Gabriel Rivi Ère

ENTROPY OF SEMICLASSICAL MEASURES FOR NONPOSITIVELY CURVED SURFACES

We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. We show that the Kolmogorov-Sinai entropy of a semiclassical measure µ for the geodesic flow g t is bounded from below by half of the Ruelle upper bound, i.e.

 and refer the reader to it for the details of several lemmas.

Introduction

Let M be a compact C ∞ Riemannian manifold. For all x ∈ M , T *

x M is endowed with a norm . x given by the metric over M . The geodesic flow g t over T * M is defined as the Hamiltonian flow corresponding to the Hamiltonian H(x, ξ) := ξ 2 x 2 . This last quantity corresponds to the classical kinetic energy in the case of the absence of potential. As any observable, this quantity can be quantized via pseudodifferential calculus and the quantum operator corresponding to H is -2 ∆ 2 where is proportional to the Planck constant and ∆ is the Laplace Beltrami operator acting on L 2 (M ). Our main concern in this note will be to study the asymptotic behavior, as tends to 0, of the following sequence of distributions: where Op (a) is a -pseudodifferential operator of symbol a [START_REF] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF] and ψ satisfies

-2 ∆ψ = ψ .
An accumulation point of such a sequence of distribution µ is called a semiclassical measure. Moreover, one knows that a semiclassical measure is a probability measure on S * M := { ξ 2 x = 1} which is invariant under the geodesic flow g t on S * M . For manifolds of negative curvature, the geodesic flow on S * M satisfies strong chaotic properties (Anosov property, ergodicity of the Liouville measure) and as a consequence, it can be shown that almost all the µ converge to the Liouville measure on S * M [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] Tartar | Verdière Ergodicité et fonctions propres du Laplacien[END_REF]. This phenomenon is known as the quantum ergodicity property. A main challenge concerning this result would be to answer the Quantum Unique Ergodicity Conjecture [START_REF] Rudnick | The behaviour of eigenstates of arithmetic hyperbolic manifolds[END_REF], i.e. determine whether the Liouville measure is the only semiclassical measure or not (at least for manifolds of negative curvature). In [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF], Anantharaman used the Kolmogorov-Sinai entropy to study the properties of semiclassical measures on manifolds of negative curvature 1 . In particular, she showed that the Kolmogorov-Sinai entropy of any semiclassical measure is positive. This result implies that the support of a semiclassical measure cannot be restricted to a closed geodesic, i.e. eigenfunctions of the Laplacian cannot concentrate only on a closed geodesic in the high energy limit. In subsequent works, with Nonnenmacher and Koch, more quantitative lower bounds on the entropy of semiclassical measures were given [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], [START_REF] Anantharaman | Entropy of eigenfunctions[END_REF]. 1 In fact, her result was about manifolds with Anosov geodesic flow, for instance manifolds of negative curvature. 1.1. Kolmogorov-Sinai entropy. Let us recall a few facts about the Kolmogorov-Sinai (also called metric) entropy (see [START_REF] Walters | An introduction to ergodic theory[END_REF] or appendix B for more details and definitions). It is a nonnegative number associated to a flow g and a g-invariant measure µ, that estimates the complexity of µ with respect to this flow. For example, a measure carried by a closed geodesic will have entropy zero. Recall also that a standard theorem of dynamical systems due to Ruelle [START_REF] Ruelle | An inequality for the entropy of differentiable maps[END_REF] asserts that, for any invariant measure µ under the geodesic flow: [START_REF] Abramov | On the entropy of a flow[END_REF] h KS (µ, g) ≤

S * M j χ + j (ρ)dµ(ρ)
with equality if and only if µ is the Liouville measure in the case of an Anosov flow [START_REF] Ledrappier | Young The metric entropy of diffeomorphisms I. Characterization of measures satisfying Pesin's entropy formula[END_REF]. In the previous inequality, the χ + j denoted the positive Lyapunov exponents of (S * M, g t , µ) [START_REF] Barreira | Pesin Lectures on Lyapunov exponents and smooth ergodic theory[END_REF]. Regarding these properties, the main result of Anantharaman-Koch-Nonnenmacher was to show that, for a semiclassical measure µ on an Anosov manifold, one has

h KS (µ, g) ≥ S * M d-1 j=1 χ + j (ρ)dµ(ρ) - (d -1)λ max 2 .
where λ max := lim t→±∞ 1 t log sup ρ∈S * M |d ρ g t | is the maximal expansion rate of the geodesic flow and the χ + j 's are the positive Lyapunov exponents [START_REF] Barreira | Pesin Lectures on Lyapunov exponents and smooth ergodic theory[END_REF]. Compared with the original result from [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF], this inequality gives a precise lower bound on the entropy of a semiclassical measure. For instance, for manifolds of constant negative curvature, this lower bound can be rewritten as d-1

2 . However, it can turn out that λ max is a very large quantity and in this case, the previous lower bound can be negative (which would imply that it is an empty result). Combining these two observations [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], they were lead to formulate the conjecture that, for any semiclassical measure µ, one has

h KS (µ, g) ≥ 1 2 S * M d-1 j=1 χ + j (ρ)dµ(ρ).
They also ask about the extension of this conjecture to manifolds without conjugate points [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF]. In recent work [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], we were able to prove that their conjecture holds for any surface with an Anosov geodesic flow (for instance surfaces of negative curvature). Regarding our proof and the nice properties of surfaces of nonpositive curvature [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF], [START_REF] Eberlein | Geodesic flows in manifolds of nonpositive curvature[END_REF], it became clear that our result can be adapted in the following way:

Theorem 1.1. Let M be a C ∞ Riemannian surface of nonpositive sectional curvature and µ a semiclassical measure. Then,

(2) h KS (µ, g) ≥ 1 2 S * M χ + (ρ)dµ(ρ),
where h KS (µ, g) is the Kolmogorov-Sinai entropy and χ + (ρ) is the upper Lyapunov exponent at point ρ.

In particular, this result shows that the support of a semiclassical measure cannot be reduced to a closed unstable geodesic. We underline that our inequality is also coherent with the quasimodes constructed by Donnelly [START_REF] Donnelly | Quantum unique ergodicity[END_REF]. In fact, his quasimodes are supported on closed stable geodesics (included in flat parts of a surface of nonpositive curvature) and have zero entropy. We can make a last observation on the assumptions on the manifold: it is not known whether the geodesic flow is ergodic or not for the Liouville measure on a surface of nonpositive curvature. The best result in this direction is that there exists an open invariant subset U of positive Liouville measure such that the restriction g |U is ergodic with respect to Liouville [START_REF] Barreira | Pesin Lectures on Lyapunov exponents and smooth ergodic theory[END_REF]. The extension of this result on the entropy of semiclassical measures raises the question of knowing whether one could obtain an analogue of this result for weakly chaotic systems. For instance, regarding the counterexamples constructed in [START_REF] Hassell | Ergodic billiards that are not quantum unique ergodic[END_REF], it would be interesting to have a lower bound for ergodic billiards. Our purpose in this note is to prove theorem 1.1. Our strategy will be the same as in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]. So we will focus on the main differences and refer the reader to [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] and [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] for the details of several lemmas. The crucial observation is that as in the Anosov case, surfaces of nonpositive curvature have continuous stable and unstable foliations and no conjugate points. This property was at the heart of the proofs in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], [START_REF] Anantharaman | Entropy of eigenfunctions[END_REF] and [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] and we will verify that even if the properties of these stable/unstable directions are weaker for surfaces of nonpositive curvature, they are sufficient to prove the conjecture of Anantharaman-Nonnenmacher in this weakly chaotic setting. In [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], [START_REF] Anantharaman | Entropy of eigenfunctions[END_REF], [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], there was a dynamical quantity which was crucially used: the unstable Jacobian of the geodesic flow. In the case of surfaces of nonpositive curvature, one can introduce an analogue of it. This quantity comes from the study of Jacobi fields and is called the unstable Riccati solution U u (ρ) [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF]. In this setting, it has been shown that the Ruelle inequality can be rewritten as follows [START_REF] Freire | On the entropy of the geodesic flow for manifolds without conjugate points[END_REF]:

h KS (µ, g) ≤ S * M U u (ρ)dµ(ρ).
So, the lower bound of theorem 1.1 can be rewritten as

(3) h KS (µ, g) ≥ 1 2 S * M U u (ρ)dµ(ρ).
The main adavantage of this new formulation is that the function in the integral of the lower bound is defined everywhere (and not almost everywhere).

Remark. One could also ask whether it would be possible to extend this result to surfaces without conjugate points. In fact, these surfaces also have a stable and unstable foliations (and of course no conjugate points). Moreover, according to Green [START_REF]Green Geodesic instability[END_REF] and Eberlein [START_REF] Eberlein | When is a geodesic flow of Anosov type I[END_REF], the Jacobi fields also satisfy a property of uniform divergence (at least in dimension 2). The main difficulty is that the continuity of U u (ρ) is not true anymore [START_REF] Ballmann | Burns On surfaces with no conjugate points[END_REF] and at this point, we do not see any way of escaping this difficulty.

1.2. Organization of the article. In section 2, we will give a precise survey2 on surfaces of nonpositive curvature and highlight the properties we will need to make the proof work. Then, in section 3, we will draw a precise outline of the proof and we will refer to [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] for the details of some lemmas. In section 4, we will explain how the main result from [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] can be adapted in the setting of surfaces of nonpositive curvature. In section 5, we follow the same strategy as in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] to derive a crucial estimate on the quantum pressures. Finally, in the appendix, we recall some results on quantum pressure from [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF].

Acknowledgements. I would like to sincerely thank my advisor Nalini Anantharaman for introducing me to this question and for encouraging me to extend the result from [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] to nonpositively curved surfaces. I also thank her for many helpful discussions about this subject.

2. Classical setting of the article 2.1. Surfaces of nonpositive curvature. In this first section, we recall some facts about nonpositively curved manifolds [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF], [START_REF] Eberlein | Geodesic flows in manifolds of nonpositive curvature[END_REF].

2.1.1. Stable and unstable Jacobi fields. We define π : S * M → M the canonical projection π(x, ξ) := x. The vertical subspace V ρ at the point ρ = (x, ξ) is the kernel of the application d ρ π. We underline that it is in fact the tangent space in ρ of the 1-dimensional submanifold S * x M . In the case of a surface, it has dimension 1. We can also define the horizontal subspace in ρ. Precisely, for Z ∈ T ρ S * M , we consider a smooth curve c(t) = (a(t), b(t)), t ∈ (-ǫ, ǫ), in S * M such that c(0) = ρ and c ′ (0) = Z. Then, we define the horizontal space H ρ as the kernel of the application

K ρ (Z) = ∇ a ′ (0) b(0) = ∇ dρπ(Z) b(0)
, where ∇ is the Levi-Civita connection. This subspace contains X H (ρ) the vector field tangent to the Hamiltonian flow. For a surface, this subspace is of dimension 2. We know that we can use these two subspaces to split the tangent space T ρ S * M = H ρ ⊕ V ρ (it is the usual way to split the tangent space in order to define the Sasaki metric on S * M [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF]). Using this decomposition, we would like to recall an important link between the linearization of the geodesic flow and the Jacobi fields on M . To do this, we underline that to each point ρ in S * M corresponds a unique unit speed geodesic γ ρ . Then we define a Jacobi field in ρ (or along γ ρ ) as a solution of the differential equation:

J"(t) + R(γ ′ ρ (t), J(t))γ ′ ρ (t) = 0,
where R(X, Y )Z is the curvature tensor applied to the vector fields X, Y and Z and J ′ (t) = ∇ γ ′ ρ (t) J(t). We recall that we can interpret Jacobi fields as geodesic variation vector fields [START_REF] Eberlein | Geodesic flows in manifolds of nonpositive curvature[END_REF]. Precisely, consider a C ∞ family of curves c s : [a, b] → M , s in (-ǫ, ǫ). We say that it is a C ∞ variation of c = c 0 . It defines a corresponding variation vector field Y (t) = ∂ ∂s (c s (t)) |s=0 that gives the initial velocity of s → c s (t). If we suppose now that c is a geodesic of M , then a C 2 vector field Y (t) on c is a Jacobi vector field if and only if Y (t) is the the variation vector field of a geodesic variation of c (i.e. ∀s ∈ (-ǫ, ǫ), c s is a geodesic of M ). For instance, γ ′ ρ (t) and tγ ′ ρ (t) are Jacobi vector fields along γ ρ . Consider now a vector (V, W ) in T ρ S * M given in the coordinates H ρ ⊕ V ρ . Using the canonical identification given by d ρ π and K ρ , there exists a unique Jacobi field J V,W (t) in ρ whose initial conditions are J V,W (0) = V and J ′ V,W (0) = W , such that

d ρ g t (V, W ) = (J V,W (t), J ′ V,W (t))
in coordinates H g t ρ ⊕ V g t ρ [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF] (lemma 1.4). We define N ρ the subspace of T ρ S * M of vectors orthogonal to X H (ρ) and H ρ the intersection of this subspace with H ρ . Using the previous property about Jacobi fields, we know that the subbundle N perpendicular to the Hamiltonian vector field is invariant by g t and that we have the following splitting [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF] (lemma 1.5):

T ρ S * M = RX H (ρ) ⊕ H ρ ⊕ V ρ .
Obviously, these properties can be extended to any energy layer E(λ) for any positive λ. Following [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF] (lemma 3.1), we can make the following construction of two particular Jacobi fields along γ ρ . We denote (γ ′ ρ (t), e(t)) an orthonormal basis defined along γ ρ (t). Given a positive T and because there are no conjugate points on the manifold M , there exists a unique Jacobi field J T (t) such that J T (0) = e(0) and J T (T ) = 0. Moreover, J T (t) is perpendicular to γ ρ (t) for all t in R [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF] (page 50). As a consequence, J T (t) can be identified with its coordinate along e(t) (as T γρ(t) M is of dimension 2). A result due to Hopf (lemma 3.1 in [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF]) tells us that the limits lim T →+∞ J T (t) and lim

T →-∞ J T (t)
exist. They are denoted J s ρ (t) and J u ρ (t) (respectively the stable and the unstable Jacobi field). They satisfy the simplified one dimensional Jacobi equation:

J"(t) + K(t)J(t) = 0,
where K(t) = K(γ ρ (t)) is the sectional curvature at γ ρ (t). They are never vanishing Jacobi fields with J * ρ (0) = e(0) and for all t in R, they are perpendicular to γ ′ ρ (t). Moreover, we have

J * ′ ρ (t) ≤ √ K 0 J * ρ (t)
for every t in R (where -K 0 is some negative lower bound on the curvature). Using the previous link between geodesic flow and Jacobi fields, we can lift these subspaces to invariant subspaces E s (ρ) and E u (ρ) called the Green stable and unstable subspaces. These subspaces have dimension 1 (in the case of surfaces) and are included in N ρ . A basis of E s (g t ρ) is given by (J s ρ (t), J s ′ ρ (t)) in coordinates H g t ρ ⊕ V g t ρ . We can underline that both subspaces are uniformly transverse to V ρ and that it can happen that they are equal to each other (which was not the case in the Anosov setting). In the case of nonpositive curvature, these subspaces depend continuously in ρ and are integrable as in the Anosov case [START_REF] Eberlein | Geodesic flows in manifolds of nonpositive curvature[END_REF].

2.1.2. Riccati equation. In the case where the Green subspaces attached to ρ are linearly independent, a splitting of N ρ is given by E u (ρ) ⊕ E s (ρ) and the splitting holds for all the trajectory. For the opposite case, we know that the Green subspaces attached to ρ (and hence to a geodesic γ ρ ) are linearly dependent if and only if the sectional curvature is vanishing at every point of the geodesic γ ρ [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF]. As a consequence, we cannot use the same kind of splitting. However, there exists a splitting of N ρ that we can use in both cases, precisely E u (ρ) ⊕ V ρ . We would like to mention that the one dimensional Jacobi equation defined earlier gives rise to the Riccati equation:

U ′ (t) + U 2 (t) + K(t) = 0,
where U (t) = J ′ (t)J(t) -1 for non vanishing J. Then we define the corresponding unstable Riccati solution associated to the unstable Jacobi field as U u ρ (t) := J u ′ ρ (t)(J u ρ (t)) -1 . It is a nonnegative quantity and it describes the growth of the unstable Jacobi field (in dimension 2) as follows:

J u ρ (t) = J u ρ (0) e R t 0 U u ρ (s)ds .
The same works for the stable Jacobi field. Both quantities are continuous 3 with respect to ρ. We underline that, we can use the previous results to obtain the bound

d ρ g t |E u (ρ) ≤ √ 1 + K 0 e R t 0 U u ρ (s)ds .
So the unstable Riccati solution describe the infinitesimal growth of the geodesic flow along the unstable direction, whereas J u (ρ) -1 used in the previous sections described the growth at time 1. More precisely, as for the unstable Jacobian, Freire and Mañé showed that the unstable Riccati solutions are related to the Lyapunov exponents. In fact, they proved that the Ruelle bound for the entropy of a g-invariant measure µ in the case of nonpositive curvature (precisely for manifolds without conjugate points) [START_REF] Freire | On the entropy of the geodesic flow for manifolds without conjugate points[END_REF] is:

h KS (µ, g) ≤ S * M U u (ρ)dµ(ρ).

Divergence of vanishing Jacobi fields.

A last point we would like to recall is a result due to Green [START_REF]Green Geodesic instability[END_REF] and to Eberlein in the general case [START_REF] Eberlein | When is a geodesic flow of Anosov type I[END_REF]. It asserts that for any positive c there exists a positive T = T (c) such that for any ρ in S * M and for any nontrivial Jacobi field J(t) along γ ρ such that J(0) = 0 and J ′ (0) ≥ 1, for all t larger than T , we have J(t) ≥ c (proposition 3.1 [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF]). This property of uniform divergence only holds in dimension 2 and as it is crucially used in the following, our proof only works for surfaces of nonpositive curvature. In larger dimensions, the same result holds but without any uniformity in ρ. Finally, all these properties allow to prove the following lemma: Lemma 2.1. Let v = (0, V ) be a unit vertical vector at ρ. Then for any c > 0, there exists T = T (c) > 0 (independent of ρ and of v) such that for any t ≥ T , d ρ g t v ≥ c.

We underline that, for t ≥ T , the angle between E u (g t ρ) and d ρ g t v is bounded by some κ(c) with κ(c) arbitrarly small as c tends to infinity.

Discretization of the unstable Riccati solution.

For θ small positive number (θ will be fixed all along the paper), one defines

E θ := H -1 (]1/2 -θ, 1/2 + θ[).
From previous section, we know that there exists a constant b 0 such that

∀ρ ∈ E θ , 0 ≤ U u (ρ) ≤ b 0 .
This function will replace the logarithm of the unstable Jacobian log J u in the proof from [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]. The situation is slightly different from the case of an Anosov flow as we do not have that U u is uniformly bounded from below by some positive constant, a property that was crucially to prove theorem 1.2 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]. We solve this problem by introducing a small positive parameter ǫ 0 and defining an auxiliary function U u 0 (ρ) := sup{U u (ρ), ǫ 0 }. We also fix ǫ and η two small positive constants lower than the injectivity radius of the manifold (that we suppose to be larger than 2). We choose η small enough to have (2 + b0 ǫ0 )b 0 η ≤ ǫ 2 (as in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], this property is only used in the proof of lemma 3.1). We underline that there exists ε > 0 such that if

∀ (ρ, ρ ′ ) ∈ E θ × E θ , d(ρ, ρ ′ ) ≤ ε ⇒ |U u (ρ) -U u (ρ ′ )| ≤ ǫ 0 ǫ. 3
The continuity in ρ is a crucial property that we will use in our proof. We underline that it is not true if we only suppose the surface to be without conjugate points [START_REF] Ballmann | Burns On surfaces with no conjugate points[END_REF].

We make the extra assumption that the small parameter ǫ used for the continuity is smaller than ǫ 0 .

Discretization of the manifold. As in the case of Anosov surfaces, our strategy to prove theorem 1.1 will be to introduce a discrete reparametrization of the geodesic flow. Regarding this goal, we cut the manifold M and precisely, we consider a partition 2 , define an open subset of T * M :

M = K i=1 O i of diameter smaller than some positive δ. Let (Ω i ) K i=1 be a finite open cover of M such that for all 1 ≤ i ≤ K, O i Ω i . For γ ∈ {1, • • • , K}
V γ := (T * Ω γ0 ∩ g -η T * Ω γ1 ) ∩ E θ .
We choose the partition (O i ) K

i=1 and the open cover

(Ω i ) K i=1 of M such that (V γ ) γ∈{1,••• ,K} 2 is
a finite open cover of diameter smaller 4 than ε of E θ . For γ := (γ 0 , γ 1 ), we define f (γ) and f 0 (γ) as in the case of an Anosov flow i.e.

f 0 (γ) := η inf{U u 0 (ρ) : ρ ∈ V γ } and f (γ) := η inf{U u (ρ) : ρ ∈ V γ }.
Compared with the Anosov case, we will have slightly different properties for the function f (γ), i.e. ( 4)

∀ρ ∈ V γ , η 0 U u ρ (s)ds -f (γ) ≤ ηǫ 0 ǫ.
We also underline that the function f 0 satisfies the following bounds, for

γ ∈ {1, • • • , K} 2 , ǫ 0 η ≤ f 0 (γ) ≤ b 0 η. Finally, let α = (α 0 , α 1 , • • • ) be a sequence (finite or infinite) of elements of {1, • • • , K}
whose length is larger than 1 and define:

(5)

f + (α) := f 0 (α 0 , α 1 ) ≤ ǫ 2 and f (α) := f (α 0 , α 1 ) ≤ ǫ 2 ,
where the upper bounds follow from the previous hypothesis. In the following, we will also have to consider negative times. To do this, we define the analogous functions, for

β := (• • • , β -1 , β 0 ) of finite (or infinite) length, f -(β) := f 0 (β -1 , β 0 ) and f (β) := f (β -1 , β 0 ).
Remark. We underline that the functions f + and f -are defined from U u 0 while f is defined from U u . This distinction will be important in the following.

Proof of theorem 1.1

Let (ψ k ) be a sequence of orthonormal eigenfunctions of the Laplacian corresponding to the eigenvalues -1/ -2 k such that the corresponding sequence of distributions µ k on T * M converges as k tends to infinity to the semiclassical measure µ. For simplicity of notations and to fit semiclassical analysis notations, we will denote tends to 0 the fact that k tends to infinity and ψ and -2 the corresponding eigenvector and eigenvalue. To prove the inequality of theorem 1.1, we will again give a symbolic interpretation of a semiclassical measure and apply results on suspension flows to this measure [START_REF] Abramov | On the entropy of a flow[END_REF]. Let ǫ ′ > 4ǫ be a positive number, where ǫ was defined in section 2.2. As in the Anosov setting, the link between the two quantities ǫ and ǫ ′ is only used to obtain theorem on product of pseudodifferential operators from sections 6 and 7 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] (here theorem 3.2). In the following of the note, the Ehrenfest time n E ( ) will be the quantity:

(6) n E ( ) := [(1 -ǫ ′ )| log |].
We underline that it is an integer time and that, compared with usual definitions of the Ehrenfest time, there is no dependence on the Lyapunov exponent. We also consider a smaller non integer time:

(7) T E ( ) := (1 -ǫ)n E ( ).
We draw now a precise outline of the proof of theorem 1.1 and refer the reader to [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] for the proof of several lemmas. The main differences with the Anosov case is that we have to indroduce a thermodynamical formalism to treat the problem.

3.1. Quantum partitions of identity. In order to find a lower bound on the metric entropy of the semiclassical measure µ, we would like to apply the uncertainty principle for quantum pressure (see appendix A) and see what informations it will give (when tends to 0) on the metric entropy of the semiclassical measure µ. To do this, we define quantum partitions of identity corresponding to a given partition of the manifold. We recall the notations from [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF].

3.1.1. Partitions of identity. In paragraph 2.2, we considered a partition of small diameter (O i ) K i=1 of M . We also defined (Ω i ) K i=1 a corresponding finite open cover of small diameter of M . By convolution of the characteristic functions 1 Oi , we obtain P = (P i ) i=1,..K a smooth partition of unity on M i.e. for all x ∈ M :

K i=1 P 2 i (x) = 1.
We assume that for all 1

≤ i ≤ K, P i is an element of C ∞ c (Ω i ).
To this classical partition corresponds a quantum partition of identity of L 2 (M ). In fact, if P i denotes the multiplication operator by P i (x) on L 2 (M ), then one has:

(8) K i=1 P * i P i = Id L 2 (M) .
3.1.2. Refinement of the quantum partition under the Schrödinger flow. Like in the classical setting of entropy, we would like to make a refinement of the quantum partition. To do this refinement, we use the Schrödinger propagation operator

U t = e ıt ∆ 2
. We define A(t) := U -t AU t , where A is an operator on L 2 (M ). To fit as much as possible with the metric entropy, we define the following operators: [START_REF] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF] τ α = P α k (kη) • • • P α1 (η)P α0 and (10)

π β = P β -k (-kη) • • • P β-2 (-2η)P β0 P β-1 (-η), where α = (α 0 , • • • , α k ) and β = (β -k , • • • , β 0 ) are finite sequences of symbols such that α j ∈ [1, K] and β -j ∈ [1, K].
We can remark that the definition of π β is the analogue for negative times of the definition of τ α . The only difference is that we switch the two first terms β 0 and β -1 . The reason of this choice relies again in the application of the quantum uncertainty principle. One can see that for fixed k, using the Egorov property:

(11) P α k (kη) • • • P α1 (η)P α0 ψ 2 → µ(P 2 α k • g kη × • • • P 2 α1 • g η × P 2 α0
) as tends to 0. This last quantity is the one used to compute h KS (µ, g η ) (with the notable difference that the P j are here smooth functions instead of characteristic functions). As in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], we will have to understand for which range of times kη, the Egorov property can be be applied. In particular, we will study for which range of times, the operator τ α is a pseudodifferential operator of symbol [START_REF] Eberlein | When is a geodesic flow of Anosov type I[END_REF]). In [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] and [START_REF] Anantharaman | Entropy of eigenfunctions[END_REF], they only considered kη ≤ | log |/λ max where λ max := lim t→±∞ 1 t log sup ρ∈S * M |d ρ g t |. This choice was not optimal and in the following, we try to define sequences α for which we can say that τ α is a pseudodifferential operator.

P α k • g kη × • • • P α1 • g η × P α0 (see
3.1.3. Index family adapted to the variation of the unstable Jacobian. Let α = (α 0 , α 1 , • • • ) be a sequence (finite or infinite) of elements of {1, • • • , K} whose length is larger than 1. We define a natural shift on these sequences

σ + ((α 0 , α 1 , • • • )) := (α 1 , • • • ).
For negative times and for β := (• • • , β -1 , β 0 ), we define the backward shift

σ -((• • • , β -1 , β 0 )) := (• • • , β -1 ).
In the following, we will mostly use the symbol x for infinite sequences and reserve α and β for finite ones. Then, using notations of section 2.1 and as described in section 4 of [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], index families depending on the value of the unstable Jacobian can be defined as follows:

(12) I η ( ) := I η (T E ( )) = (α 0 , • • • , α k ) : k ≥ 3, k-2 i=1 f + σ i + α ≤ T E ( ) < k-1 i=1 f + σ i + α , ( 13
) K η ( ) := K η (T E ( )) = (β -k , • • • , β 0 ) : k ≥ 3, k-2 i=1 f -σ i -β ≤ T E ( ) < k-1 i=1 f -σ i -β .
We underline that f + , f -≥ ǫ 0 η ensures that we consider finite sequences. These sets define the maximal sequences for which we can expect to have Egorov property for the corresponding τ α .

The sums used to define these sets were already used in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]. We can again think of the time |α|η as a stopping time for which property [START_REF] Eberlein | When is a geodesic flow of Anosov type I[END_REF] will hold (for a symbol τ α corresponding to α).

A good way of thinking of these families of words is by introducing the sets

Σ + := {1, • • • , K} N and Σ -:= {1, • • • , K} -N .
Once more, the sets I η ( ) (resp. K η ( )) lead to natural partitions of Σ (resp. Σ -). Families of operators can be associated to these families of index: (τ α ) α∈I η ( ) and (π β ) β∈K η ( ) . One can show that these partitions form quantum partitions of identity (lemma 5.1 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]):

α∈I η ( ) τ * α τ α = Id L 2 (M) and β∈K η ( ) π * β π β = Id L 2 (M) .
3.2. Symbolic interpretation of semiclassical measures. Now that we have defined these partitions of variable size, we want to show that they are adapted to compute the pressure of a certain measure with respect to some reparametrized flow associated to the geodesic flow. To do this, we proceed as in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] and provide a symbolic interpretation of the quantum partitions. We denote Σ + := {1, • • • , K} N . We also denote C i the subset of sequences (x n ) n∈N such that x 0 = i. Define also:

[α 0 , • • • , α k ] := C α0 ∩ • • • ∩ σ -k + C α k , where σ + is the shift σ + ((x n ) n∈N ) = (x n+1
) n∈N (it fits the notations of the previous section). The set Σ + is then endowed with the probability measure (not necessarily σ-invariant):

µ Σ+ ([α 0 , • • • , α k ]) = µ Σ+ C α0 ∩ • • • ∩ σ -k + C α k = P α k (kη) • • • P α0 ψ 2 .
Using the property of partition of identity, it is clear that this definition assures the compatibility conditions to define a probability measure:

α k+1 µ Σ+ ([α 0 , • • • , α k+1 ]) = µ Σ+ ([α 0 , • • • , α k ]) .
Then, we can define the suspension flow, in the sense of Abramov, associated to this probability measure. To do this, the suspension set is defined as:

(14) Σ + := {(x, s) ∈ Σ + × R + : 0 ≤ s < f + (x)}.
Recall that the roof function f + is defined as f + (x) := f + (x 0 , x 1 ). We define a probability measure µ Σ+ on Σ + :

(15)

µ Σ+ = µ Σ+ × dt α∈{1,••• ,K} 2 f + (α) P α ψ 2 = µ Σ+ × dt α∈{1,••• ,K} 2 f + (α)µ Σ+ ([α]
) .

The suspension semi-flow associated to σ + is for time s:

(16) σ s + (x, t) :=   σ n-1 + (x), s + t - n-2 j=0 f + σ j + x   ,
where n is the only integer such that

n-2 j=0 f + σ j + x ≤ s + t < n-1 j=0 f + σ j + x .
Remark. We underline that we used the fact that f + > 0 to define the suspension flow. If we had considered f , we would not have been able to construct the suspension flow as f could be equal to 0.

A notable difference with the Anosov setting is that we will not consider time 1 of the suspension of the flow. Instead of it, we fix a large integer N 0 (such that 5 ǫ ′ ≪ 1/N 0 ≪ ǫ 0 ) and consider time 1/N 0 of the flow and its iterates.

Remark. It can be underlined that the same procedure holds for the partition (π β ). The only differences are that we have to consider Σ

-:= {1, • • • , K} -N , σ -((x n ) n≤0 ) = (x n-1
) n≤0 and that the corresponding measure is, for k ≥ 1:

µ Σ-([β -k , • • • , β 0 ]) = µ Σ-σ -k -C β -k ∩ • • • ∩ C β0 = P β -k (-kη) • • • P β0 P β-1 (-η)ψ 2 .
For k = 0, one should take the only possibility to assure the compatibility condition:

µ Σ-([β 0 ]) = K j=1 µ Σ-([β -1 , β 0 ]) .
The definition is quite different from the positive case but in the semiclassical limit, it will not change anything as P β0 and P β-1 (-η) commute. Finally, the "past" suspension set can be defined as

Σ -:= {(x, s) ∈ Σ -× R + : 0 ≤ s < f -(x)}.
Now let α be an element of I η ( ). Define: we now have to apply the uncertainty principle to these partitions of variable size. The main difference with [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] is that we will apply it for quantum pressures (see section A). We introduce the weights

(17) C+ α := C α0 ∩ • • • ∩ σ -k + C α k . This new
W + α := exp   1 2 k-1 j=1 f (σ j + α)   and W - β := exp   1 2 k-1 j=1 f (σ j -β)   .
We underline that the weights depends on f and not f + or f -. It cames from the fact that f is the function that appears in theorem 4.1. We introduce the associated quantum pressure 6 :

(18) p µ Σ+ , C + := H µ Σ+ , C + -2 α∈I η ( ) µ Σ+ C + α log W + α and (19) p µ Σ-, C -:= H µ Σ-, C --2 β∈K η ( ) µ Σ-C - β log W - β .
We follow then the procedure of section 5 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] to apply the entropic uncertainty principle (i.e. apply it K 2 times and not 1 time as in [START_REF] Anantharaman | Entropy of eigenfunctions[END_REF]) and we use the main estimate on the norms of the quantum partitions (see theorem 4.1) to derive that 5 To summarize the relations between the different parameters, we have ǫ 4 < ǫ ′ ≪ 1 N 0 ≪ ǫ 0 . Moreover η depends on ǫ and ǫ 0 and tends to 0 when ǫ tends to 0 and ǫ 0 is fixed. 6 We refer the reader to appendix B for the definition of H.

(20) p µ Σ+ , C + + p µ Σ-, C -≥ -log C -(1 + ǫ ′ + 4ǫ)n E ( ),
where C is a constant that does not depend on .

Remark. This last inequality is a crucial step to prove theorem 1.1. We will recall how one can get such a lower bound in section 5. This inequality corresponds to proposition 5.3 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]. The strategy of the proof is exactly the same except that we have to deal with quantum pressures and not quantum entropies (see section 5). However, we can follow the same lines as in section 5.3.2 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] (i.e. apply K 2 times the uncertainty principle) and obtain a lower bound that depends on the bound from theorem 4.1. At this point, there is a difference because theorem 4.1 was proved in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] for Anosov manifolds. In section 4, we will show that the proof of theorem 4.1 from [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] can be adapted in the setting of nonpositively curved surfaces.

The problem of expression ( 20) is that it is not exactly the pressure of a refined partition. As in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], one can prove the following lemma: Lemma 3.1. Let N 0 be a positive integer defined as previously. There exists an explicit

C + N0 of Σ + , independent of such that ∨ nE ( )N0-1 i=0 σ -i N 0 + C + is a refinement of the partition C + .
Moreover, let n be a fixed positive integer. Then, an atom of the refined partition ∨ n-1 i=0 σ

-i N 0 + C + is of the form [α] × B(α), where α = (α 0 , • • • , α k ) is a k + 1-uple such that (α 0 , • • • , α k ) verifies n N0 (1 -ǫ) ≤ k-1 j=0 f + σ j + α ≤ n N 0 (1 + ǫ) and B(α) is a subinterval of [0, f + (α)[.
This lemma is the exact analogue of lemma 4.1 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] and its proof is the same: the only difference is that we consider times 1/N 0 instead of time 1. In particular, in the proof, the partition C + N0 is constructed from 7 I η (1/N 0 ) and not from I η (1). As in the Anosov case, we would like to use this lemma to rewrite the quantum pressure in terms of the pressure of a refined partition. To do this, we use basic properties of the classical entropy (see appendix B) to find that:

H µ Σ+ , C + ≤ H N0nE ( ) µ Σ+ , σ 1 N 0 + , C + N0 .
Consider now an atom A of the partition

∨ nE ( )N0-1 j=0 σ -j N 0 + C + N0 .
To this atom, it corresponds an unique family (γ 0 , • • • , γ nE ( )N0-1 ) in I η (1/N 0 ) N0nE ( ) and we define the corresponding weight as

W + A := N0nE ( )-1 j=0 W + γj .
With these notations, we introduce the refined pressure at times n:

p n µ Σ+ , σ 1 N 0 + , C + N0 := H n µ Σ+ , σ 1 N 0 + , C + N0 -2 A∈∨ n-1 j=0 σ - j N 0 C + N 0 µ Σ+ (A) log W + A .
One can then write the following inequality -2

α∈I η ( ) µ Σ+ C + α log W + α ≤ -2 A∈∨ N 0 n E ( )-1 j=0 σ - j N 0 C + N 0 µ Σ+ (A) log W + A + 2 b 0 ǫ 0 b 0 ηN 0 n E ( ).
The correction term in the last expression comes from the fact that, for each atom A in the partition ∨ N0nE ( )-1 j=0 σ -j N 0 C, one has an unique α ′ in I η ( ) and the corresponding W + α ′ is not exactly equal to W + A . Finally, the previous inequalities can be summarized as follows: (21)

-4 b 0 N 0 n E ( ) ǫ 0 ǫ-log C-(1+ǫ ′ +4ǫ)n E ( ) ≤ p nE ( )N0 µ Σ+ , σ 1 N 0 + , C + N0 +p nE ( )N0 µ Σ-, σ 1 N 0 -, C - N0 .
This estimate is crucial in our proof as we have derived from a quantum relation a lower bound on the classical pressure of a dynamical system associated to the geodesic flow. 7 We recall that I η (t) was defined as the set of words

n α = (α 0 , • • • , α k ) : k ≥ 3, P k-2 i=1 f + `σi + α ´≤ t < P k-1 i=1 f + `σi + α ´o.
3.3. Subadditivity of the quantum pressure. As in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], we would like to let tends to 0 in inequality [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF]. The main difficulty to do this is that everything depends on . So, once more, we have to prove a subadditivity property for the quantum pressure:

Theorem 3.2. Let C + N0 be the partition of lemma 3.1. There exists a function R(n 0 , ) on N×(0, 1] and R(N 0 ) independent of n 0 such that

∀n 0 ∈ N, lim sup →0 |R(n 0 , )| = R(N 0 ).
Moreover, for any ∈ (0, 1] and any n 0 , m ∈ N such that n 0 + m ≤ N 0 n E ( ), one has:

p n0+m µ Σ+ , σ 1 N 0 , C + N0 ≤ p n0 µ Σ+ , σ 1 N 0 , C + N0 + p m µ Σ+ , σ 1 N 0 , C + N0 + R(n 0 , ).
Proof. To prove this subadditivity property, we will prove subadditivity of the quantum entropy and subadditivity of the pressure term. As in section 6 from [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], we write for the entropy part that:

H n0+m µ Σ+ , σ 1 N 0 , C + N0 ≤ H σ m N 0 + ♯µ Σ+ , ∨ n0-1 j=0 σ -j N 0 C + N0 + H m µ Σ+ , σ 1 N 0 , C + N0 .
So, as in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], we have to show that the measure of the atoms of the partition are almost invariant under σ

1 N 0
+ for the range of times we have considered (proposition 6.1 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]). Consider now the pressure term in the quantum pressure. Using the multiplicative structure of the W + A , one has

A∈∨ n 0 +m-1 j=0 σ - j N 0 C + N 0 µ Σ+ (A) log W + A = A∈∨ m-1 j=0 σ - j N 0 C + N 0 µ Σ+ (A) log W + A + A∈∨ n 0 -1 j=0 σ - j N 0 C + N 0 σ m N 0 + ♯µ Σ+ (A) log W + A + A∈C + N 0 σ m N 0 + ♯µ Σ+ (A) log W + A .
So, once more, the additivity property of the pressure term derives from the almost invariance of the measure for the range of times we consider 8 . Precisely, according to the last two inequalities, we only need to verify that proposition 6.1 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] remains true for the partition C + N0 in the setting of surfaces of nonpositive curvature. We will not reproduce the proof here: it is the same one. We recall that this proposition relied on a theorem for products of pseudodifferential operators (theorem 7.1 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]) and we need to verify that the proof we gave still works in the case of surfaces of nonpositive curvature. The key point of the proof of this theorem is that in the allowed range of times |d ρ g t | is bounded by some -ν (with ν < 1/2) (see section 7.2 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]). We know that to each ρ we can associate a word α of length k. The range of times we will consider will be 0 ≤ t ≤ kη. To prove previous property in the case of surfaces of nonpositive curvature, we use the splitting of T ρ S * M given by RX H (ρ) ⊕ E u (ρ) ⊕ V ρ . These three subspaces are uniformly transverse so we only have to give an estimate of d ρ g t E→T * g t ρ M when E is one of them. In the case where E = RX H (ρ), it is bounded by 1 and in the case where

E = E u (ρ), it is bounded by √ 1 + K 0 e R t 0 U u ρ (s)ds .
In the last case, lemma 2.1 tells us that the spaces d ρ g t V ρ and E u (g t ρ) become uniformly close (in direction) to each other. Then, we consider e 0 a unit vector in V ρ and for 0 ≤ p ≤ k -1, we define the e pη as the unit vector dρg pη e0 dρg pη e0 . We can write:

d ρ g kη e 0 = | d ρ g kη e 0 , e kη | = | d g (k-1)η ρ g η e (k-1)η , e kη • • • d ρ g η e 0 , e η |.
We also define the corresponding sequence e u pη :=

dρg pη e u 0 dρg pη e u 0 of unit unstable vectors, where e u 0 :=

(J u ρ (0),J u ′ ρ (0)) (J u ρ (0),J u ′ ρ (0))
. From lemma 2.1, we know that e pη becomes uniformly close (in ρ) to e u pη . So, up to an error term of order Ce kηδ (with C uniform in ρ and δ arbitrarly small), we have:

d ρ g kη e 0 ≤ Ce kηǫ0ǫ | d g (k-1)η ρ g η e u (k-1)η , e u kη • • • d ρ g η e u 0 , e u η | = Ce kηδ d ρ g kη |E u (ρ)
. 8 We underline that R(N 0 ) will be equal to sup A∈C + N 0 log W + A which only depends on N 0 .

Finally, taking δ = ǫ 0 ǫ, we have that d ρ g kη is bounded by Ce kηǫ0ǫ e R kη 0 U u ρ (s)ds (with C uniform in ρ). For the allowed words, e kηǫ0ǫ is of order -ǫ (as kηǫ 0 ≤ 1/2n E ( )). To conclude, we can estimate:

kη 0 U u ρ (s)ds - k-1 j=0 f (σ j α) ≤ k-1 j=0 (j+1)η jη U u ρ (s)ds -f (σ j α) .
To bound this sum, we can use the continuity of U u (see inequality ( 4)) to show that this quantity is bounded by ǫ| log |. By definition of the allowed words α, we know that k-1 j=0 f (σ j α) ≤ 1/2n E ( ). This allows to conclude that |d ρ g t | is bounded by some C -ν (with C independent of ρ and ν < 1/2).

Remark. We underline that here we need to use the specific properties of surfaces of nonpositive curvature to prove this theorem. It is not really surprising that theorem 7.1 from [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] can be extended in our setting as the situation can only be less 'chaotic'. We also mention that we have to use the continuity of U u (ρ) which is for instance false for surfaces without conjugate points [START_REF] Ballmann | Burns On surfaces with no conjugate points[END_REF].

3.4. The conclusion.

3.4.1.

Applying the Abramov theorem. Thanks to the subadditivity property of the quantum pressure, we can proceed as in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] and write, for a fixed n 0 , the euclidean division N 0 n E ( ) = qn 0 + r. Using the same method, we find, after applying the subadditivity property and letting tends to 0,

-4 b 0 ǫ 0 ǫ - R(N 0 ) n 0 - 1 N 0 (1 + ǫ ′ + 4ǫ) ≤ 1 n 0 p n0 µ Σ+ , σ 1 N 0 + , C + N0 + p n0 µ Σ-, σ 1 N 0 -, C - N0
.

As in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], we can replace the smooth partitions by true partitions of the manifold in the previous inequality. We would like now to transform the previous inequality on the metric pressure into an inequality on the Kolmogorov-Sinai entropy. To do this, we write the multiplicative property of W A to write:

A∈∨ n 0 -1 j=0 σ - j N 0 + C + N 0 µ Σ + (A) log W + A = A0,••• ,An 0 -1∈C + N 0 µ Σ + (A 0 ∩ • • • ∩ σ -n 0 -1 N 0 A n0-1 ) n0-1 j=0 log W + Aj .
After simplification and using the fact C + N0 is a partition of Σ + , we find that this last inequality can be rewritten as follows

A∈∨ n 0 -1 j=0 σ - j N 0 + C + N 0 µ Σ + (A) log W + A = n 0 A∈C + N 0 µ Σ + (A) log W + A
The same property holds for the backward side. After letting n 0 tends to infinity, we find that:

-4 b 0 ǫ 0 ǫ - 1 N 0 (1 + ǫ ′ + 4ǫ) + 2    A∈C + N 0 µ Σ + (A) log W + A + A∈C - N 0 µ Σ - (A) log W - A    ≤ 1 N 0 h KS µ Σ+ , σ + + h KS µ Σ-, σ -.
We now underline that, by construction (see the proof of lemma 4.1 in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]) and by invariance of the measure µ Σ+ , one has:

A∈C + N 0 µ Σ + (A) log W + A + A∈C - N 0 µ Σ - (A) log W - A = 2 γ ′ ∈{1,••• ,K} 2 f 0 (γ ′ )µ Σ ([γ ′ ]) γ∈I η (1/N0) f 0 (γ)µ Σ ([γ]) log W γ .
We use this last property and combine it with the Abramov theorem [START_REF] Abramov | On the entropy of a flow[END_REF]. We find then

γ ′ ∈{1,••• ,K} 2 f 0 (γ ′ )µ Σ ([γ ′ ]) -2 b 0 N 0 ǫ 0 ǫ - 1 2 (1 + ǫ ′ + 4ǫ) +2N 0 γ∈I η (1/N0) f 0 (γ)µ Σ ([γ]) log W γ ≤ ηh KS (µ, g).

3.4.2.

The different small parameters tend to 0. We have obtained a lower bound on the Kolmogorv-Sinai entropy of the measure µ. This lower bound depend on several small parameters that are linked to each other in the following way:

ǫ < 4ǫ ′ ≪ 1 N 0 ≪ ǫ 0 .
Moreover the small parameter η depends on ǫ and ǫ 0 . For a fixed ǫ 0 , it tends to 0 when ǫ tends to 0. We have now to be careful to transform our lower bound on the entropy of µ into the expected lower bound. To do this, we use the notations of section 2.1 and introduce, for ρ ∈ S * M , the application F 0 (ρ) :=

γ∈I η (1/N0) f 0 (γ) log W γ 1 Oγ 0 (ρ) • • • 1 Oγ k • g kη (ρ).
We underline that for each ρ in S * M , there exists an unique γ in

I η (1/N 0 ) such that 1 Oγ 0 (ρ) • • • 1 Oγ k • g kη (ρ
) is non zero (it is then equal to 1). With this new function, the lower bound on the Kolmogorov-Sinai entropy can be rewritten as follows:

γ ′ ∈{1,••• ,K} 2 f 0 (γ ′ )µ Σ ([γ ′ ]) -2 b 0 N 0 ǫ 0 ǫ - 1 2 (1 + ǫ ′ + 4ǫ) + 2N 0 S * M F 0 (ρ)dµ(ρ) ≤ ηh KS (µ, g).
We define then

X 0 := ρ ∈ S * M : ∀0 ≤ t ≤ 1 N 0 ǫ 0 , U u (g t ρ) > 2ǫ 0 .
We can verify that

F 0 (ρ) ≥ (1/N 0 ) γ0,γ1 f 0 (γ)1 X0 (ρ)1 Oγ 0 (ρ)1 Oγ 1 • g η (ρ) for all ρ in E θ . In fact, one has, for ρ ∈ X 0 (otherwise the inequality is trivial), log W γ = 1 2 k-1 j=1 f (σ j γ), where ρ belongs to O γ0 ∩ • • • g -kη O γ k and γ satisfies k-2 j=1 f 0 (σ j γ) ≤ 1 N 0 < k-1 j=1 f 0 (σ j γ).
In particular, one has (k -2)ηǫ 0 ≤ 1/N 0 . Using the relation of continuity (4) and the fact that

U u 0 (g t ρ) = U u (g t ρ) on X 0 , one find that, for ρ ∈ X 0 ∩ O γ0 ∩ • • • g -kη O γ k , log W γ ≥ - 2ǫ 0 N 0 + 1 2 k-1 j=1 f 0 (σ j γ) ≥ 1 2 -2ǫ 0 1 N 0 .
We use this function 1 X0 (ρ) in our lower bound on the entropy of µ. We let the diameter of the partition tends to 0 and we divide by η. This gives us

-2 b 0 N 0 ǫ 0 ǫ - 1 2 (1 + ǫ ′ + 4ǫ) S * M U u 0 (ρ)dµ(ρ) + (1 -4ǫ 0 ) S * M U u 0 (ρ)1 X0 (ρ)dµ(ρ) ≤ h KS (µ, g).
Finally, we let ǫ and ǫ ′ tend to 0 (in this order). We find the following bound on the entropy of µ:

- 1 2 S * M U u 0 (ρ)dµ(ρ) + (1 -4ǫ 0 ) S * M U u 0 (ρ)1 X0 (ρ)dµ(ρ) ≤ h KS (µ, g).
We let now N 0 tend to infinity and then ǫ 0 tend to 0 (in this order). We find finally the expected lower bound:

1 2 S * M U u (ρ)dµ(ρ) ≤ h KS (µ, g).
4. Proof of the main estimate from [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] In the previous section, we have been able to apply the method we used for Anosov surfaces in order to prove theorem 1.1. As in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], the strategy relied on a careful adaptation of an uncertainty principle. In particular, to derive inequality [START_REF] Ruelle | An inequality for the entropy of differentiable maps[END_REF], we had to use the following equivalent of theorem 3.1 from [3]: Theorem 4.1. Let M be a surface of nonpositive sectional curvature and ǫ, ǫ 0 and η be small positive parameters as in section 2.2. For every K > 0 (K ≤ C δ0 ), there exists K and C K (ǫ, η, ǫ 0 ) such that uniformly for all ≤ K , for all k ≤ K| log |, for all α = (α 0 , • • • , α k ), ( 22)

P α k U η P α k-1 • • • U η P α0 Op (χ (k) ) L 2 (M) ≤ C K (ǫ, η, ǫ 0 ) -1 2 -cδ0 e 2kηǫ0ǫ exp   - 1 2 k-1 j=0 f (σ j + α)   ,
where c depends only on the riemannian manifold M .

Remark. We underline two facts about this theorem. The first one is that Op (χ (k) ) is a cutoff operator that was already defined in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] (section 5.3) and in the appendix of [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF]. We describe briefly its construction in section 5.1. The second one is that it is function f and not f + that appears in the upper bound.

This theorem is the analogue for surfaces of nonpositive of a theorem from [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF]. As the geometric situation is slightly different from [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], we will recall the main lines of the proof where the geometric properties appear and focus on the differences. We refer the reader to [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] for the details 9 . On [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], the proof of the analogue of theorem 4.1 (section 3 and more precisely corollary 3.5) relies on a study of the action of P α k U η P α k-1 • • • U η P α0 on a particular family of Lagrangian states. This reduction was possible because of the introduction of the cutoffs operators Op (χ (k) ) (see section 3 in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] for the details).

4.1. Evolution of a WKB state. Consider u (0, x) = a (0, x)e ı S(0,x) a Lagrangian state, where a (0, •) and S(0, •) are smooth functions on a subset Ω in M and a (0, •) ∼ k k a k (0, •). This represents a Lagrangian state which is supported on the Lagrangian manifold L(0) := {(x, d x S(0, x) : x ∈ Ω}. According to [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], if we are able to understand the action of

P α k U η P α k-1 • • • U η P α0
on Lagrangian states (with specific initial Lagrangian manifolds: see next paragraph), then we can derive our main theorem. A strategy to estimate this action is to use a WKB Ansatz. Recall that if we note ũ(t) := U t U (0), then, for any integer N , the state ũ(t) can be approximated to order N by a Lagrangian state u(t) of the form u(t, x) := e ı S(t,x) a (t, x) = e ı S(t,x)

N -1 K=0 k a k (t, x).
As u is supposed to solve ı ∆ 2 u = ∂ t u (up to an error term of order N ), we know that S(t, x) and the a k (t, x) satisfy several partial differential equations. In particular, S(t, x) must solve the Hamilton-Jacobi equation

∂S ∂t + H(x, d x S) = 0.
Assume that, on a certain time interval (for instance s ∈ [0, η]), the above equations have a well defined smooth solution S(s, x), meaning that the transported Lagrangian manifold L(s) = g s L(0) is of the form L(s) = {(x, d x S(s, x))}, where S(s) is a smooth function on the open set πL(s).

As in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], we shall say that a Lagrangian manifold L is "projectible" if the projection π : L → M is a diffeomorphism onto its image. If the projection of L to M is simply connected, this implies L is the graph of dS for some function S: we say that L is generated by S. Suppose now that, for s ∈ [0, η], the Lagrangian L(s) is "projectible". Then, this family of Lagrangian manifolds define an induced flow on M , i.e.

g t S(s) : x ∈ πL(s) → πg t (x, d x S(s, x)) ∈ πL(s + t). This flow satifies a property of semi-group as follows: g t S(s+τ ) • g τ S(s) = g t+τ S(s) . Using this flow, we define an operator that sends functions on πL(s) into functions on πL(s + t):

T t S(s) (a)(x) := a • g -t S(s+t) (x) J -t S(s+t) (x) 1 2
, 9 We assume the reader is familiar with the proof of [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF].

where J t S(s) (x) is the Jacobian of the map g t S(s) at point x (w.r.t. the riemannian volume). This operator allows to give an explicit expression for all the a k (t) [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], i.e. a k (t) := T t S(0) a 0 (0) and a k (t) := T t S(0) a k (0) +

t 0 T t-s S(s) ı∆a k-1 (s) 2 ds.
Regarding the details of the proof in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], we know that there are two main points where the dynamical properties of the manifold are used:

• the evolution of the Lagrangian manifold under the action of P α k U η P α k-1 • • • U η P α0 (section 3.4.1 in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF]); • the value of J t S(0) for large t (section 3.4.2 in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF]). We will discuss these two points in the two following paragraphs. We will recall what was proved for these two questions in section 3.4 of [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] and see how it can be translated in the setting of surfaces of nonpositive curvature. 4.2. Evolution of the Lagrangian manifolds. The first thing we need to understand is how the Lagrangian manifolds evolve under the action of the operator P α k U η P α k-1 • • • U η P α0 . According to [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], we know that the introduction of the cutoff operator Op (χ) implies that we can restrict our selves to a particular family of Lagrangian states. Precisely, we fix some small parameter η 1 and we know that they must be localized on a piece of Lagrangian manifold L 0 (0) which is included in the set ∪ |τ |≤η g τ S * z,η1 M (where S * z,η1 M := {(z, ξ) :

ξ 2 z = 1 + 2η 1 }).
If we follow the method developped in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], we are given a sequence of Lagrangian manifolds L j (0) as follows:

∀t ∈ [0, η], ∀j, L 0 (t) := g t L 0 (0) and L j (t) := g t L j-1 (η) ∩ T * Ω αj .

The manifold L j (0) is obtained after performing P αj U η P α k-1 • • • U η P α0 on the initial Lagrangian state. To show that the procedure from [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] is consistent (i.e. performing several WKB Ansatz), we need to verify that the Lagrangian manifold L j (t) does not develop caustics and remains "projectible". The only geometric properties which were used to derive these two properties were:

• M has no conjugate points (to derive that S j will not develop caustics);

• the injectivity radius is larger than 2 (to ensure the "projectible" property). In our setting, these two properties remain true (in particular, a surface of nonpositive curvature has no conjugate points [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF]). Finally, we undeline that, thanks to the construction of the strong unstable foliation for surfaces of nonpositive curvature, any vector in S * z,η1 M becomes uniformly close to the unstable subspace under the action of d ρ g t (see lemma 2.1). As a consequence, under the geodesic flow, a piece of sphere becomes uniformly close to the unstable foliation as j tends to infinity. This point is the main difference with [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF]. In fact, if we consider an Anosov geodesic flow, we have the stronger property that a piece of sphere becomes exponentially close to the unstable foliation, as j tends to infinity. However, we will check that this property is sufficient for our needs.

Remark. At this point of the proof, we can ask about an extension of these results to manifolds without conjugate points. According to [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF], the 'uniform divergence' property (given by lemma 2.1) is true for surfaces without conjugate points. We mention that this property fails in higher dimension for manifolds without conjugate points.

4.3.

Estimates on the induced Jacobian. As was already mentioned, the Jacobian J t S j of the map g t sj appears in the WKB expansion of a Lagrangian state evolved under the operator P αj U η P αj-1 • • • U η P α0 . Precisely, by iterating the WKB Ansatz, we have to estimate the following quantity (see equation 3.22 in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF]):

(23) J k (x) := J -η S k-1 (x)J -η S k-2 (g -η S k (x)) • • • J -η S 1 (g (-k+2)η S k (x)) 1 2
.

This Jacobian appears in each term of the WKB expansion of a Lagrangian state evolved under the operator P α k U η P α k-1 • • • U η P α0 (see the formulas for the a p ). It is necessary to provide a way to bound this quantity as it will appear in the control of every derivatives of the WKB expansion.

According to the proof in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], if we are able to bound uniformly this quantity, the bound we will obtain is the one that will appear in theorem 4.1. This point of the proof is the main difference with the proof in the Anosov case. So, our goal in this paragraph is to provide an upper bound on [START_REF] Sjöstrand | Zworski Asymptotic distribution of resonances for convex obstacles[END_REF]. This last quantity can be rewritten

J k (x) := exp 1 2 log J -η S k-1 (x) + log J -η S k-2 (g -η S k (x)) • • • + log J -η S 1 (g (-k+2)η S k (x)) .
As the Lagrangian L j become uniformly close to the unstable foliation when j tends to infinity, we know that, for every ε ′ > 0, there exists some integer j(η, ε ′ ) such that ∀j ≥ j(η, ε ′ ), ∀ρ = (x, ξ) ∈ L j (0), | log J -η S j (x)log J -η S u (ρ) (x)| ≤ ε ′ , where S u (ρ) generates the local unstable manifold at point ρ (which is a Lagrangian submanifold). Therefore, we find that there exists a constant C(ε ′ , η) (depending only on ε ′ and η) such that, uniformly with respect to k and to ρ in L k (0),

J k (x) ≤ C(ε ′ , η)e kε ′ k-1 j=0 J -η S u (g (-j+1)η ρ) (g (-j+1)η S k (x)) = C(ε ′ , η)e kε ′ J (1-k)η S u (ρ) (x).
The Jacobian J -η S u (ρ) measures the contraction of g -η along the unstable direction. From the construction of the unstable Riccati solution U u ρ (s), we know that U u ρ (s) also measures the contraction of g -η along E u (ρ). In fact, according to section 2.1, one has

d ρ g -t |E u (ρ) ≤ 1 + K 0 e R -t 0 U u ρ (s)ds .
As a consequence, there exists an uniform constant C (depending only on the manifold) such that:

J (1-k)η S u (ρ) (x) ≤ Ce R (1-k)η 0 U u ρ (s)ds .
Using then relation (4) between the discrete Riccati solution f and the continuous one, we find that there exists a constant C(ǫ, η, ǫ 0 ) such that, uniformly in k,

sup x∈πL k (0) J k (x) ≤ C(ǫ, η, ǫ 0 )e 2kηǫǫ0 exp   - 1 2 k-1 j=0 f (σ j α)   .
Finally, this last inequality gives us a bound on the quantity [START_REF] Sjöstrand | Zworski Asymptotic distribution of resonances for convex obstacles[END_REF]. This estimate is not as sharp as the one derived in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] (equation 3.23 for instance) however it is sufficient as the correction term is not too large: it is of order -ǫ .

Remark. We underline that we used the continuity of U u to go from the continuous representation of the upper bound of J k to the one in terms of the discrete Riccati solution. We underline again that this property fails for surfaces without conjugate points [START_REF] Ballmann | Burns On surfaces with no conjugate points[END_REF].

Applying the uncertainty principle for quantum pressures

In this section, we would like to prove inequality (20) which was a crucial step of our proof. To do this, we follow the same lines as in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] (section 5.3) and prove the following proposition: Proposition 5.1. With the notations of section 3, one has:

(24) p µ Σ+ , C + + p µ Σ-, C -≥ -log C -(1 + ǫ ′ + 4ǫ)n E ( ),
where p is defined by [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF] and where C ∈ R * + does not depend on (but depends on the other parameters (ǫ, ǫ 0 , η)).

To prove this result, we will proceed in three steps. First, we will introduce an energy cutoff in order to get the sharpest bound as possible in our application of the uncertainty principle. Then, we will apply the uncertainty principle and derive a lower bound on p µ

Σ+ , C + + p µ Σ-, C -.
Finally, we will use sharp estimates of theorem 4.1 to conclude.

5.1. Energy cutoff. Before applying the uncertainty principle, we proceed to sharp energy cutoffs so as to get precise lower bounds on the quantum pressure (as it was done in [START_REF] Anantharaman | Entropy and the localization of eigenfunctions[END_REF], [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] and [START_REF] Anantharaman | Entropy of eigenfunctions[END_REF]). These cutoffs are made in our microlocal analysis in order to get as good exponential decrease as possible of the norm of the refined quantum partition. This cutoff in energy is possible because even if the distributions µ are defined on T * M , they concentrate on the energy layer S * M . The following energy localization is made in a way to compactify the phase space and in order to preserve the semiclassical measure. Let δ 0 be a positive number less than 1 and χ δ0 (t) in C ∞ (R, [0, 1]). Moreover, χ δ0 (t) = 1 for |t| ≤ e -δ0/2 and χ δ0 (t) = 0 for |t| ≥ 1. As in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], the sharp -dependent cutoffs are then defined in the following way:

∀ ∈ (0, 1), ∀n ∈ N, ∀ρ ∈ T * M, χ (n) (ρ, ) := χ δ0 (e -nδ0 -1+δ0 (H(ρ) -1/2)).
For n fixed, the cutoff χ (n) is localized in an energy interval of length 2e nδ0 1-δ0 centered around the energy layer E. In this paper, indices n will satisfy 2e nδ0 1-δ0 << 1. It implies that the widest cutoff is supported in an energy interval of microscopic length and that n ≤ K δ0 | log |, where K δ0 ≤ δ -1 0 . Using then a non standard pseudodifferential calculus (see [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] for a brief reminder of the procedure from [START_REF] Sjöstrand | Zworski Asymptotic distribution of resonances for convex obstacles[END_REF]), one can quantize these cutoffs into pseudodifferential operators. We will denote Op(χ (n) ) the quantization of χ (n) . The main properties of this quantization are recalled in the appendix of [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]. In particular, the quantization of these cutoffs preserves the eigenfunctions of the Laplacian: Proposition 5.2. [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] For any fixed L > 0, there exists L such that for any ≤ L , any n ≤ K δ | log | and any sequence β of length n, the Laplacian eigenstate verify

1 -Op χ (n) π β ψ ≤ L ψ .

5.2.

Applying theorem A.1. Let ψ = 1 be a fixed element of the sequence of eigenfunctions of the Laplacian defined earlier, associated to the eigenvalue -1 2 . To get bound on the pressure of the suspension measure, the uncertainty principle should not be applied to the eigenvectors ψ directly but it will be applied several times. Precisely, we will apply it to each P γ ψ := P γ1 P γ0 (-η)ψ where γ = (γ 0 , γ 1 ) varies in {1, • • • , K} 2 . In order to apply the uncertainty principle to P γ ψ , we introduce new families of quantum partitions corresponding to each γ. Let γ = (γ 0 , γ 1 ) be an element of {1, • • • , K} 2 . We define γ.α ′ = (γ 0 , γ 1 , α ′ ). Introduce the following families of indices:

I (γ) := {(α ′ ) : γ.α ′ ∈ I η ( )} , K (γ) := {(β ′ ) : β ′ .γ ∈ K η ( )} .
We underline that each sequence α of I η ( ) can be written under the form γ.α ′ where α ′ ∈ I (γ). The same works for K η ( ). The following partitions of identity can be associated to these new families, for α ′ ∈ I (γ) and 2η). The families (τ α ′ ) α ′ ∈I (γ) and (π β ′ ) β ′ ∈I (γ) form quantum partitions of identity [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF]. Given these new quantum partitions of identity, the unceratinty principle should be applied for given initial conditions γ = (γ 0 , γ 1 ) in times 0 and 1. We underline that for α ′ ∈ I (γ) and β ′ ∈ K (γ): [START_REF]Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces[END_REF] τα ′ U -η P γ = τ γ.α ′ U -η and πβ ′ P γ = π β ′ .γ , where γ.α ′ ∈ I η ( ) and β ′ .γ ∈ K η ( ) by definition. In equality [START_REF]Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces[END_REF] appears the fact that the definitions of τ and π are slightly different (see [START_REF] Dimassi | Spectral Asymptotics in the Semiclassical Limit[END_REF] and ( 10)). It is due to the fact that we want to compose τ and π with the same operator P γ . Suppose now that P γ ψ is not equal to 0. We apply the quantum uncertainty principle A.1 using that • (τ α ′ ) α ′ ∈I (γ) and (π β ′ ) β ′ ∈K (γ) are partitions of identity;

β ′ ∈ K (γ), τα ′ = P α ′ n (nη) • • • P α ′ 2 (2η), πβ ′ = P β ′ -n (-nη) • • • P β ′ -2 (-
• the cardinal of I (γ) and K (γ) is bounded by N ≃ -K0 where K 0 is some fixed positive number (depending on the cardinality of the partition K, on a 0 , on b 0 and η); • Op(χ (k ′ ) ) is a family of bounded bounded operators O β ′ (where k ′ is the length of β ′ );

• the constants W + γ.α ′ and W - β.γ are bounded by -b 0 2ǫ 0 ;

• the parameter δ ′ can be taken equal to P γ ψ -1 L where L is such that L-K0-b 0 2ǫ 0 ≪ e 2kηǫǫ0 -1/2-cδ0 for every k ≪

W + γ.α ′ W - β ′ .γ τα ′ U -η π * β ′ Op(χ (k ′ ) ) .
Under this form, the quantity P γ ψ -1 appears several times and we would like to get rid of it. First, remark that the quantity c γ χ (U -η ) can be easily replaced by

W + γ.α ′ W - β ′ .γ τα ′ U -η π * β ′ Op(χ (k ′ ) ) ,
which is independent of γ. Then, one has the following lower bound:

(27) -2 log c γ χ (U -η ) + L-K0 P γ ψ -1 ≥ -2 log c χ (U -η ) + L-K0-b 0 2ǫ 0 + 2 log P γ ψ 2 .

as P γ ψ ≤ 1. Now that we have given an alternative lower bound, we rewrite the entropy term h τ (U -η ψ ) of the quantum pressure p τ (U -η ψ ) as follows:

h τ (U -η ψ ) = - α ′ ∈I (γ)
τα ′ U -η ψ 2 log τα ′ U -η P γ ψ 2 + α ′ ∈I (γ)

τα ′ U -η ψ 2 log P γ ψ 2 .

Using the fact that ψ is an eigenvector of U η and that (τ α ′ ) α ′ ∈I (γ) is a partition of identity, one has:

h τ (U -η ψ ) = - 1 P γ ψ 2 α ′ ∈I (γ) τ γ.α ′ ψ 2 log τ γ.α ′ ψ 2 + log P γ ψ 2 .
The same holds for the entropy term h π ( ψ ) of the quantum pressure p π( ψ ) (using here equality (25)):

h π ( ψ ) = - 1 P γ ψ 2 β ′ ∈K (γ) π β ′ .γ ψ 2 log π β ′ .γ ψ 2 + log P γ ψ 2 .
Combining these last two equalities with (27), we find that (28) -

α ′ ∈I (γ) τ γ.α ′ ψ 2 log τ γ.α ′ ψ 2 -2 α ′ ∈I (γ) τ γ.α ′ ψ 2 log W + γ.α ′ - β ′ ∈K (γ) π β ′ .γ ψ 2 log π β ′ .γ ψ 2 -2 β ′ ∈K (γ) π β ′ .γ ψ 2 log W - β ′ .γ ≥ -2 P γ ψ 2 log c χ (U -η ) + L-K0-b 0 2ǫ 0 .
This expression is very similar to the definition of the quantum pressure. We also underline that this lower bound is trivial in the case where P γ ψ is equal to 0. Using the following numbers:

(29) c γ.α ′ = c β ′ .γ = c γ = f (γ)

γ ′ ∈{1,••• ,K} 2 f (γ ′ ) P γ ′ ψ 2 , one can derive, as in [START_REF] Rivière | Entropy of semiclassical measures in dimension[END_REF], the following property: As expected, by a careful use of the entropic uncertainty principle, we have been able to obtain a lower bound on the pressures of the measures µ Σ+ and µ Σ-.

The conclusion.

To conclude the proof of proposition 5.1, we use theorem 4.1 to give an upper bound on c χ (U -η ). From our assumption on L, we know that L-K0-b 0 2ǫ 0 ≪ c χ (U -η ). As kη ≤ n E ( )/ǫ 0 , we also have that c χ (U -η ) ≤ C K (ǫ, η, ǫ 0 ) -1 2 -cδ0 e 4ǫnE ( ) . For δ 0 small enough, we find the expected property.

H(µ, Q) ≤ H(µ, P ). Otherwise, one denotes P ∨ Q := (P α ∩ Q β ) α∈I,β∈K their join (which is still a partition) and one has H(µ, P ∨ Q) ≤ H(µ, P ) + H(µ, Q) (subadditivity property). Let T be a measure preserving transformation of X. The n-refined partition ∨ n-1 i=0 T -i P of P with respect to T is then the partition made of the atoms (P α0 ∩ • • • ∩ T -(n-1) P αn-1 ) α∈I n . We define the entropy with respect to this refined partition: Using the subadditivity property of entropy, we have for any integers n and m:

(33) H n+m (µ, T, P ) ≤ H n (µ, T, P ) + H m (T n ♯µ, T, P ) = H n (µ, T, P ) + H m (µ, T, P ).

For the last equality, it is important to underline that we really use the T -invariance of the measure µ. A classical argument for subadditive sequences allows us to define the following quantity:

(34) h KS (µ, T, P ) := lim n→∞ H n (µ, T, P ) n .

It is called the Kolmogorov Sinai entropy of (T, µ) with respect to the partition P . The Kolmogorov Sinai entropy h KS (µ, T ) of (µ, T ) is then defined as the supremum of h KS (µ, T, P ) over all partitions P of X.

  ∀a ∈ C ∞ o (T * M ), µ (a) = T * M a(x, ξ)dµ (x, ξ) := ψ , Op (a)ψ L 2 (M) ,

1

 1

  family of subsets forms a partition of Σ + . Then, a partition C + of Σ + can be defined starting from the partition C and [0, f + (α)[. An atom of this suspension partition is an element of the form C + α = C+ α × [0, f + (α)[. For Σ -(the suspension set corresponding to Σ -), we define an analogous partition C -= ([β] × [0, f -(β)[) β∈K η ( ) . As in the case of the Anosov geodesic flows,

1 Corollary 5 . 3 .P γ ψ - 1 ,

 1531 ǫη | log | (see proposition 5.2 and the upper bound in theorem 4.1); • U -η is an isometry; • ψ := Pγ ψ Pγ ψ is a normalized vector.Applying the uncertainty principle A.1 for quantum pressures, one gets: Suppose that P γ ψ is not equal to 0. Then, one hasp τ (U -η ψ ) + p π ( ψ ) ≥ -2 log c γ χ (U -η ) + where c γ χ (U -η ) = max α ′ ∈I (γ),β ′ ∈K (γ)

  (26) c χ (U -η ) := max γ∈{1,••• ,K} 2 max α ′ ∈I (γ),β ′ ∈K (γ)

Corollary 5 . 4 .

 54 One has:(30) p µ Σ+ , C + + p µ Σ-, C -≥ -2 log c χ (U -η ) +

  (32)H n (µ, T, P ) = -|α|=n µ(P α0 ∩ • • • ∩ T -(n-1) P αn-1 ) log µ(P α0 ∩ • • • ∩ T -(n-1) P αn-1).

We refer the reader to[START_REF] Eberlein | Geodesic flows in manifolds of nonpositive curvature[END_REF] or[START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points[END_REF] for more details.

Appendix A. Uncertainty principle for the quantum pressure

In [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF], generalizations of the entropic uncertainty principle were derived for quantum pressures. We saw that the use of this thermodynamic formalism was crucial in our proof and we recall in this section the main results from [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] (section 6) on quantum pressures. Consider two partitions of identity (π k ) N k=1 and (τ j ) M j=1 on L 2 (M ), i.e.

We also introduce two families of positive numbers: (V k ) N k=1 and (W j ) M j=1 . We denote A := max k V k and B := max j W j . One can then introduce the quantum pressures associated to these families, for a normalized vector ψ in L 2 (M ),

The main result on these quantities that was derived in [START_REF] Anantharaman | Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold[END_REF] was theorem 6.5:

Theorem A.1. Under the previous setting, suppose U is an isometry of L 2 (M ) and suppose (O k ) N k=1 is a family of bounded operators. Let δ ′ be a positive number and ψ be a vector in

Appendix B. Kolmogorov-Sinai entropy

Let us recall a few facts about Kolmogorov-Sinai (or metric) entropy that can be found for example in [START_REF] Walters | An introduction to ergodic theory[END_REF]. Let (X, B, µ) be a measurable probability space, I a finite set and P := (P α ) α∈I a finite measurable partition of X, i.e. a finite collection of measurable subsets that forms a partition. Each P α is called an atom of the partition. Assuming 0 log 0 = 0, one defines the entropy of the partition as: Given two measurable partitions P := (P α ) α∈I and Q := (Q β ) β∈K , one says that P is a refinement of Q if every element of Q can be written as the union of elements of P and it can be shown that


A Boutayeb
email: ayoub.boutayeb@st.com

C Theodorou

D Golanski

P Batude

L Brunet

D Bosch

F Guyader

S Joblot

F Ponthenier

J Lacord

Extracting Edge Conduction around Threshold in mesa-isolated SOI MOSFETs

This work proposes a methodology to decompose the center and edge current contributions around threshold in mesa-isolated SOI MOSFETs using 3D TCAD simulations. Applied to pMOS measurements, it reveals that the subthreshold regime is driven by the active edge, whatever the device width. It also explains why the threshold voltage modulation by the back-gate bias depends on the device width, as well as why these effects are worse for high channel doping values.

Introduction

3D sequential integration (3DSI) consists in vertically stacking active device layers, in a successional manner [1]. For example, mesa-isolated stackable 2.5V Silicon-On-Insulator (SOI) devices used for imager applications were demonstrated in [2]. The mesa isolation process is an easy and cost-friendly solution compared to low temperature Shallow Trench Isolation STI. An unusual transfer characteristic has been already observed in LOCal Oxidation of Silicon (LOCOS) SOI MOSFETs [START_REF] Pretet | IEEE International SOI Conference[END_REF], and on mesa-isolated SOI MOSFETs with five terminals, for which the corner threshold voltage was predicted using a model that uses the body potential value [START_REF] Matloubian | [END_REF]. The specific behavior of these devices was explained in [5] by the irregular shape of the corners, the gate oxide thickness variation and the formation of a sharp beak at the edges, due to the accelerated corner oxide growth. A similar study done on Tri-Gate (TG) MOSFETs demonstrated that the low corner threshold voltage could be eliminated by using an undoped body [6]. In this work, the mesa impact on the subthreshold regime conduction in four terminals SOI MOSFETs is studied using an original method that isolates the active region's edge and center contributions. The dependence of those effects on the channel doping and on back-gate bias is investigated as well.

Device under test & Observations

The SOI p-MOSFET configuration was defined such as to fulfill the requirements for imager applications [2]: relatively high operating voltage (V dd = 2.5 V), SiO 2 -polysilicon gate stack, 23 nm silicon film thickness (T Si ), 25 nm buried oxide thickness (T BOX ), 6 nm SiO 2 gate oxide and 250 nm nominal gate length (L g ). A TEM scan image is shown in Figure 1.a.

Drain current versus gate voltage characteristic (I d -V g ) at V d = -0.1 V on undoped channel pMOS for various device widths (W) are presented in Figure 1.b. It can be observed that there is a current proportionality to the device width at each part of the curve, starting from the subthreshold regime going up to the strong inversion regime. The transconductance (g m = dI d /dV g ) derivative, dg m /dV g , is known to exhibit a local maximum around a MOSFET's threshold voltage. This behavior is verified here, because as shown in Figure 1.c, dg m /dV g shows one clear hump meaning that there is only one channel conduction.

However, a different behavior is observed for the I d -V g of the p-MOSFET with doped channel (around 8.10 17 at/cm 3 ) in linear regime, which was also measured for various device widths (W). The corresponding curves are presented in Figure 1.d. Contrary to the undoped channel device, we find the current proportionality to the width only in the strong inversion region, nevertheless the subthreshold regime is the same for all device widths. In fact, a knee appears for the wide channel devices, between the subthreshold and moderate inversion regions. As a result, the dg m /dV g plotted in Figure 1.e, exhibits two humps: this indicates two independent conductions at different threshold voltages, V t , the first around V g = -0.5 V and the second around V g = -0.7 V. The observed effects are more noticeable in the case of large devices. However, this is often related to parasitic channel conduction on active edge, which is more visible in narrow devices [7,8].

Based on the TEM cross section, we defined a geometrical 3D TCAD structure with the same thicknesses values, a constant channel doping (8.10 17 at/cm 3 ), and with a gate oxide and polysilicon all around the active region (Figure 2

Decomposition methodology

From the 3D TCAD simulations, we have analyzed the potential in the width direction and at 1nm below the gate oxide, to understand the reason behind the early edge conduction. We extracted the potential variation, from active edge to the center in subthreshold regime (V g =-0.1V) in Figure 3.a. For narrower devices, the potential has a "bell-shape" and its value in the center reduces with W. For W > 0.2 µm however, this value exhibits a plateau at ≈0.1µm from the edges, and across the center region of the channel.

So, if the device width is higher than 0.2µm (W>0.2µm), the mesa-isolated transistor can be seen as two transistors in parallel (Figure 3.b): the edge transistor with a constant, W-independent, current (in red) and the center one, similar to a planar SOI transistor, (in blue) (1), with width equal to W-2α, where 2α is the part of the top which is dominated by edge conduction.

I d tot = J d center W center + J d,edge W edge , W center =W-2α & W edge =2T Si +2α (1)
where I d,tot is the total current, J d,center the center transistor current density, J d,edge the edge transistor current density, and α is the center region portion governed by the edges. Thus, to evaluate each contribution, we can use the I d -V g curve of W min = 0.2 μm as a reference:

I d,edge = J d,edge W edge = [I d,tot (W) -I d,tot (W = 0.2) W-2𝛼 0.2-2𝑎 ] 0.2-2α 0.2-W (2)
Our methodology is used with 3D TCAD simulation on Figure 4 with α=90nm (W=0.2μm in Figure 3.a). It demonstrates that the subthreshold regime is fully driven by the edge transistor, regardless the device width. It shows as well that the narrower structure is closer to a tri-gate architecture and is completely driven by the edge transistor. Figure 4 explains as well that the knee appears only for wide devices, due to the difference in terms of current value at strong inversion between the several device widths W. Additional TCAD simulations are performed to check our decomposition methodology validity with channel doping and back gate voltage V b variation, and their impact on α. Figure 5 shows the potential variation in the active from the center to the edge (Δψ). Higher channel doping and/or higher V b increase the potential difference between center and edge: the V t difference between the 2 contributions increases and the double hump is more visible. The inset figures show also that α, extracted as the position where Δψ saturates and becomes negligible, i.e. smaller than -0.5mV, at the center of the device increases with increasing doping and V b , which indicates that the edge device width W edge increases. Moreover, it indicates that the "edge/center" transistor model is valid down to a minimum device width W min , which increases with increasing the back-gate bias and the channel doping.

The potential variation versus V g for several V b values (Figure 6.a), is plotted for both components: edge (at 10nm far from the edge and 1nm below the gate oxide) and center (at 100nm far from the edge and 1nm below the gate oxide). First, we consider the edge component, dash curves on Figure 6.a. The gate voltage at which we reach a threshold potential of -0.4V (where we are not impacted by the surface potential saturation), is extracted for each back-bias polarization (dashed curves). Then, the extracted values are used to extract the difference of potential to the extracted value for V b =0V "Δψ", in order to perform the comparison to the ΔV t extracted by our methodology in Then, the edge and center transistors I d -V g are plotted for α>0.04µm (Figure 7.b), from where we can conclude that if α is sufficiently high (α > α min ), the subthreshold regime is not impacted. This proves that our methodology can be used to analyze this regime.

Experimental application

The threshold voltage V t of doped channel devices was extracted at constant current (10 -7 (W/L)), with W the width of each current contributor) (Figure 8). The total current threshold voltage V t,tot for the narrowest structures is closer to that of edge component one V t,edge . It shows that V t,tot (total transistor) depends on W, and it is closer to V t,center (center component) for higher W, because the extraction is done on a zone above the knee. This reveals that the threshold voltage is not impacted by the parasitic edge transistor, for the largest device (W=10µm) when the extraction is performed at a constant current equal to 10 -7 (W/L), and the problem remains if the extraction is done at a lower threshold current. This shows that the knee caused by the parasitic edge transistor skews the V t extraction at a constant current. This also proves that the center transistor behaves like a classical planar SOI one, because the center transistor V t variation with V b corresponds to the theorical planar SOI trend from [START_REF] Lim | Threshold voltage of thin-film Silicon-on-insulator (SOI) MOSFET's[END_REF] (dashed curves in Figure 8).

For a pMOS with a doped channel device, the threshold voltage is then plotted versus the gate length for each component: total, edge and center in Figure 9. The difference of the threshold voltage to its value with a long channel device L=1µm is then plotted to quantify the short channel effects (ΔV t,SCE ). This demonstrates that the center transistor is the most impacted by the short channel effects.

Conclusions

In this paper, we have demonstrated the presence of a parasitic transistor at the edges of SOI pMOSFETs due to the mesa isolation. This results in an early conduction which impacts the subthreshold region of the transfer characteristic. We have presented a methodology to decompose edge and center contributions in mesa-isolated SOI pMOSFETs, in order to properly study the operation in threshold regime. It was revealed that the subthreshold regime is always driven by the edges in a pMOS transistor with a doped channel. This explains why the threshold voltage V t is less modulated by the back-bias voltage. These effects are worsened when increasing the back-bias voltage V b , or for higher channel dopant concentration as well. We have shown as well that the center transistor is more impacted by the short channel effects. Finally, the proposed model could be further developed to analyze the Floating Body effects, especially occurring in doped channel devices.

  .a): the structure is equivalent to a wide tri-gate MOSFET. Basic 3D TCAD simulations[START_REF]TCAD Sentaurus software -TCAD Sentaurus U[END_REF] were performed with only Poisson equation, drift-diffusion, SRH recombination and constant mobility. As our TCAD I d -V g and dg m /dV g for various device widths reproduce the trend of the experimental results concerning the behavior with varying W (Figure2.b & c), it means that we can safely assume that the Weffect is only related to electrostatic (charge) and not transport effects.

Figure 1 :Figure 2 :

 12 Figure 1 : a) mesa-isolated pMOS TEM cross section along W, and measured drain current and transconductance derivative (b,c) for undoped channel pMOS and doped channel pMOS (d,e), for various W and V d =-0.1V

Figure 3 :

 3 Figure 3 : a) Potential variation along W from 3D TCAD pMOS for various W and b) Device components schematic: Center and Edge operating in parallel a) b)

Figure 6 .

 6 b. At a second time the same work is done for the center component. This demonstrates that the V t extractions for the edge and the center follow the potential variations coming from 3D TCAD (Figure6.b).

Figure 5 :

 5 Figure 5: 3D TCAD pMOS W=0.5µm Δψ vs position along W (1nm below gate oxide at (V g =V t ) and extracted alpha value at Δψ=-0.5mV a) vs channel doping b) vs back gate bias

Figure 4 :Figure 6 :

 46 Figure 4 : 3D TCAD I d -V g in a linear regime (V d =-0.1V): Extraction of center and edge contributions for devices with 3 W values 0.2, 0.5 and 10µm

Figure 7 :

 7 Figure 7 : pMOS 3D TCAD: a) Edge device transconductance g m,edge -V g extracted for α (0 to 100nm) and b) pMOS 3D TCAD I d -V g at V d =-0.1V for α from 50 to 100nm Edge & Center transistor (W=0.5µm)

Figure 8 :

 8 Figure 8 : pMOS measurement. Extracted threshold voltage in a linear regime V t versus backgate polarization for edge, center and total transistor for several W values: a)W=0.5µm, b) W=10µm

Figure 9 :

 9 Figure 9: pMOS transistor W=1µm , a) extracted threshold voltage for each component total, edge and total versus gate length and b) difference of the threshold voltage to its value with a long channel device L=1µm