ i ∈ W n ; Γ j ∈ Λ c n ; ∀i > j, Γ i ∈ Λ n ,
and accordingly ( 26)

Γ∈(Λ k n ) c Π Γ = k-1 j=0 Γ j+1 ,...,Γ k-1 ∈Λn Π Γ k-1 • • • • • • Π Γ j+1 Γ j ∈Λ c n Π Γ j Γ 0 ,...,Γ j-1 ∈W n Π Γ j-1 • • • • • • Π Γ 0 .
Using this equality and property [START_REF] De Verdière | Parisse Équilibre instable en régime semi-classique. I. Concentration microlocale[END_REF], we are lead to ( 27)

Γ∈(Λ k n ) c Π Γ ψ ≤ k-1 j=0 γ∈Λn Π γ k-j-1 γ∈Λ c n Π γ ψ e jnn0 Im(z( )) + O( ∞ ) .
We will show in section 4 (more precisely in Eq. ( 34)) that there exists a constant c > 0, such that for small enough one has

γ∈Λn Π γ ≤ c e nn0β .
This bound uses the fact that we uniformly control the averaged damping on cylinders of Λ n , see lemma 3.3; in particular it uses the assumption [START_REF] Bowen | Ruelle The ergodic theory of Axiom A flows[END_REF].

Remark 3.7. In our argument below, we will crucially use the fact that the previous bound is ce nn0β and not ce n(n0β+ǫ) (even an arbitrary small ǫ > 0 is not a priori be sufficient for our proof).

For that purpose, it was important to restrict ourselves to cylinders of trajectories that remain very close to the set Λ. If we have used all cylinders in W n (instead of Λ n ), we would have get a bound of order ce n(n0β+ǫ) which would have not been sufficient for the end of our proof.

7 In [START_REF] Nonnenmacher | Zworski Quantum decay rates in chaotic scattering[END_REF], the hyperbolic estimates were valid for cylinders in a small vicinity of the trapped set -see section 7 of this reference. 8 A similar property already appeared in the selfadjoint case treated in [1, §2.2].

4.1.

Egorov property for long times. In this paragraph, we recall an Egorov property for times of order κ 0 | log |, where κ 0 is a small enough constant that we will not try to optimize. Consider q 1 and q 2 two symbols belonging to S 0,0 (T * M ) (for the sake of simplicity, we also assume that these symbols depend smoothly on ∈ (0, 1]). In this article, we will use the symbols q i equal to 2z( )a, -2z( )a or 0 -see paragraph 4.2 below.

4.1.1. The case of fixed times. We consider a smooth function b on T * M which is compactly supported in a neighborhood of S * M , say supp(b) ⊂ {(x, ξ) : ξ 2 ∈ [1/2, 3/2]} and which belongs to S -∞,0 (T * M ). The following operator is a pseudodifferential operator, for every t ∈ R,

B(t, b) = e -ıt - 2 ∆ 2 -ı Op (q1) * Op (b)e -ıt - 2 ∆ 2 -ı Op (q2) .
We briefly recall how such a fact can be proved by a direct adaptation of the arguments used in the selfadjoint case [START_REF] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF][START_REF] Bouzouina | Robert Uniform semiclassical estimates for the propagation of quantum observables[END_REF][START_REF] Royer | Analyse haute fréquence de l'équation de Helmholtz disipative[END_REF]. Take q = q 1 + q 2 , and introduce, for t, s ∈ R, the symbol

B t (s) := b • g t-s exp - t-s 0 q • g τ dτ .
To alleviate our notations, we call

U s (q i ) := e -ıs - 2 ∆ 2 -ı Op (qi) , i = 1, 2 ,
so that the operator B(t, b) = (U t (q 1 )) * Op (b) U t (q 2 ). Fixing t, we then introduce the auxiliary operators R( , s) = (U s (q 1 )) * Op (B t (s)) U s (q 2 ). Like in the classical proof of the Egorov Theorem (i.e. in the selfadjoint case), one can compute the derivative of R( , s):

d ds (R( , s)) = (U s (q 1 )) * ı - 2 ∆ 2 , Op (B t (s)) -Op (q 1 ) * Op (B t (s)) -Op (B t (s)) Op (q 2 ) U s (q 2 ) -(U s (q 1 )) * Op ({p 0 , B t (s)}) -Op (B t (s)(q 1 + q 2 )) U s (q 2 ).
We integrate this equality between 0 and t [6]:

U t (q 1 ) * Op (b) U t (q 2 ) = Op b • g t e -t 0 q•g τ dτ + t 0 (U s (q 1 )) * R( , s)U s (q 2 )ds,
where R( , s) is a pseudodifferential operator in Ψ -∞,-1 (M ) thanks to pseudodifferential rules. Proceeding by induction and using pseudodifferential calculus perfomed locally on each chart [START_REF] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF] (respectively Chapter 7 and 4) and the fact that U s (q 2 ) is a bounded operator (with a norm depending 9 on q 2 and s), one in fact finds that (U t (q 1 ))

* Op (b)U t (q 2 ) is a pseudodifferential operator in Ψ -∞,0 (M ), (30)
U t (q 1 ) * Op (b)U t (q 2 ) = Op ( b(t)) + O( ∞ ), where b(t) ∼ j≥0 j b j (t), b 0 (t) = B t (0) = b • g t exp - t 0 (q 1 + q 2 ) • g τ dτ ,
and all the higher order terms (b j (t)) j≥1 in the asymptotic expansion depend on b, t, q 1 , q 2 and the choice of coordinates on the manifold. Moreover, for a fixed t ∈ R, one can verify that every term b j (t) is supported in g -t supp(b). Each b j (t) can be written as c j (t) exp -t 0 (q 1 + q 2 ) • g τ dτ , where c j (t) ∈ S -∞,0 (T * M ). The Calderón-Vaillancourt Theorem [START_REF] Zworski | Semiclassical analysis[END_REF]Chap.5] tells us that there exist constants C b,t and C ′ b,t (depending on b, q 1 , q 2 , t and M ) such that

U t (q 1 ) * Op (b) U t (q 2 ) L 2 (M)→L 2 (M) ≤ C b,t b 0 (t) ∞ , 9
It is in fact bounded by a constant of order e |s| q 2 ∞ . and also

(31) U t (q 1 ) * Op (b)U t (q 2 ) -Op (b 0 (t)) L 2 (M)→L 2 (M) ≤ C ′ b,t .
4.1.2. The case of logarithmic times. All the above discussion was done for a fixed t ∈ R. In this article, we needed to apply Egorov property for long range of times of order κ 0 | log | [START_REF] Bouzouina | Robert Uniform semiclassical estimates for the propagation of quantum observables[END_REF][START_REF] Anantharaman | Half delocalization of eigenfunctions of the Laplacian on an Anosov manifold[END_REF]. This can be achieved as all the arguments above can be adapted if we use more general classes of symbols, i.e. S -∞,0 ν (T * M ) where ν < 1/2 is a fixed constant 10 . In particular, one can show that, for b ∈ S -∞,0 (T * M ) supported near S * M as above and κ 1 small enough (depending on the support of b, on ν, on q 1 and on q 2 ), the operator

B(t, b) is a pseudodifferential operator in Ψ -∞,0 ν (M ) for all |t| ≤ κ 1 | log |.
Precisely, its symbol has an asymptotic expansion of the same form as in the case of fixed times, except that for every j ≥ 0 the symbol c j (t) belongs to S -∞,kj ν (T * M ) for every |t| ≤ κ 1 | log |, where jk j is an increasing sequence of real numbers converging to infinity as j → +∞.

We also mention that all the seminorms of the symbols c j (t) can be bounded uniformly for |t| ≤ κ 1 | log |. Finally, using pseudodifferential calculus (performed locally on every chart), one can verify that the following uniform estimates hold: Proposition 4.1. There exist constants κ 1 > 0 and ν 0 > 0 (depending only on q 1 , q 2 , ν and M ) such that for every smooth function b compactly supported in {(x, ξ) :

ξ 2 ∈ [1/2, 3/2]}, there exists a constant C b > 0 such that for every |t| ≤ κ 1 | log |, one has U t (q 1 ) * Op (b) U t (q 2 ) L 2 (M)→L 2 (M) ≤ C b b 0 (t) ∞ , and
U t (q 1 ) * Op (b) U t (q 2 ) -Op (b 0 (t)) L 2 (M)→L 2 (M) ≤ C b ν0 .
Remark 4.2. We will mostly use evolutions involving the propagator U t of (4). Then, the expression (U t ) * Op (b) U t has the form of ( 30), with q 1 = q 2 = 2z( )a. As a result, in this case the principal symbol is b 0 (t) = b • g t e -2 t 0 a•g τ dτ . Another operator will be used: (U t ) -1 Op (b) U t also has the form [START_REF] Sjöstrand | Asymptotic distributions of eigenfrequencies for damped wave equations[END_REF], now with q 1 = -√ 2za, q 2 = 2z( )a. In this case, the principal symbol b 0 (t) = b • g t . 4.2. Sums of long products of pseudodifferential operators. In this paragraph, we make a few observations on "long" product of pseudodifferential operators (with ≍ | log | factors), that we used at different stages of our proof -e.g. in paragraphs 3.2.3 and 3.2. [START_REF] Anantharaman | Half delocalization of eigenfunctions of the Laplacian on an Anosov manifold[END_REF].

The open cover and the time n 0 of paragraph 2.2 (and their corresponding quantum partition near E δ ) are fixed in this paragraph.

We would like to use the above results to show that, for κ 0 > 0 small enough, for 0 ≤ p ≤ κ 0 | log | and for any subset X p ⊂ W p of p-cylinders, the operator

Π Xp := γ∈Xp Π γ
is a pseudodifferential operator, with a principal symbol in a "good" symbol class. Using the composition rule for pseudodifferential operators in Ψ -∞,0 ν (M ) and proposition 4.1, there exist ν 0 > 0 and κ 0 > 0 such that, for every 0 ≤ p ≤ κ 0 | log | and for every γ ∈ W p , Π γ -Op ( P γ )

L 2 (M) = O( ν0 ),
where the remainder can be bounded uniformly for every 0 ≤ p ≤ κ 0 | log | and for every cylinder γ ∈ W p .

Remark 4.3. The constants ν 0 and κ 0 appearing here are a priori smaller than the one from proposition 4.1.

This observation leads us to the bound where K = |W |. Hence, for κ 0 small enough, the remainder is of the form O( ν ′ 0 ) for some positive ν ′ 0 > 0. We underline that the constant in the remainder is uniform w.r.to 0 ≤ p ≤ κ 0 | log | and

X p ⊂ W p .
We can also verify that there exists κ 0 > 0 small enough and ν < ν ′ < 1/2 such that the function P Xp belongs to the symbol class S -∞,0 ν ′ (T * M ), and such that the seminorms (defining this class) can be bounded uniformly w.r.to 0 ≤ p ≤ κ 0 | log | and X p ⊂ W p . In particular, one can apply semiclassical calculus to this operator. For instance, the Calderón-Vailancourt Theorem tells us that

(33) Op ( Π Xp ) L 2 →L 2 = O(1),
where the constant in the remainder is uniform w.r.to 0 ≤ p ≤ κ 0 | log | and X p ⊂ W p .

Remark 4.4. When proving the subadditive property, we also needed to bound from above the norm of

Q Xp := e -ıpn 0 ∆ 2 γ∈X Π γ , for a subset X p ⊂ Λ p .
Using the notations of §4.1, this operator can be written

Q Xp = U (0) -pn0 Π Xp U ( √ 2za) pn0
Hence, using [START_REF]Zelditch Recent developments in mathematical quantum chaos[END_REF] and the Egorov type estimate of Proposition 4.1, one obtains, for κ 0 small enough,

Q Xp -Op P Xp • g pn0 e -pn 0 0 a•g s ds L 2 (M) = O( ν ′ 0 ),
for some ν ′ 0 > 0. The symbol P Xp • g pn0 e -pn 0 0 a•g s ds = e -pn 0 0 a•g s ds γ∈Xp P γ p-1 • g (p-1)n0 . . . P γ 1 • g n0 P γ 0 belongs to a class S -∞,0 ν ′ (T * M ). In particular, since X p ⊂ Λ p , one can combine lemma 3.3 with the Calderón-Vaillancourt Theorem in order to derive that, for κ 0 small enough and for any 0 ≤ p ≤ κ 0 | log |, one has the norm estimate (34)

Q Xp L 2 = O(e pn0β ),
where the implied constant is uniform in p, X p ⊂ Λ p and depends on a, on the choice of the open cover and on n 0 .

Remark 4.5. Even if we did not mention it at every stage of the proof, the remainders due to the semiclassical approximation depend on the choice of the open cover and on n 0 that were introduced in paragraph 2.2.

Pseudodifferential calculus on a manifold

In this last section, we review some basic facts on semiclassical analysis that can be found for instance in [START_REF] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF]. 5.1. General facts. Recall that we define on R 2d the following class of symbols:

S m,k (R 2d ) := (b (x, ξ)) ∈(0,1] ∈ C ∞ (R 2d ) : |∂ α x ∂ β ξ b | ≤ C α,β -k ξ m-|β| .
Let M be a smooth Riemannian d-manifold without boundary. Consider a smooth atlas (f l , V l ) of M , where each f l is a smooth diffeomorphism from

V l ⊂ M to a bounded open set W l ⊂ R d . To each f l correspond a pull back f * l : C ∞ (W l ) → C ∞ (V l
) and a canonical map fl from T * V l to T * W l : fl : (x, ξ) → f l (x), (Df l (x) -1 ) T ξ .

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 b in C ∞ (T * M ) can be decomposed as follows: b = l b l , where b l = bφ l . Each b l belongs to C ∞ (T * V l ) and can be pushed to a function bl = ( f -1 l ) * b l ∈ C ∞ (T * W l ). As in [START_REF] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF], define the class of symbols of order m and index k

(35) S m,k (T * M ) := (b (x, ξ)) ∈(0,1] ∈ C ∞ (T * M ) : |∂ α x ∂ β ξ b | ≤ C α,β -k ξ m-|β| .
Then, for b ∈ S m,k (T * M ) and for each l, one can associate to the symbol bl ∈ S m,k (R 2d ) the standard Weyl quantization

Op w ( bl )u(x) := 1 (2π ) d R 2d e ı x-y,ξ bl x + y 2 , ξ; u(y)dydξ,
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 b ∈ S m,k (T * M ) is then defined in the following way (see chapter 14 in [START_REF] Zworski | Semiclassical analysis[END_REF]):

(36) Op (b)(u) := l ψ l × f * l Op w ( bl )(f -1 l ) * (ψ l × 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] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF][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 σ 0 : Ψ m,k (M ) → S m-1,k /S m-1,k-1 (T * M ) is then 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] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF], for 0

≤ ν < 1/2, ( 37
) S m,k ν (T * M ) := (b ) ∈(0,1] ∈ C ∞ (T * M ) : |∂ α x ∂ β ξ b | ≤ C α,β -k-ν|α+β| ξ m-|β| .
Results of [START_REF] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF] can be applied to this new class of symbols. For example, a symbol of S 0,0 ν (T * M ) gives a bounded operator on L 2 (M ) (with norm uniformly bounded with respect to ).

Positive quantization.

Even if the Weyl procedure is a natural choice to quantize an observable b on R 2d , it is sometimes preferrable to use a quantization procedure Op that satisfies the property : Op (b) ≥ 0 if b ≥ 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 b 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 (38) Op w (b) -Op AW (b) L 2 ≤ C |α|≤D |α|+1 2 ∂ α db ,
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 (36). This new choice of quantization is well defined for every element 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 to

C ∞ o (T * M )∩ S 0,0 ν (T * M ).
Appendix A. Inverse logarithmic "spectral gap" under a pressure condition By Stéphane Nonnenmacher and Gabriel Rivière

In this appendix, we consider the problem (2) in the case where the damping function a(x) ≥ 0 does not identically vanish. We also make the assumption that the set of undamped trajectories

N = ρ ∈ S * M : a • g t (ρ) = 0, t ∈ R
is not empty. In this case, it is generally not known whether there exists a strip of fixed width below the real axis without eigenvalues of (2). Lebeau showed [START_REF] Lebeau | Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli[END_REF] the existence of an exponentially thin strip, meaning that there exists C > 0 such that all eigenvalues τ = 0 satisfy

Im τ ≤ - 1 C e -C|τ | .
Lebeau also constructed a geometric situation where this upper bound is sharp. Yet, it is natural to ask whether additional assumptions on the manifold M and on the set N allow to improve this upper bound. In this appendix, we apply the techniques developed above to prove the following criterium for an inverse logarithmic gap.

Theorem A.1. Assume the set of undamped trajectories N is a hyperbolic set, and satisfies the pressure condition

(39) P top N , g t , 1 2 log J u < 0.
Then, there exists a constant C > 0 such that for the following resolvent estimate holds:

(-∆ -2iaτ -τ 2 ) -1 ≤ C(log(Re τ )) 2 Re τ , uniformly for τ ∈ Re τ ≥ C, | Im τ | ≤ C -1 log(Re τ ) . (40)
As a consequence, there is a C > 0 such that any eigenvalue τ n = 0 of the problem (2) satisfies

(41) Im τ n ≤ - C log(1 + |τ n |) .
This inverse logarithmic spectral gap was recently obtained in [14, Thm. 5.5] using a different approach, and under the slightly stronger assumption that π(N ) ∩ supp(a) = ∅ where π : S * M → M is the canonical projection on M (in our setting, N is allowed to intersect supp a ∩ a -1 (0)). However, the resolvent estimate obtaind in [14, Thm. 5.5] is of order log(Re τ ) Re τ , which is sharper (by a logarithmic factor) than the one we obtain above. We believe that this loss of a logarithmic factor is due to our method of proof, and that the upper bound log(Re τ ) Re τ should hold under our conditions as well.

A similar result had been proved by Christianson in [START_REF] Christianson | Semiclassical nonconcentration near hyperbolic orbits[END_REF], under the assumption that N consists in a single hyperbolic closed geodesic, and extended in [START_REF] Christianson | Quantum Monodromy and Non-concentration Near a Closed Semi-hyperbolic Orbit[END_REF] to the case of a (single) semihyperbolic closed geodesic 11 satisfying a nonresonance assumption. In [START_REF] Rivière | Delocalization of slowly damped eigenmodes on Anosov manifolds[END_REF] the same spectral gap was proved under the assumption that the geodesic flow on M is Anosov [START_REF] Katok | Hasselblatt Introduction to the Modern Theory of Dynamical Systems[END_REF]. The above Theorem thus generalizes the results of [START_REF] Christianson | Semiclassical nonconcentration near hyperbolic orbits[END_REF][START_REF] Rivière | Delocalization of slowly damped eigenmodes on Anosov manifolds[END_REF], and it cannot be improved without additional assumptionssee the example announced in [START_REF] Burq | Christianson Imperfect control for the damped wave equation[END_REF].

In order to get a larger gap, one can try to make global assumptions on the geodesic flow on M , for instance assume it is of Anosov type. It was conjectured in [START_REF] Nonnenmacher | Spectral theory of damped quantum chaotic systems[END_REF] that if the geodesic flow is Anosov and N satisfies the condition (39), then there should be a finite spectral gap, namely all eigenvalues τ = 0 of the problem (2) should satisfy Im τ ≤ -γ for some γ > 0. We refer the reader to [START_REF] Schenck | Exponential stabilization without geometric control[END_REF][START_REF] Nonnenmacher | Spectral theory of damped quantum chaotic systems[END_REF] for partials results in favor of this conjecture.

The references [START_REF] Lebeau | Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli[END_REF][START_REF] Christianson | Applications of Cutoff Resolvent Estimates to the Wave Equation[END_REF] show how to connect resolvent estimates with the decay of the energy

E(v(t)) def = 1 2 ∇v(t) 2 + ∂ t v(t) 2
of a wave v(x, t) satisfying (1). With our dynamical conditions one obtains a stretched exponential decay (see [START_REF] Christianson | From resolvent estimates to damped waves[END_REF]Cor. 5.2]):

Corollary A.2. Assume the same geometric conditions as in Thm A.1. For any s > 0 there exists C s > 0, such that for any initial data (v(0), ∂ t v(0)) ∈ H s+1 (M ) × H s (M ), the energy of the wave v(t) solving (1) with those data satisfies ∀t ≥ 0, E(v(t)) ≤ C s e -t 1/2 /Cs v(0) 2 H s+1 + ∂ t v(0) 2 H s .

Remark A.3. The undamped set N can be "lifted" to nearby energy shells, and we will often consider N δ defined as in [START_REF] Burq | Christianson Imperfect control for the damped wave equation[END_REF]. Due to the homogeneity of the geodesic flow, the condition (39) is satisfied on all nonzero energy shells when it is on S * M = p -1 0 (1/2). We now give the proof of Theorem A.1.

Proof. Using the semiclassical notations of the introduction, we need to establish the existence of constants δ 0 > 0, and C > 0 such that, for > 0 small enough, (42)

∀z ∈ 1 2 -δ 0 , 1 2 + δ 0 + ı -C -1 | log | , C -1 | log | , (P( , z) -z) -1 L(L 2 (M)) ≤ C| log | 2 .
Translating back to the original setting of (2), this resolvent estimate implies (40).

In order to prove (42), we proceed by contradiction. Namely, we assume that there exist a sequence of parameters ( l ց 0) l∈N , of spectral parameters z( l ) ∈ C and of normalized quasimodes ψ l ∈ L 2 (M ), so that, when l tends to infinity,

P(z( l ), l )ψ l = z( l )ψ l + o( l | log l | -2 ), z( l ) = 1 2 + o(1), Im z( l ) l = o(| log l | -1 ) . ( 43
)
To alleviate the notations we will omit the parameter l and just use , z, ψ . A notable difference with the proof of Theorem 1.1 is that we need to deal with quasimodes, instead of eigenmodes (considering only eigenmodes would allow to prove the inverse logarithmic gap (41), but not the resolvent estimate (40)).

The assumptions (43) imply the following estimates, that we will frequently use in our proof. For any K > 0, the following estimates hold uniformly for times |t| ≤ K| log | U t e itz/ ψ def = e -ıt (P( ,z)-z) ψ = ψ + o(|t|| log | -2 ), and Applying the Egorov estimate [START_REF] Sjöstrand | Asymptotic distributions of eigenfrequencies for damped wave equations[END_REF], in particular the case described in remark 4. Since the family (P α ) α∈W forms a resolution of identity near E δ/2 , we have for any t ∈ R This lower bound establishes the contradiction with Eq. ( 46), and shows that our assumption (43) cannot be verified. This proves the resolvent estimate (42), and our theorem.

Remark A.5. Provided that we consider a sequence of o( | log | -2 ) quasimodes, the above logarithmic lower bound on ψ , Op (P ∞ ) ψ holds as well in the selfadjoint case for a smooth cutoff function 1 -P ∞ around an hyperbolic subset Λ satisfying P top (Λ, g t , log J u /2) < 0. In fact, its proof only used the fact that Im z = o( | log | -1 ) and a ≥ 0. In this case, this lower bound generalizes the concentration results obtained in [START_REF] De Verdière | Parisse Équilibre instable en régime semi-classique. I. Concentration microlocale[END_REF][START_REF] Toth | L p norms of eigenfunctions in the completely integrable case[END_REF][START_REF] Burq | Zworski Geometric control in the presence of a black box[END_REF][START_REF] Christianson | Semiclassical nonconcentration near hyperbolic orbits[END_REF] for hyperbolic closed geodesics (yet, the required precision of our quasimode is stronger than the one used in [START_REF] Christianson | Semiclassical nonconcentration near hyperbolic orbits[END_REF]; besides, our result does not encompass the case of a semihyperbolic orbit treated in [START_REF] Christianson | Quantum Monodromy and Non-concentration Near a Closed Semi-hyperbolic Orbit[END_REF]).

( 32 )

 32 Π Xp -Op ( P Xp ) L 2 (M) = O(K p ν0 ), P Xp := γ∈Xp P γ

et-- 1

 1 Im z = 1 + o(|t|| log | -1 ). (45) Hence, even for |t| ≍ | log | both remainders are o →0 (1). Applying the quasimode equation and (44), we obtain, for every fixed t > 0, Im z = --1 Im ψ , P( , z)ψ + o(| log | -1 ) = ψ , a ψ + O( ) + o(| log | -1 ) = e -2t Im z ψ , (U t ) * a U t ψ + o t (| log | -1 ) .

-- 1 aP- 1 P

 11 2, and averaging over t ∈ [-T, T ], we get--1 Im z = ψ , Op 1 2T T -T a • g t e -2t Im z -2 t 0 a•g s ds dt ψ + o T (| log | -1 ).Using the fact that the quantization procedure is almost positive -see §5.2 -and the identity (45), one gets the bound Im z≥ (1 + o T (1))e -2T a ∞ ψ , • g t dt ψ + o T (| log | -1 ).We now use the cutoff functionP ∞ ∈ C ∞ c (V ∞ , [0, 1]) introduced in §3.1.2: notice that its support is at positive distance from N δ . Using again that Op is approximately positive, one finds that--1 Im z ≥ (1 + o T (1))e -2T a ∞ ψ , Op P ∞ × 1 2T T -T a • g t dt ψ + o T (| log | -1 ).Since P ∞ is supported away from the undamped set N δ , there exists T > 0 and a 0 > 0 (independent of ) such that inf ρ∈suppP∞1 2TT -T a • g t (ρ)dt ≥ a 0 , γ P ∞ • g -(n-p+1)n0 γ∈W pγ • g -(n-p+1)n0 .

γ∈W n-p P γ • g t ≤ 1, γ∈W p- 1 P 2 .

 12 γ • g t = 1, near E δ/The approximate positivity of Op implies Opγ∈(W n ) c P γ ψ , ψ ≤ n p=1 Op (P ∞ • g (p-n-1)n0 )ψ , ψ + O( ν0 ), so from (47) we get (c k k) -2 (1 + o(1)) ≤ n p=1 Op P ∞ • g -pn0 ψ , ψ + O( ν0 ).We now again combine the fact that ψ is an quasimode (via equation (44)) with the Egorov theorem, and obtain(c k k) -2 (1 + o(1)) ≤ n p=1 Op P ∞ e -2pn0 Im z -2 pn 0 0 a•g s ds ψ , ψ + O( ν0 ) + o(n 2 | log | -2 ). A last application of the fact that a ≥ 0, Im z = o( | log | -1 ), n = O(| log |) and that Op is almost positive implies that (c k k) -2 (1 + o(1)) ≤ n (1 + o(1)) Op (P ∞ )ψ , ψ + O( ν0 ) + o(1).Hence, for n = [κ 0 | log |] we end up with (c k k) -2 κ 0 | log | (1 + o(1)) ≤ Op (P ∞ )ψ , ψ .

For simplicity of exposition, we only deal with operators of this form. However, our approach could in principle be adapted to treat the case of more general families of nonselfadjoint operators like the ones considered in[START_REF] Sjöstrand | Asymptotic distributions of eigenfrequencies for damped wave equations[END_REF], §1.

As pointed out at the end of the appendix, our proof also allows to recover (and to generalize) this result.

In the selfadjoint case (a ≡ 0, β = 0), Colin de Verdière & Parisse's example satisfies such a property.

In order to avoid too many indices, we take the same ν as in the definition of Θ Λ, ,ν .

This work has been partially supported by the grant ANR-09-JCJC-0099-01 of the Agence Nationale de la Recherche.

Then, the assumption (13) on z( ) shows that the above right hand side is smaller than c e nn0 Im z (1 + o(1)), therefore ( 27) becomes ( 28)

Combining this inequality with the lower bound [START_REF] Pesin | Dimension Theory in Dynamical Systems: Contemporary Views and Applications[END_REF], one obtains (29)

This lower bound is our desired lower bound for a "short" logarithmic time.

Remark 3.8. We underline again that the constants c and k do not depend on the sequence (ψ ), but only on δ, P 0 , n 0 , the choice of open cover and κ 0 . Thus, it depends only on Λ, a and ν and it will be this constant (c k k) -1 that will play the role of c Λ,a,ν in (14).

3.2.4. Using semiclassical calculus. Since ψ is an eigenstate of U , the inequality (29) can be rewritten as

Using the observations of paragraph 4.2, and the fact that κ 0 has been chosen small enough, for

is approximately the quantization of the symbol P Λ c n := γ∈Λ c n P γ , which belongs to the symbol class S -∞,0 ν ′ (T * M ) for some ν ′ ∈ (0, 1/2). Using also the composition rule in Ψ -∞,0

), for some ν 0 > 0. By construction, the function P Λ c n takes values in [0, 1]. Because the quantization Op is approximately positive for symbols in this class -see paragraph 5.2 -one finds that

). Remark 3.9. The value of ν 0 > 0 can be different from the one appearing above: we have just kept the largest remainder term.

We now split the above left hand side into two parts, using the cutoff function Θ ,Λ,ν . It remains to estimate (A) := Op P Λ c n (1 -Θ ,Λ,ν ) ψ , ψ , and (B) := Op P Λ c n Θ ,Λ,ν ψ , ψ . Using again the fact that Op is almost positive, and that

On the other hand, the definition of Λ c n implies that (B) = 0. This leads to lim inf

which concludes the proof of Theorem 1.1. The lower bound depends only on Λ, a and ν -see remark 3.8.

Long products of pseudodifferential operators

In this section, we describe some properties of long products of pseudodifferential operators evolved under the quantum propagator. For that purpose, we recall first a few facts on the Egorov property for nonselfadjoint operators and then we apply them to our problem. which implies

In particular, from our assumption on Im z( ) we get

.

To obtain a contradiction we will prove an inverse logarithmic lower bound for the above left hand-side. This can be achieved by adapting the argument of Theorem 1.1. We will use the notations introduced in §3.1.2. Instead of considering the subset of cylinders Λ n ⊂ W n in the argument of §3.2, we will use the full family W n , and obtain an upper bound for

time, for which we may apply Egorov's Theorem and the pseudodifferential calculus.

Remark A.4. In §3 the restriction to cylinders in Λ n had allowed to show that the Birkhoff averages -nn0 0 a • g s (ρ)ds were bounded above by βnn 0 + O(1), a property which was crucially used in §3.2.3. We are now interested in the case β = 0, and the upper bound -nn0 0 a • g s (ρ)ds ≤ 0 obviously holds for every point ρ ∈ T * M since a is nonnegative.

Using the hyperbolic dispersive estimate [START_REF] Nonnenmacher | Spectral theory of damped quantum chaotic systems[END_REF] and taking the sum over W n , we can prove the inequality [START_REF] Pesin | Dimension Theory in Dynamical Systems: Contemporary Views and Applications[END_REF] for our quasimode ψ . Using (45) and the fact that the time knn 0 = O(| log |), we get

Implementing the same subadditivity argument as in §3.2.3, we find

Thanks to the upper bound [START_REF] Zworski | Semiclassical analysis[END_REF] and the subunitarity bound U n ≤ 1, we verify that γ∈(W n ) c Π γ = O(1). We now use the identities (44,45) one more time and we obtain

Like in §3.2.4 and using again (44,45), this inequality can be rewritten as

and then analyzed through the pseudodifferential calculus like in the proof of Theorem 1.1. We obtain The set (W n ) c consists in the cylinders in W n with at least one index γ j = ∞, so it can be split into

12 Like in paragraph 3.2.4, the parameter ν 0 > 0 will change from line to line, meaning that we keep the worst remainder term.