Distorted plane waves on manifolds of nonpositive curvature

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1 Distorted plane waves on manifolds of nonpositive curvature Maxime Ingremeau December 6, 2016 arxiv: v2 [math-ph] 3 Dec 2016 Abstract We will consider the high frequency behaviour of distorted plane waves on manifolds of nonpositive curvature which are Euclidean or hyperbolic near infinity, under the assumption that the curvature is negative close to the trapped set of the geodesic flow and that the topological pressure associated to half the unstable Jacobian is negative. We obtain a precise expression for distorted plane waves in the high frequency limit, similar to the one in [GN14] in the case of convex co-compact manifolds. In particular, we will show L loc bounds on distorted plane waves that are uniform with frequency. We will also show a small-scale equidistribution result for the real part of distorted plane waves, which implies sharp bounds for the volume of their nodal sets. 1 Introduction Consider a Riemannian manifold X, g) such that there exists X 0 X and R 0 > 0 such that X\X 0, g) and R d \B0, R 0 ), g eucl ) are isometric we shall say that such a manifold is Euclidean near infinity). The distorted plane waves on X are a family of functions E h x; ξ) with parameters ξ S d 1 the direction of propagation of the incoming wave) and h a semiclassical parameter corresponding to the inverse of the square root of the energy) such that and which can be put in the form h 2 1)E h x; ξ) = 0, 1.1) E h x; ξ) = 1 χ)e ix ξ/h + E out. 1.2) Here, χ Cc is such that χ 1 on X 0, and E out is outgoing in the sense that it is the image of a function in Cc R d ) by the outgoing resolvent h i0) 1. It can be shown cf. [Mel95]) that there is only one function E h ; ξ) such that 1.1) is satisfied and which can be put in the form 1.2). In [Ing15], the author studied the behaviour as h 0 of E h, under some assumptions on the geodesic flow Φ t : S X S X. The trapped set is defined as K := {x, ξ) S X; Φ t x, ξ) remains in a bounded set for all t R}. The main result of [Ing15] was that provided that K is non-empty, that the dynamics is hyperbolic close to K, and that some topological pressure is negative, then E h ; ξ) can be written as 1

2 a convergent sum of Lagrangian states associated to Lagrangian manifolds which are close to the unstable directions of the hyperbolic dynamics. In this paper, we will show that this result can be made more precise if we work on a manifold of nonpositive curvature. In this framework, we shall show that all the Lagrangian states which make up E h ; ξ) are associated to Lagrangian manifolds which can be projected smoothly on the base manifold X. As a consequence, we will deduce the following estimates on the C l norms of the distorted plane waves. We refer to section 2 for the general assumptions we need, and to section 3.2 for the theorem concerning the decomposition of distorted plane waves into a sum of Lagrangian states. In this section, we will also be able to describe the semiclassical measures associated to the distorted plane waves. Theorem 1.1. Let X, g) be a Riemannian manifold which is Euclidean near infinity. We suppose that X, g) has nonpositive sectional curvature, and that it has strictly negative curvature near K. We also suppose that Hypothesis 2.3 on topological pressure is satisfied. Let ξ S d 1, l N and χ Cc X). Then there exists C l,χ > 0 such that, for any h > 0, we have χe h, ξ) C l C l,χ h l. Small-scale equidistribution If x 0 X and r > 0, let us write Bx 0, r) for the geodesic ball centred at x 0 of radius r. The following result, which tells us that RE h has L 2 norm bounded from below on any ball of radius larger than Ch for C large enough, can be seen as a small-scale equidistributuon result. Note that the upper bound in 1.3) is just a consequence of Theorem 1.1. Theorem 1.2. Let X, g) be a Riemannian manifold which is Euclidean near infinity. We suppose that X, g) has nonpositive sectional curvature, and that it has strictly negative curvature near K. We also suppose that Hypothesis 2.3 on topological pressure is satisfied. Let ξ S d 1, and χ Cc X). There exist constants C, C 1, C 2 > 0 such that the following holds. For any x 0 X such that χx 0 ) = 1, for any sequence r h such that 1 r h > Ch, we have for h small enough: C 1 rh d RE h 2 x)dx C 2 rh. d 1.3) Bx 0,r h ) In particular, for any bounded open set U X, there exists cu) > 0 and h U > 0 such that for all 0 < h < h U, we have RE h 2 cu). 1.4) U Here, we could have considered the imaginary part of E h instead, or even E h itself, and we would have obtained the same result. Nodal sets Theorem 1.2 has interesting applications to the study of the nodal volume of the real part of the distorted plane waves. Namely, let us fix a compact set K and a ξ S d 1 and consider N K,h := {x K; RE h )x, ξ) = 0}. Then we have the following estimate. We refer once again to section 3.2 for the precise assumptions we make. 2

3 Corollary 1.1. We make the same assumptions as in Theorem 1.1. Then there exist C K, C K > 0 such that C K h Haus d 1N K,h ) C K h, 1.5) where Haus d 1 denotes the d 1)-dimensional Hausdorff measure. Here, again we could have considered the imaginary part of E h instead, and we would have obtained the same result. The lower bound in 1.5) could be deduced from [Log16b], but we will give a proof of this fact in section 6.3 since it is easy to deduce from 1.4). As for the upper bound, it follows from the recent work [Hez16a], which says precisely that the upper bound in 1.5) can be deduced from small-scale equidistribution 1.3). We refer to this paper for more details. For other applications of small-scale equidistribution, see [Hez16b] and [Hez16c]. Comparison of the results with the case of compact manifolds Nodal sets, small-scale behaviour and L p norms of eigenfunctions of the Laplace-Beltrami operator on compact manifolds have been actively studied recently. Let us recall what is known in this framework. Let X, g) be a d-dimensional closed compact, without boundary) manifold. Then there exists a sequence h n of positive numbers going to zero and an L 2 orthonormal basis of real-valued) eigenfunctions φ n such that h 2 n φ n = φ n. The following estimate on the L p norm of φ n was proven in [Sog88]: ) p φ n L p C h σp) n where σp) = d 1 d 1 2, and σp) = 2 d p estimates are sharp if no further assumption is made on the manifold. if 2 p 2d+1) d 1, if 2d+1) d 1 p. These However, if X, g) has negative curvature, these bounds were slightly improved in [HR14], [HT15] and [Sog15]. We refer to these papers and to the references therein for precise statements and for more historical background. These estimates are far from showing that φ n is bounded uniformly in L : actually, it is not clear if such a bound should hold. The estimates given by Theorem 1.1 for distorted plane waves are therefore much better than what is available in the case of compact manifolds. Small scale equidistribution results, similar to 1.3) were obtained in [Han15] and in [HR14] for a density one sequence of eigenfunctions on the Laplace-Beltrami operator on compact manifolds of negative curvature and for sequences of the form r h Cα log h for any 0 < α < 1/2d). Small scale α equidistribution results similar to 1.3) were also obtained on the torus see [LR15] and the references therein). We may conjecture that on compact manifolds of negative curvature, an inequality like 1.3) should hold for any sequence r h >> h, for some density 1 subsequence of eigenfunctions of the Laplace-Beltrami operator. As for nodal sets, let us write N hn := {x X; φ n x) = 0}. The following conjecture was made in [Yau93]. Conjecture Yau). There exists c 1, c 2 > 0 such that c 1 h n Haud d 1 N hn ) c 2 h n. 3

4 This conjecture was proven for analytic manifolds in [DF88], and the lower bound was proven in dimension 2 in [Brü78]. A breakthrough was made recently in [Log16b] and [Log16a], where the lower bound was proved and a polynomial upper bound was established respectively, in any dimension. Corollary 1.1 can be seen as an analogue of Yau s conjecture in the case of non-compact manifolds. Relation to other works An important part of this paper consists in describing the semiclassical measures associated to distorted plane waves. It can thus be considered as a partial) generalization of [GN14], where the authors describe the semiclassical measures associated to eigenfunctions of the Laplace-Beltrami operator on manifolds of infinite volume, with sectional curvature constant equal to 1 convex co-compact hyperbolic manifolds). While the proofs in [GN14] rely heavily on the quotient structure of constant curvature hyperbolic manifolds, our proofs are based on the properties of the classical dynamics, being thus more versatile. Semiclassical measures associated to distorted plane waves were studied in [DG14] under very general assumptions, and we follow their approach on many points. Semiclassical measures for Eisenstein series associated to specral parameters away from the spectrum of the Laplacian were also studied in [Dya11] and [Bon14] on noncompact manifolds with finite volume manifolds with cusps) using methods similar to ours. In all these papers, distorted plane waves are seen as the propagation in the long-time limit of usual plane waves or hyperbolic waves). However, the reason for the convergence in the long time limit is very different in these papers: in [Dya11] and [Bon14], the convergence happens because the energy parameter is away from the real axis. In [DG14], convergence occurs because the authors average on all directions and on a small energy layer. In this paper, just as in [Ing15], convergence takes place because of a topological pressure assumption, as in [NZ09]. We will often use the methods and results of [NZ09] to take advantage of the hyperbolicity and topological pressure assumptions. Nodal sets of distorted plane waves on manifolds of infinite volume were studied for the first time in [JN15] in the framework of Eisenstein series on convex co-compact hyperbolic surfaces. Since convex co-compact manifolds are analytic, the proof of [DF88], which is purely local, gives an analogue of Yau s conjecture. The main results in [JN15] concern the counting of the number of intersections of the nodal sets N K,h with a given geodesic. Some of the results in [JN15] should still work on a manifold on nonpositive curvature under the assumptions made in this paper. This will be pursued elsewhere. Organisation of the paper In section 2, we will state the general assumptions we need on the manifold X, g) and on the generalised eigenfunctions E h. In section 3, we will recall the main results from [Ing15], and state the new results we obtain. In section 4, we will prove results about the propagation of Lagrangian manifolds. In section 5, we will prove results about distorted plane waves, including Theorem 1.1. Finally, section 6 is devoted to the proof of Theorem 1.2, and of Corollary 1.1. Appendix A simply recalls a few classical facts from semiclassical analysis. In Appendix B, we show that the general hypotheses formulated in [Ing15] and used in the present paper are fulfilled on manifolds that are hyperbolic near infinity. Acknowledgements The author would like to thank Stéphane Nonnenmacher for his supervision and advice during this work. He would also like to thank Frédéric Naud for suggesting to study nodal sets of distorted plane waves, as well as for fruitful discussion during the writing of this 4

5 paper. Finally, the author would like to thank the anonymous referee for his many suggestions and comments. The author is partially supported by the Agence Nationale de la Recherche project GeRaSic ANR-13-BS ). 2 General assumptions In this section, we will state the main assumptions under which our results apply. The assumptions in section 2.1 concern the background manifold X, g), while the assumptions in section 2.2 concern the distorted plane waves. Most of these assumptions were already made in [Ing15], in the framework of potential scattering. The additional assumptions which allow us to obtain more precise results than those of [Ing15] were regrouped in sections and Assumptions on the manifold Let X, g) be a noncompact complete Riemannian manifold of dimension d, and let us denote by p the classical Hamiltonian p : T X x, ξ) ξ 2 x R. For each t R, we denote by Φ t : T X T X the geodesic flow generated by p at time t. We will write by the same letter its restriction Φ t : S X S X to the energy layer px, ξ) = 1. Given any smooth function f : X R, it may be lifted to a function f : T X R, which we denote by the same letter. We may then define f, f C T X) to be the derivatives of f with respect to the geodesic flow. fx, ξ) := d fφ t d x, ξ)), fx, 2 t=0 ξ) := dt t=0 dt 2 fφ t x, ξ)) Hypotheses near infinity We suppose the following conditions are fulfilled. Hypothesis 2.1 Structure of X near infinity). We suppose that the manifold X, g) is such that the following holds: 1) There exists a compactification X of X, that is, a compact manifold with boundaries X such that X is diffeomorphic to the interior of X. The boundary X is called the boundary at infinity. 2) There exists a boundary defining function b on X, that is, a smooth function b : X [0, ) such that b > 0 on X, and b vanishes to first order on X. 3) There exists a constant ɛ 0 > 0 such that for any point x, ξ) S X, if bx, ξ) ɛ 0 and ḃx, ξ) = 0 then bx, ξ) < 0. Part 3) in the hypothesis implies that any geodesic ball with a large enough radius is geodesically convex. Example 2.1. R d fulfils the Hypothesis 2.1, by taking the boundary defining function bx) = 1 + x 2 ) 1/2. X can then be identified with the closed unit ball in R d. Example 2.2. The Poincaré space H d also fulfils the Hypothesis 2.1. Indeed, in the ball model B 0 1) = {x R d ; x < 1}, where denotes the Euclidean norm, then H d compactifies to the closed unit ball, and the boundary defining function bx) = 2 1 x 1+ x fulfils conditions 2) and 3). 5

6 Example 2.3. Let e 1 R 2, and consider X = R 2 /Ze 1 ), the flat two-dimensional cylinder. It may be compactified in the e 2 -direction by setting bx) = 1 + x 2 2 ) 1/2. X may then be identified with S 1 S 1, and b is a boundary defining function. However, part 3) of the hypothesis is not satisfied. Indeed, for any ɛ > 0, the set b = ɛ contains a closed geodesic whose trajectory is just a circle). On this geodesic, we have ḃ = 0 and b = 0. We will write X 0 := {x X; bx) ɛ 0 /2}. We will call X 0 the interaction region. We will also write V 0 := T X\X 0 ) = {ρ T X; bρ) < ɛ 0 /2}. 2.1) By possibly taking ɛ 0 smaller, we may ask that ρ S X\X 0 ), bφ 1 ρ)) < ɛ ) Definition 2.1. If ρ = x, ξ) S X, we say that ρ escapes directly in the forward direction, denoted ρ DE +, if bx) < ɛ 0 and ḃx, ξ) 0. If ρ = x, ξ) S X, we say that ρ escapes directly in the backward direction, denoted ρ DE, if bx) < ɛ 0 and ḃx, ξ) 0. Note that we have {ρ S X; bρ) < ɛ 0 } = DE DE Hyperbolicity Let us now describe the hyperbolicity assumption we make. For ρ S X, we will say that ρ Γ ± if {Φ t ρ), ±t 0} is a bounded subset of S X; that is to say, ρ does not go to infinity, respectively in the past or in the future. The sets Γ ± are called respectively the outgoing and incoming tails. The trapped set is defined as K := Γ + Γ. It is a flow invariant set, and it is compact by the geodesic convexity assumption. Hypothesis 2.2 Hyperbolicity of the trapped set). We assume that K is non-empty, and is a hyperbolic set for the flow Φ t. That is to say, there exists an adapted metric g ad on a neighbourhood of K included in S X, and λ > 0, such that the following holds. For each ρ K, there is a decomposition T ρ S X) = R Φ t ρ) ) E ρ + Eρ t such that dφ t ρv) gad e λ t v gad for all v Eρ, ±t 0. The spaces E ± ρ are respectively called the unstable and stable spaces at ρ. We may extend g ad to a metric on S X, so that outside of the interaction region, it coincides with the restriction of the metric on T X induced from the Riemannian metric on X. From now on, we will denote by d ad the Riemannian distance associated to the metric g ad on S X. 6

7 Figure 1: A surface which has negative curvature close to the trapped set of the geodesic flow, and which is isometric to two copies of R 2 \B0, R 0 ) outside of a compact set. It satisfies Hypothesis 2.2 near the trapped set which consists of a single geodesic) and Hypothesis 2.1 near infinity. by Any ρ K admits local strongly un)stable manifolds W ± ɛ ρ), defined for ɛ > 0 small enough W ± ɛ ρ) = {ρ E; dφ t ρ), Φ t ρ )) < ɛ for all ± t 0 and lim t dφt ρ ), Φ t ρ)) = 0}. Note that E ± ρ is the d 1)-dimensional) tangent space of W ± ɛ at ρ. We also define the weakly unstable manifolds by We call W ±0 ɛ ρ) = {ρ E; dφ t ρ), Φ t ρ )) < ɛ for all ± t 0}. E +0 ρ := E + ρ RH p ρ), E 0 ρ := E ρ RH p ρ), the weak unstable and weak stable subspaces at the point ρ respectively, which are respectively the tangent spaces to W ±0 ɛ at ρ. Adapted coordinates To state our result concerning the propagation of Lagrangian manifolds, we need adapted coordinates close to the trapped set. These coordinates, constructed in [Ing15, Lemma 2], satisfy the following properties: 7

8 For each ρ K, we build an adapted system of symplectic coordinates y ρ, η ρ ) on a neighbourhood of ρ in T X, such that the following holds. For j = 1,..., d 1, write i) ρ 0, 0) ii) E ρ + = span{ y ρ ρ), i i = 2,..., d}, iii) Eρ = span{ η ρ ρ), i i = 2...d}, iv) η ρ 1 = p 1 is the energy coordinate, v) y ρ ρ), i y ρ ρ) g ad ρ) = δ i,j, j i, j = 2,..., d. u ρ j := yρ j+1, and sρ j := ηρ j ) Let us now introduce unstable Lagrangian manifolds, that is to say, Lagrangian manifolds whose tangent spaces form small angles with the unstable space at ρ. Definition 2.2. Let Λ S X be an isoenergetic Lagrangian manifold not necessarily connected) included in a small neighbourhood W of a point ρ K, and let γ > 0. We will say that Λ is a γ-unstable Lagrangian manifold or that Λ is in the γ-unstable cone) in the coordinates y ρ, η ρ ) if it can be written in the form Λ = {y ρ 1, uρ, 0, F y ρ 1, uρ )); y ρ 1, uρ ) D}, where D R d, is an open subset with finitely many connected components, and with piecewise smooth boundary, and F : R d R d is a smooth function with df C 0 γ. Let us note that, since Λ is isoenergetic and is Lagrangian, an immediate computation shows that F does not depend on y ρ 1, so that Λ can actually be put in the form Λ = {y ρ 1, uρ, 0, fu ρ )); y ρ 1, uρ ) D}, where f : R d 1 R d 1 is a smooth function with df C 0 γ. Note that, since f is defined on R d 1, a γ-unstable manifold may always be seen as a submanifold of a connected γ-unstable Lagrangian manifold Topological pressure We shall now give a definition of topological pressure, so as to formulate Hypothesis 2.3. Recall that the distance d ad was defined in section 2.1.2, and that it was associated to the adapted metric. We say that a set S K is ɛ, t)-separated if for ρ 1, ρ 2 S, ρ 1 ρ 2, we have d ad Φ t ρ 1 ), Φ t ρ 2 )) > ɛ for some 0 t t. Such a set is necessarily finite.) The metric g ad induces a volume form Ω on any d-dimensional subspace of T T R d ). Using this volume form, we will define the unstable Jacobian on K. For any ρ K, the determinant map Λ n dφ t ρ) E +0 ρ : Λ n E +0 ρ Λ n E +0 Φ t ρ) 8

9 can be identified with the real number det ) dφ t ) Ω Φ ρ) E +0 ρ := t ρ) dφ t v 1 dφ t v 2... dφ t v n, Ω ρ v 1 v 2... v n ) where v 1,..., v n ) can be any basis of E +0 ρ From there, we take. This number defines the unstable Jacobian: ). 2.4) exp λ + t ρ) := det dφ t ρ) E +0 ρ Z t ɛ, s) := sup S exp sλ + t ρ)), where the supremum is taken over all ɛ, t)-separated sets. The pressure is then defined as ρ S 1 Ps) := lim lim sup ɛ 0 t t log Z tɛ, s). This quantity is actually independent of the volume form Ω and of the metric chosen: after taking logarithms, a change in Ω or in the metric will produce a term O1)/t, which is not relevant in the t limit. Hypothesis 2.3. We assume the following inequality on the topological pressure associated with Φ t on S X: P1/2) < ) Additional assumptions on the manifold In order to obtain stronger results than in [Ing15], we will need the following additional assumptions on X, g). Hypothesis 2.4. From now on, we will suppose that i) X, g) has nonpositive sectional curvature. ii) The sectional curvatures are all bounded from below by some constant b 0, with b 0 0, ). iii) The injectivity radius goes to infinity at infinity, in the following sense : for all sequences of points x n ) X such that bx n ) goes to 0, we have r i x n ). Recall that if x X, the injectivity radius of x, denoted by r i x), is the largest number r > 0 such that the exponential map at x is injective on the open ball B0, r). On a manifold of nonpositive curvature, saying that r i x) < means that there exists y X such that d X x, y) = r i x) and there exist two different unite-speed minimizing geodesics from x to y. Part iii) of Hypothesis 2.4 implies 1 that r i := inf x X r ix) > ) Example 2.4. The manifold represented on Figure 1 is a simple example of a manifold which fulfils Hypothesis 2.4. Example 2.5. Any quotient of the hyperbolic space by a convex co-compact group of isometries satisfies Hypothesis 2.4. If we perturb slightly the metric on a compact set of such a manifold, it will still satisfy Hypothesis 2.4. Manifolds which are hyperbolic near infinity will be considered in more detail in Appendix B. 1 Actually, one can easily show that 2.6) is implied by Hypothesis 2.1 and part i) of Hypothesis

10 2.2 Hypotheses on the distorted plane waves Hypotheses on the incoming Lagrangian manifold Let us consider an isoenergetic Lagrangian manifold L 0 S X of the form L 0 := {x, ϕx)), x X 1 }, where X 1 is a closed subset of X with finitely many connected components and piecewise smooth boundary, and ϕ : X 2 x ϕx) T x X is a smooth co-vector field defined on some neighbourhood X 2 of X 1. We make the following additional hypothesis on L 0 : Hypothesis 2.5 Invariance hypothesis). We suppose that L 0 satisfies the following invariance properties. t 0, Φ t L 0 ) DE = L 0 DE. 2.7) t 0, Φ t L 0 ) DE + = L 0 DE ) Example 2.6. Suppose that X\X 0, g) = R d \B0, R), g Eucl ) for some R > 0. Given a ξ R d with ξ 2 = 1, the Lagrangian manifold fulfils Hypothesis 2.5. S X Λ ξ := {x, ξ); x / X 0 } Example 2.7. Suppose that X\X 0, g) = R d \B0, R), g Eucl ) for some R > 0. Then the incoming spherical Lagrangian, defined by fulfills Hypothesis 2.5. Λ sph := {x, x ); x > R}, x We also make the following transversality assumption on the Lagrangian manifold L 0. It roughly says that L 0 intersects the stable manifold transversally. Hypothesis 2.6 Transversality hypothesis). We suppose that L 0 is such that, for any ρ K, for any ρ L 0, for any t 0, we have for ɛ > 0 small enough, Φ t ρ ) W ɛ ρ) = W ɛ ρ) and Φ t L 0 ) intersect transversally at Φ t ρ ), that is to say T Φt ρ )L 0 T Φt ρ )W ɛ ρ) = T Φt ρ )S X. 2.9) Note that 2.9) is equivalent to T Φt ρ )L 0 T Φt ρ )W ɛ ρ) = {0}. In general, this assumption is not easy to check. However, we will show in Proposition 3.1 that it is always satisfied if the hypotheses of sections and are satisfied. 10

11 2.2.2 Assumptions on the generalized eigenfunctions We consider a family of smooth functions E h C X) indexed by h 0, 1] which satisfy where P h 1)E h = 0, P h = h 2 c 0 h 2. Here, c 0 > 0 is a constant which is equal to 0 in the case of Euclidean near infinity manifolds, and to d 1) 2 /4 on manifolds that are hyperbolic near infinity. We will furthermore assume that these generalized eigenfunctions may be decomposed as follows. For the definitions of Lagrangian states, tempered distributions and wave-front sets, we refer the reader to Appendix A Hypothesis 2.7. We suppose that E h can be put in the form E h = E 0 h + E 1 h, 2.10) where Eh 0 is a Lagrangian state associated to a Lagrangian manifold L 0 which satisfies Hypothesis 2.5 of invariance, as well as Hypothesis 2.6 of transversality, and where Eh 1 is a tempered distribution such that for each ρ W F h Eh 1), we have ρ S X. Furthermore, we suppose that Eh 1 is outgoing in the sense that there exists ɛ 2 > 0 such that for all x, ξ) T X such that bx) < ɛ 2, we have ρ W F h E 1 h) ρ DE ) Remark 2.1. Note that 2.11) implies that for any χ C c X), we may find ˆχ C c X) such that ˆχ 1 on suppχ) {x X; bx) ɛ 2 }, and such that the support of ˆχ1 ˆχ) is small enough so that for any t 1, we have We will often use this consequence of 2.11). Φ t W F h 1 ˆχ)E 1 h ) ) T suppˆχ) =. 2.12) Example 2.8. In [Ing15], it is explained how distorted plane waves enter in this framework on manifold that are Euclidean near infinity. In Appendix B, we will show that distorted plane waves on manifolds that are hyperbolic near infinity do also satisfy this assumption Additional assumptions on the Lagrangian manifold L 0 From now on, we will denote by d X the Riemannian distance on the base manifold. It should not be confused with the distance d ad on the energy layer which was introduced in section 2.1.2, and which we will sometimes use too. If ρ, ρ T X, we will write d X ρ, ρ ) for dπ X ρ), π X ρ )), where π X denotes the projection on the base manifold. We will need the following assumptions on the incoming Lagrangian manifold L 0. First of all, we require that L 0 does not expand when propagated in the past. Hypothesis 2.8. We suppose that L 0 is such that ρ 1, ρ 2 L 0, t 0, d X Φ t ρ 1 ), Φ t ρ 2 )) d X ρ 1, ρ 2 ). 11

12 Example 2.9. Without Hypothesis 2.8, Theorem 1.1 might not be satisfied. For example, on X = R d the Lagrangian manifold Λ circ from Example 2.7 does not satisfy part of Hypothesis Consider E h x) := e h S iθ x/h dθ. One can show, by stationary phase see [Mel95, 2]) d 1)/2 d 1 that C E h x) = e ir/h + e ir/h) + R x d 1)/2 h, 2.13) where R h x) goes to zero when h/ x goes to zero. If χ Cc R d ) is such that χx) = 1 for x R for some R > 0, then Eh 0 x) := 1 χx))ei x /r is a Lagrangian state associated to Λ circ. Eh 1 := E h Eh 0 is a tempered distribution, and Hypothesis C 2.7 is satisfied. However, E h 0) =, so that E h d 1)/2 h is not bounded in L independently of h. Therefore, Hypothesis 2.8 is essential for Theorem 1.1 to hold. For Theorem 1.2, we also require a sort of completeness assumption for L 0, which is as follows. Note that Hypothesis 2.9 is not required for Theorem 1.1 to hold. Hypothesis 2.9. We suppose that L 0 is such that for all ρ DE, we have [ ] ρ L 0, t 0, d X Φ t ρ), Φ t ρ )) d X ρ, ρ ) = ρ L 0. Beware that if L 0 satisfies Hypothesis 2.9, a subset of L 0 may not satisfy Hypothesis 2.9, even if it satisfies the invariance property of Hypothesis Main results In this section, we state our main results concerning distorted plane waves on manifolds of nonpositive curvature. Before doing so, we recall the main results of [Ing15], so as to introduce some useful notations, and since we will need them in the proofs in sections 4 and Recall of the main results from [Ing15] Let us recall the main result from [Ing15]. The definitions of pseudo-differential operators and of Fourier integral operators are recalled in appendix A. Theorem 3.1. Suppose that the manifold X, g) satisfies Hypothesis 2.1 near infinity, and that the geodesic flow Φ t ) satisfies Hypothesis 2.2 on hyperbolicity and Hypothesis 2.3 concerning the topological pressure. Let E h be a generalized eigenfunction of the form described in Hypothesis 2.7, where Eh 0 is associated to a Lagrangian manifold L 0 which satisfies the invariance Hypothesis 2.5 as well as the transversality Hypothesis 2.6. Then there exists a finite set of points ρ b ) b B1 K and a family Π b ) b B1 of operators in X) microsupported in a small neighbourhood of ρ b such that b B 1 Π b = I microlocally on a neighbourhood of K in T X such that the following holds. Let U b : L 2 X) L 2 R d ) be a Fourier integral operator quantizing the symplectic change of local coordinates κ b : x, ξ) y ρ b, η ρ b ), and which is microlocally unitary on the microsupport of Π b. For any r > 0 and l N, there exists M r,l > 0 such that we have as h 0: Ψ comp h U b Π b E h y ρ b ) = M r,l log h n=0 β B n e iφβ,by ρ b )/h a β,b y ρ b ; h) + R r,l, 3.1) 12

13 where the a β,b S comp R d ) are classical symbols in the sense of Definition A.1, and each φ β,b is a smooth function independent of h, and defined in a neighbourhood of the support of a β,b. Here, B n is a set of words with length close to n; hence its cardinal behaves like some exponential of n. We have the following estimate on the remainder R r,l C l = Oh r ). For any l N, ɛ > 0, there exists C l,ɛ such that for all n 0, for all h 0, h 0 ], we have β B n a β,b C l C l,ɛ e np1/2)+ɛ). 3.2) Let us recall in a very sketchy way the idea behind the proof of Theorem 3.1. Since h 2 1)E h = 0, we have formally that e itph/h E h = e it/h E h, where e itph/h is the Schrödinger propagator. Of course, this statement can only be formal, since E h is not in L 2, but by working with cut-off functions, we can make it rigorous up to a Oh ) remainder. By using some resolvent estimates and hyperbolic dispersion estimates, one can show that if we propagate E h by the Schrödinger flow during a long enough time of the order of some logarithm of h), the term involving Eh 1 becomes smaller than any power of h. Hence we only have to study the propagation during long times of Eh 0. Since, by assumption, E0 h is a Lagrangian state, we can use the WKB method to study its propagation. The main part in the WKB analysis is to understand the Lagrangian manifold Φ t L 0 ), especially for large values of t. This is the content of Theorem 3.2 below. Before stating it, we recall a few notations. Let us fix γ uns > 0 3.3) small enough. In [Ing15], we built V b ) b B a finite open cover of S X in T X depending on γ uns ) such that Theorem 3.2 below holds. This open cover had the following properties. We have B = B 1 B 2 {0}, where V 0 is as in 2.1). For each b B 1, V b ) b B1 is an open cover of K in T X, such that for every b B 1, there exists a point ρ b V b K, and such that the adapted coordinates y b, η b ) centred on ρ b are well defined on V b for every b B 1. The sets V b are all bounded for b B 1 B 2. Truncated Lagrangians Let N N, and let β = β 0, β 1...β N 1 B N. Let Λ be a Lagrangian manifold in T X. We define the sequence of possibly empty) Lagrangian manifolds Φ k β Λ)) 0 k N 1 by recurrence by: If β B N, we will often write Φ 0 βλ) = Λ V β0, Φ k+1 β Λ) = V βk+1 Φ 1 Φ k βλ)). 3.4) Φ β Λ) := Φ N 1 β Λ). For any β B N such that β N 1 0, we will define τβ) := max{1 i N 1; β i = 0} 3.5) 13

14 if there exists 1 i N 1 with β i = 0, and τβ) = 0 otherwise. The sets Φ N β L 0) are related to the result of Theorem 3.1 as follows. For all N N, β B N and b B 1, we have, using the notations of Theorem 3.1: κ b Φ N β L 0 ) ) {y ρ b, φ β,b y ρ b )); y ρ b Ω β,b }, 3.6) where Ω β,b R d is an open set containing the support of a β,b. The properties of the sets Φ N β L 0) are described in the following theorem. Theorem 3.2. Suppose that, the manifold X, g) satisfies Hypothesis 2.1 at infinity, that the Hamiltonian flow Φ t ) satisfies Hypothesis 2.2, and that the Lagrangian manifold L 0 satisfies the invariance Hypothesis 2.5 as well as the transversality Hypothesis 2.6. There exists N uns N such that for all N N, for all β B N and all b B 1, then V b Φ β L 0 ) is either empty, or can be written in the coordinates y b, η b ) as V b Φ β L 0 ) = {y b 1, u b, 0, f b,β u b )); y b 1, u b ) D b,β }, where f b,β : R d 1 R d 1 is a smooth function. and D b,β R d is a bounded open set. For each α N d 1, there exists a constant C α such that for all N N, for all β B N and all b B 1, we have α f b,β C 0 C α. Furthermore, if N τβ) N uns, then df b,β C 0 γ uns. 3.2 New results in nonpositive curvature The results of [Ing15] can be improved in the case of geometric scattering in nonpositive sectional curvature, for Lagrangian manifolds that are non-expanding in the past, as we shall now describe Results on the propagation of L 0 The first consequence of Hypotheses 2.4 and 2.8 is the following lemma, which guarantees that Hypothesis 2.6 concerning transversality is always satisfied. Proposition 3.1. Suppose that X, g) satisfies Hypothesis 2.1 near infinity, Hypothesis 2.4, as well as Hypothesis 2.2 on hyperbolicity, and that L 0 is a Lagrangian manifold which satisfies Hypothesis 2.5 of invariance, as well as Hypothesis 2.8. Then L 0 satisfies Hypothesis 2.6 on transversality. To state our main result concerning the propagation of L 0, which is an improvement on Theorem 3.2, we need the following definition. Definition 3.1. If X is a d-dimensional submanifold of T X, we shall say that X projects smoothly on X if it is contained in a smooth section of T X. That is to say, X can be written in the form X = {x, fx)), x Ω}, 3.7) where Ω is an open subset of X, and f is smooth. 14

15 Theorem 3.3. Suppose that X, g) satisfies Hypothesis 2.1 near infinity, Hypothesis 2.4, as well as Hypothesis 2.2 on hyperbolicity, and that L 0 is a Lagrangian manifold which satisfies Hypothesis 2.5 of invariance, as well as Hypothesis 2.8. Then there exists a γ > 0 such that, if we take γ uns γ in 3.3), the following holds. Let O X be an open set which is small enough so that we may define local coordinates on it. Then for any N N and any β B N, Φ β L 0 ) S O) is a Lagrangian manifold which may be projected smoothly on X. In particular, in local coordinates, the manifold Φ N β L 0) T O may be written in the form Φ N β L 0 ) T O {x, x ϕ β,o x)); x O β }, where O β is an open subset of O. Furthermore, for any l N, there exists a C l,o > 0 such that for any N N, β B N, we have Quantum results x ϕ β,o C l C l,o. 3.8) Theorem 3.4. Let X be a manifold which is Euclidean or hyperbolic near infinty, and which satisfies Hypothesis 2.4. Suppose that the geodesic flow Φ t ) satisfies Hypothesis 2.2 on hyperbolicity, Hypothesis 2.3 concerning the topological pressure. Let E h be a generalized eigenfunction of the form described in Hypothesis 2.7, where Eh 0 is associated to a Lagrangian manifold L 0 which satisfies the invariance Hypothesis 2.5 as well as Hypothesis 2.8. Let K X be compact. There exists ε K > 0 such that for any χ Cc X) with a support in K of diameter smaller than ε K, the following holds. There exists a set B χ and a function ñ : B χ N such that the number of elements in { β B χ ; ñ β) N} grows at most exponentially with N. For any r > 0, l > 0, there exists M r,l > 0 such that we have as h 0: χe h x) = β B χ ñ β) M r,l log h e iϕ β x)/h a βx; h) + R r,l, 3.9) where a β S comp X) is a classical symbol in the sense of Definition A.1, and each ϕ β is a smooth function defined in a neighbourhood of the support of a β. We have R r,l C l = Oh r ). For any l N, ɛ > 0, there exists C l,ɛ such that β B χ ñ β)=n a β C l C l,ɛ e np1/2)+ɛ). 3.10) Furthermore, there exists a constant C 1 such that for all β, β B χ, we have where b 0 is as in Hypothesis 2.4. ϕ βx) ϕ β x) C 1 e b 0 maxñ β),ñ β )), 3.11) 15

16 The link between this theorem and the previous one is as follows: let χ Cc X) be as in the theorem and O be a small open set such that suppχ) O. As defined in section 5.1.1, the set B χ is a set of equivalence classes of a subset of n N B n. Let β B χ and let β B n be a representative of β. We may consider Φñβ L 0). We have: x suppϕ β,o ) suppϕ β), ϕ β,o x) = ϕ βx). Therefore, locally, the gradient of the phases described in Theorem 3.3 and 3.4 are the same. Remark 3.1. Note that in Theorem 3.4, the assumption that χ has a small support is important only to obtain 3.11). If χ is any function in Cc X), we may use Theorem 3.4 combined with a partition of unity argument to write χe h as a decomposition similar to 3.9), with an estimate as in 3.10). Actually, this will be done in a more direct way in the proof of Theorem 3.4 see 5.1) and the discussion which follows). As a corollary of Theorem 3.4, we may deduce the following generalisation of Theorem 1.1. Corollary 3.1. We make the same hypotheses as in Theorem 3.4. Let l N and χ Cc Then there exists C l,χ > 0 such that, for any h > 0, we have X). χe h C l C l,χ h l. In particular, the sequence E h ) h is uniformly bounded with respect to h in L loc. Proof. The corollary follows from the decomposition 3.9) along with the estimates 3.10) and 3.8). Corollary 3.2. We make the same hypotheses as in Theorem 3.4. Let χ Cc X) and let ɛ > 0. Then there exists a finite measure µ χ on S X such that we have for any ψ S comp S X) Op h ψ)χe h, χe h = ψx, ξ)dµ χ x, ξ) + O h min 1, ɛ) ) P1/2) 2 b 0, T X where b 0 is the minimal value taken by the sectional curvature on X. If K is a compact set and if the support of χ is in K and of diameter smaller than ε K, we have dµ χ x, ξ) = β Bχ a 0 β) 2 x)δ {ξ= ϕ βx)}dx, where a β is as in 3.9), and a 0 β is its principal symbol as defined in Definition A.1. Furthermore, if L 0 satisfies Hypothesis 2.9, then for every N N, there exists c N > 0 such that for any x X such that χx) = 1, we have a 0 β 2 x) c N. 3.12) β B χ ñ β) N We will prove this corollary in section 5.2. Let us finally state a generalization of Theorem 1.2. We will prove it in section

17 Theorem 3.5. Let X be a manifold which satisfies Hypothesis 2.1 near infinity, and which satisfies Hypothesis 2.4. Suppose that the geodesic flow Φ t ) satisfies Hypothesis 2.2 on hyperbolicity, Hypothesis 2.3 concerning the topological pressure. Let E h be a generalized eigenfunction of the form described in Hypothesis 2.7, where Eh 0 is associated to a Lagrangian manifold L 0 which satisfies the invariance Hypothesis 2.5, part iii) of Hypothesis 2.4 and Hypotheses 2.8 and 2.9. Let χ Cc X). Then there exist constants C, C 1, C 2 > 0 such that the following result holds. For all x 0 X such that χx 0 ) = 1, for any sequence r h such that 1 >> r h > Ch, we have for h small enough: C 1 rh d RE h 2 x)dx C 2 rh. d Bx 0,r h ) In particular, for any bounded open set U X, there exists cu) > 0 and h U > 0 such that for all 0 < h < h U, we have RE h 2 cu). U 4 Proofs of the results concerning the propagation of L General facts concerning manifolds of nonpositive curvature Growth of the distance between points on manifolds of nonpositive curvature. In this paragraph, we will recall a few facts about the way the distances d X and d ad between Φ t ρ 1 ) and Φ t ρ 2 ) depend on time. Remember that the distance d ad was introduced in section 2.1.2, while d X was defined in section The easiest bound simply comes from the compactness of S X 0, and the geodesic convexity of X 0 : we may find a constant µ > 0 such that for any ρ, ρ S X 0 and for any t 0 such that Φ t ρ), Φ t ρ ) S X 0, we have d ad Φ t ρ), Φ t ρ )) e µt d ad ρ, ρ ). 4.1) In all the sequel, we will shrink the sets V b ) b B2 appearing in Theorems 3.1 and 3.2 so that the following holds: the sets V b, b B 1 B 2 have a diameter smaller than some constant ɛ max such that x, y X 0, d ad x, y) < ɛ max = d X Φ 1 x), Φ 1 y)) < e µ r i, 4.2) where µ is as in 4.1). Remark 4.1. Actually, when we work close to the trapped set, using the bounds on the growth of Jacobi fields which may be found in [Ebe01, III.B], one can show that there exists C > 0 such that if ρ, ρ S X and T 0 are such that for all t [0, T ], we have Φ t ρ) b B 1 V b and Φ t ρ ) b B 1 V b, then we have d ad Φ t ρ), Φ t ρ )) Ce b 0t d X ρ, ρ ), where b 0 is the lowest value taken by the sectional curvature on X as in Hypothesis 2.4. On the other hand, if two points are on the same local stable manifold, they will approach each other exponentially fast in the future. This is the point of the following classical lemma, whose proof can be found in [KH95, Theorem )]. 17

18 Lemma 4.1. There exists C, λ > 0 such that for any ρ K and ρ 1, ρ 2 W ɛ ρ) for some ɛ > 0 small enough, we have d ad Φ t ρ 1 ), Φ t ρ 2 )) C e λt d ad ρ 1, ρ 2 ). On a manifold of nonpositive curvature, the square of the distance d X between two points will be convex with respect to time, as long as these two points remain close enough to each other. This is the content of the following lemma, whose proof may be found in [Jos08, 4.8]. Lemma 4.2. Let γ 1, γ 2 be two geodesics on a manifold of nonpositive sectional curvature X which is simply connected. Then t d 2 X γt), γ t)) is a convex function. Furthermore, if γ 1 and γ 2 are different geodesics, there exist t 1 < t 2 + such that for all t t 1, t 2 ), γ 1 t) and γ 2 t) belong to a region of X where sectional curvature is strictly negative, then on t 1, t 2 ), t d 2 X γ 1t), γ 2 t)) is strictly convex. From now on, we will denote by X the universal covering of X. Corollary 4.1. Suppose that ρ 1, ρ 2 X and t 1 < t 2 + are such that for all t t 1, t 2 ), we have d X Φ t ρ 1 ), Φ t ρ 2 )) < r i. Then t 1, t 2 ) t d 2 X Φt ρ 1 ), Φ t ρ 2 )) is a convex function. Furthermore, if ρ 1 and ρ 2 do not belong to the same geodesic and if for all t t 1, t 2 ), Φ t ρ 1 ) and Φ t ρ 2 ) belong to a region of X where sectional curvature is strictly negative, then on t 1, t 2 ), t d 2 X Φt ρ 1 ), Φ t ρ 2 )) is strictly convex. Proof. Using the fact that the exponential map is a covering map on balls of radius smaller than the injectivity radius, we can push back the curves Φ t ρ i ), i = 1, 2 to geodesics Φ t ρ i ) on the universal cover X of X such that d X Φ t ρ 1 ), Φ t ρ 2 )) = d X Φ t ρ 1 ), Φ t ρ 2 )). We may then conclude by Lemma Covering of X by t>0 Φt L 0 ) The following lemma can be found in [Ebe01, IV.A]. Lemma 4.3. Let X satisfy Hypothesis 2.4, and let ρ T X and x X. Then there exists a unique ξ T x X such that d XΦ t ρ), Φ t x, ξ)) remains bounded as t. In particular, by Lemma 4.2, t d XΦ t ρ), Φ t x, ξ)) is non-decreasing. Consider the set Φ L 0 ) := t>0 Φ t L 0 ). The following corollary of Lemma 4.3 says that if Hypothesis 2.9 is satisfied, then X is covered infinitely many times by Φ L 0 ). Note that this is the only place where we actually need Hypothesis 2.9 to hold. Corollary 4.2. Let X be a manifold satisfying Hypothesis 2.1 near infinity and Hypothesis 2.4 of nonpositive curvature, and such that the trapped set K is non-empty and satisfies Hypothesis 2.2 of hyperbolicity. Let L 0 be a Lagrangian manifold which satisfies Hypotheses 2.5, 2.8 and 2.9. Let x X. Then there exist infinitely many ξ T x X such that x, ξ) Φ L 0 ). 18

19 Proof. Fix some x 0, ξ 0 ) = ρ 0 L 0 DE and some x X. Since K is non-empty and hyperbolic, it contains at least a closed orbit. Therefore, π 1 X) is not trivial, hence infinite. Indeed, by [Ebe01, III.G], any non trivial element in the fundamental group of a manifold of nonpositive curvature has infinite order.) Therefore, ρ 0 has infinitely many pre-images by the projection S X = S X S X. Let us denote them by ρ i 0) i I = x i 0, ξ i 0) i I, and let us fix a point x such that π X X x) = x. Thanks to Lemma 4.3, for each i I, there is a ξ i S x X such that d XΦ t ρ 0 i ), Φt x, ξ i )) remains bounded as t. Let us write x, ξ i ) = π T X T X x, ξ i ). The ξ i and ξ i may of course be identified.) For each t 0, we have d X Φ t ρ 0 ), Φ t x, ξ i )) d XΦ t ρ i 0), Φ t x i, ξ i )), which is bounded as t by assumption. Therefore, in the topology of the compactification of X given by Hypothesis 2.1, Φ t ρ 0 ) and Φ t x, ξ i ) are approaching each other as t. Since we took ρ 0 DE, we also have Φ t x, ξ i ) DE if t is large enough. Therefore, by Hypothesis 2.9, Φ t x, ξ i ) L 0 if t is large enough. To prove the corollary, we have to check that the ξ i are all distinct, and hence that the ξ i are all distinct. It suffices to show that for i i, we have d XΦ t ρ i 0), Φ t ρ i 0 )) is unbounded as t. 4.3) Suppose for contradiction that 4.3) were not true for some i i. Consider, as in Figure 2, the geodesic γ in X such that γ0) = x i 0 and γ1) = x i 0. According to Lemma 4.3, for each s [0, 1], there exists a unique direction ξ s such that d XΦ t ρ i 0), Φ t γs), ξ s )) is bounded as t. By the uniqueness part in Lemma 4.3, and because we assumed that 4.3) is not satisfied, we have that ξ i 1 = ξ 0. Let us consider γs), ξ s ) := π S X S X γs), ξ s ). We have γ0), ξ 0 ) = ρ 0 by assumption, and γ0) = γ1). Furthermore, d X Φ t ρ 0 ), Φ t γs), ξ s )) d XΦ t γ0), ξ 0 ), Φ t γs), ξ s )), which is bounded as t by construction of ξ s. Let us write γ t s) = Φ t γs), ξ s ). This curve has a length bounded independently of t by what precedes, and its points are going to infinity as t. Furthermore, for each t 0, γ t is a closed curve which is not contractible, since it joins two different points in X. Therefore, for each t 0, there must be a s [0, 1] such that d X γ t 0), γ t s)) r i γ t 0)). Indeed, if this were not true, γ t would be contained in a chart where the exponential map is injective, and it would be contractible. But since γ t has a length bounded independently of t, and since γ t 0) goes to infinity with t, we obtain a contradiction with part iii) of Hypothesis 2.4. Remark 4.2. The end of the proof actually shows that, if we denote by L0 j)j J the different preimages of L 0 by the projection π S X S X, then we have for each j j J, L0 j L0 j =. 19

20 Figure 2: What happens if 4.3) is not true 4.2 Smooth projection and Transversality General criteria Recall that smoothly projecting manifolds were introduced in Definition 3.1. We shall now rephrase the notion of projecting smoothly on X in terms of transversality. Let L 1, L 2 be two submanifolds of S X and let ρ L 1 L 2. Recall that we say that L 1 and L 2 intersect transversally at ρ if T ρ S X) = T ρ L 1 T ρ L 2. Lemma 4.4. Let X be a submanifold of S X which can be written in the form X = {x, fx)), x Ω}, where Ω is an open subset of X, and where f is C 0. Suppose furthermore that for every x Ω, the manifolds X and S xx intersect transversally at x, fx)). Then X projects smoothly on X. Proof. Let us write κ : X x x, fx)) X and π : X x, fx)) x X. We have of course π κ = Id. The transversality assumption ensures us that dπ is invertible wherever it is defined. By the inverse function theorem, κ is C. Therefore, since f is the second component of κ, it is also smooth. The following lemma gives us a criterion to check that two manifolds intersect transversally, which we shall use several times. 20

21 Figure 3: Two non-transverse manifolds Lemma 4.5. Let L 1, L 2 be two submanifolds of S X with dimension d and d 1 respectively, and let ρ L 1 L 2. We assume that i) is satisfied, as well as ii) or ii ), where the assumptions i), ii) and ii ) are as follows. i) For all ρ 1 L 1, the function t d X Φ t ρ 1 ), Φ t ρ)) is non-increasing for t 0. ii) For all ρ 2 L 2, we have d X ρ 2, ρ) = 0 that is to say, L 2 S xx for some x X). ii ) Set x = π S X Xρ). The manifolds L 1 and S xx are transverse at ρ. Furthermore, there exists ν > 0, C > 0 and ɛ > 0 such that for all ρ 2 L 2 and for all t 0 such that d ad Φ t ρ), Φ t ρ 2 )) ɛ, we have: Then L 1 and L 2 intersect transversally at ρ. d X Φ t ρ 2 ), Φ t ρ)) Ce νt d X ρ 2, ρ). Proof. Suppose for contradiction that L 1 and L 2 do not intersect transversally at ρ. This means that there exists v T ρ L 1 ) T ρ L 2 ), v 0. Therefore, there exist ρ n 1 L 1, ρ n 2 L 2 such that d ad ρ n 1, ρ) = 1/n and d ad ρ n 2, ρ) = 1/n but d ad ρ n 1, ρ n 2 ) = o1/n), as in as in Figure 3. We will find a contradiction by finding a time t > 0 such that d X Φ t ρ), Φ t ρ n 1 )) > d X ρ, ρ n 1 ), 4.4) which will contradict assumption i). We have d X Φ t ρ n 1 ), Φ t ρ n 2 )) d ad Φ t ρ n 1 ), Φ t ρ n 2 )) e µt o1/n). 4.5) 21

22 Suppose first that assumption ii) is satisfied, and write ρ = x, ξ) and ρ n 2 = x, ξ2 n ). Then, since d ad ρ n 2, ρ) = 1/n and since the distances induced by all metrics on S X are equivalent in a neighbourhood of ρ, there exists a constant c > 0 such that ξ ξ2 n c/n for all n large enough. d Therefore, d X Φ t ρ), Φ t ρ n 2 )) c/n. Since the second derivative with respect to time t=0 dt of the distance between geodesics only involve local terms and since the trajectories of Φ t ρ) and Φ t ρ n 1 ) are close to each other for t small enough and n large enough, we may find a time t 0 > 0 independent of n and a c > 0 such that for all n large enough and all t [0, t 0 ], we have d X Φ t ρ), Φ t ρ n 2 )) > c t/n. We hence have d X Φ t0 ρ), Φ t0 ρ n 1 )) d X Φ t0 ρ), Φ t0 ρ n 2 )) d X Φ t0 ρ n 2 ), Φ t0 ρ n 1 )) On the other hand, we have c t 0 /n + o1/n). d X ρ n 1, ρ) d X ρ n 2, ρ) + d X ρ n 1, ρ n 2 ) = d X ρ n 1, ρ n 2 ) Cd ad ρ n 1, ρ n 2 ) = o1/n). Therefore, taking t n = t 0 gives 4.4). Suppose now that ii ) is satisfied. Since we have assumed that the manifolds L 1 and S xx are transverse at ρ, there exists a constant c > 0 such that for all n large enough, we have cd ad ρ, ρ n 1 ) d X ρ, ρ n 1 ). Therefore, d X ρ n 1, ρ) c/n. Let us consider a t 0 such that Ce νt > 2. If n is large enough, we have d ad Φ t ρ), Φ t ρ n 2 )) ɛ. We hence have On the other hand, d X Φ t ρ), Φ t ρ n 2 )) Cc eνt n 2c n. 1 d X Φ t ρ n 2 ), Φ t ρ n 1 )) e µt d ad ρ n 2, ρ n 1 ) = o. n) We deduce from this that d X Φ t ρ), Φ t ρ n 1 )) 2c n d XΦ t ρ n 2 ), Φ t ρ n 1 )) > c n. This gives us 4.5), and concludes the proof of the lemma Three applications of Lemma 4.5 As a first application of Lemma 4.5, let us now state a useful lemma. It seems rather classical, but since we could not find a proof of it in the literature, we recall it. It is likely that this result and most of the results of this section) still holds if we suppose that X, g) has no conjugate points, instead of supposing that it has non-positive curvature. We refer the reader to [Rug07] for an overview of the properties of such manifolds. 22

23 Lemma 4.6. Let X be a manifold of non-positive sectional curvature, such that K is a hyperbolic set. Let ρ K. Then there exists ɛ > 0 such that W ±0 ɛ ρ) projects smoothly on X. Proof. We shall only prove the lemma for W ɛ +0. The result for Wɛ 0 unstable manifolds are exchanged by changing the sign of the speeds ξ. Suppose that ρ i = x, ξ i ) S X W +0 ɛ will follow, since stable and We want to apply Lemma 4.4. Let us first check that W ɛ +0 can indeed be written as a graph. for i = 1, 2. Then if gt) = d X Φ t ρ 1 ), Φ t ρ 2 )), we have g0) = 0 and lim gt) < by definition of the stable manifold. But, if ɛ is small enough, we have t gt) r i for all t 0, so that by Corollary 4.1, we have ξ 1 = ξ 2. Therefore, W ɛ +0 for ɛ small enough as W +0 ɛ = {x, fx)); x Ω} can be written for some open set Ω X. The fact that W ɛ +0 is a connected manifold implies that f is continuous. We now have to prove the transversality condition of Lemma 4.4, by applying Lemma 4.5 for L 1 = Wɛ and L 2 = SxX. ii) is trivially satisfied. Let us check that i) is satisfied. Take ρ 1, ρ 2 W ɛ +0 ρ), and write gt) = d X Φ t ρ 1 ), Φ t ρ 2 )). We have lim gt) <, and if ɛ is taken t + small enough, we may assume that gt) r i for all t 0. Therefore, by Corollary 4.1, g is nonincreasing for t 0, and i) is satisfied. Hence we may apply Lemma 4.5 and Lemma 4.4 to conclude the proof. Remark 4.3. Actually, with the same proof, we can prove the following statement, saying that if we propagate in the past resp. future) a small piece of the stable resp. unstable) manifold, it projects smoothly on X: Let ρ K. Then there exists ɛ > 0 such that for all ±t 0 and ρ W ɛ ±0 ρ), there exists ɛ > 0 such that Φ t {ρ W ɛ ±0 ρ); d ad ρ, ρ ) < ɛ } projects smoothly on X. We now prove a lemma which is the first step in the proof of Theorem 3.3. Lemma 4.7. Let τ 0 and let ρ L 0. For ɛ > 0, the manifold Φ τ {ρ L 0 ; d ad ρ, ρ ) < ɛ} projects smoothly on X. Proof. We have to check that the assumptions of Lemma 4.4 are satisfied. If ɛ is chosen small enough, then for all t τ and for all ρ L 0 such that d ad ρ, ρ ) < ɛ, we have that d ad Φ t ρ ), Φ t ρ) ) r i thanks to Hypothesis 2.8. This implies that Φ τ {ρ L 0 ; d ad ρ, ρ ) < ɛ} can be written as a graph above X. Indeed, suppose that this manifolds contains two points ρ 1 = x, ξ 1 ) and ρ 2 = x, ξ 2 ). Then, by Lemma 4.1, t d 2 X Φt ρ 1 ), Φ t ρ 2 )} is a convex function on ; 0). By Hypothesis 2.8, this function goes to a constant as t goes to. Hence it must be constant equal to zero, and ρ 1 = ρ 2. We now have to check that this graph is transversal to the vertical fibres. To do so, we want to apply Lemma 4.5 for L 1 = Φ τ {ρ L 0 ; d ad ρ, ρ ) < ɛ} and L 2 = SxX. Hypothesis ii) of Lemma 4.5 is then trivially satisfied. As for hypothesis i), it is satisfied by Hypothesis 2.8 combined with Lemma 4.1. Therefore, we may apply Lemma 4.5 and Lemma 4.4 to conclude the proof of the Lemma. As a last application of Lemma 4.5, we shall prove Proposition 3.1. Recall that this proposition says that: Proposition. Suppose that X satisfies Hypothesis 2.4, as well as Hypothesis 2.2 on hyperbolicity, and that L 0 satisfies Hypothesis 2.5. Then the Lagrangian manifold L 0 automatically satisfies Hypothesis 2.6 on transversality. 23

24 Proof. Let ρ K, let τ 0 and let ρ Φ τ L 0 ) W ɛ ρ). We want to apply Lemma 4.5 to Φ τ L 0 ) and W ɛ ρ), or at least to the restriction of these manifolds to a small neighbourhood of ρ. If we take a small enough neighbourhood V of ρ, then by Lemma 4.1 and Hypothesis 2.8, we have that L 1 = Φ τ L 0 ) V satisfies assumption i) of Lemma 4.5. Let us check that assumption ii ) is satisfied. The fact that L 1 projects smoothly on X comes from Lemma 4.7. As for the second condition, we know by Lemma 4.1 that there exists C, λ > 0 such that for any ρ K and ρ 1, ρ 2 W ɛ ρ), we have for all t 0: d ad Φ t ρ 1 ), Φ t ρ 2 )) Ce λt d ad ρ 1, ρ 2 ). We therefore have, for all t 0 such that Φ t ρ i ) W ɛ ρ) for i = 1, 2 : d ad ρ 1, ρ 2 ) Ce λt d ad Φ t ρ 1 ), Φ t ρ 2 )). But, since Wɛ 0 ρ) projects smoothly on X for ɛ small enough by Lemma 4.6, there exists a constant C such that for all ρ 1, ρ 2 Wɛ ρ), we have 1 C d Xρ 1, ρ 2 ) d ad ρ 1, ρ 2 ) C d X ρ 1, ρ 2 ). Therefore, hypothesis ii ) of Lemma 4.5 is satisfied, and we can use Lemma 4.5 and Lemma 4.4 to conclude the proof. 4.3 Proof of Theorem 3.3 Proof. First of all, let us make a few remarks to show that we don t need to consider all sequences β. We have L 0 = L 0 DE ) L 0 DE + ). We therefore have ) ) Φ N β L 0 ) T O = Φ N β L 0 DE + ) T O Φ N β L 0 DE ) T O. By hypothesis 2.8), we have Φ N β L 0 DE + ) L 0. Consequently, we only have to focus on the propagation of L 0 DE. Let us note that by 2.8), we have that for all k 1, Φ k 0,...,0L 0 DE ) Φ 1 0L 0 DE ). We may hence, without loss of generality, restrict ourselves in what follows to sequences β such that β 2 0. By the geodesic convexity assumption, for all bounded open set O, we may find a N O such that for all N N O and for all β B N such that β 2 0 and Φ N β L 0) O =, we have k = 2...N N O, β k 0. Now, since the sets V b, b B 2 are at positive distance from the trapped set, we may find a N 1 such that for all N N O + 2N 1 and for all β B N such that β 2 0 and Φ N β L 0) O, we have k = N 1...N N 1 N O, β k B 1. Let us now check that a single Lagrangian manifold Φ N β L 0) projects smoothly on the base manifold. Lemma 4.8. Let O X be an open bounded set. Then for any N N, β B N, Φ N β L 0) S O) projects smoothly on X. 24

25 Proof. The proof follows from Lemma 4.7, if we can check that Φ N β L 0) S O) can be written as a graph on X. Let ρ i = x, ξ i ) Φ N β L 0), i = 1, 2. Thanks to condition 4.2), we see that Φ t ρ 1 ) and Φ t ρ 2 ) remain at a distance smaller than r i from each other for all t 0. Therefore, by Lemma 4.2, we have Φ t ρ 1 ) Φ t ρ 2 ), and ξ 1 = ξ 2, which concludes the proof. Let us fix a small open set O and a local chart on it. Thanks to Lemma 4.8, in these local coordinates, the manifold Φ N β L 0) T O, if nonempty, may be written in the form Φ N β L 0 ) T O {x, x ϕ β,o x)); x O β }, 4.6) where O β is an open subset of O, and x ϕ β,o is a smooth function. We will now show that the functions ϕ β,o are bounded independently of β and N in any C l norm. Let us start by working close to the trapped set, that is to say, by considering the case where O = O b = π X V b ) for some b B 1. Let us denote by κ b the symplectomorphism sending x, ξ) to y b, η b ) in a neighbourhood of ρ b. In the notations of Theorem 3.2, we have {x, x ϕ β,ob x)); x O β b } = κ 1 b {y b 1, u b, 0, f b,β u b )); y b 1, u b ) D b,β,n } ). We will write F b,β := 0, f b,β ), and we will denote by ˆx b and ˆξ b the components of κ 1 We shall consider the map κ β,b sending x O β b to yb R d such that κ b x, ϕ β,ob x)) = y b, F b,β y b )). Lemma 4.9. If we take γ uns small enough in 3.3), we may find a constant c > 0 such that for any b B 1, for any N N and for any β B N, we have Proof. We have hence κ 1 β,b y b = ˆx b b. κ 1 β,b > c. 4.7) y b κ 1 β,b yb ) = ˆx b y b, F b,β y b )), 4.8) y yb, F b,β y b )) + F b,βy b ) y b ˆx b η yb, F b,β y b )). Now, ˆx b y yb, 0) is invertible at every y b, because the unstable manifold projects smoothly on the base manifold, hence we may find a c > 0 such that x b y yb, 0) > 2c for any b B 1 and any y b. Using the fact that F b,β C 1 γ uns for τβ) large enough, the lemma follows for τβ) large enough, that is to say, for N large enough since we have assumed that β 2 0. For finite values of N, inequality 4.7) is true thanks to Lemma 4.7. The statement follows. Let us come back to the proof of Theorem 3.3. Equation 4.8) ensures us that κ 1 β,b is bounded independently of β in any C l norm. Indeed, when we differentiate it, it depends on β only through F b,β, which is bounded independently of β in any C l norm. 25

26 Therefore, we deduce from Lemma 4.9 and the chain rule that we may bound the C l norm of the functions κ β,b independently of β. Next, we check that ϕ β,ob is bounded independently of β in any C l norm. Indeed, we have ϕ β,ob x) = ˆξ b κβ,b x), F b,β κ β,b x)) ). When we differentiate this expression, we see that the only terms which depend on β involve some derivatives of F b,β and some derivatives of κ β,b, both of which are bounded independently of β. This proves the theorem in the case where O πv b ) for some b B 1. Let us now consider an arbitrary O. We may suppose that N N O +2N 1, since for finite values of N, the result just follows from Lemma 4.8. Let β B N be such that β 2 0 and Φ N β L 0) O. By the preliminary remarks of the proof, there exists k N such that β k B 1. We may suppose that N k N O for a N O which does not depend on N or k. We then have Φ N β L 0 ) T O {x, x ϕ β,o x))} = Φ N k β {x, ϕ β,o βk x )} ) T O, where β = β 0...β k 1, β = β k...β N 1 and where the x belong to a subset of O b. Let us denote by Φ β,k the map sending x to the x such that x, x ϕ β,o x)) = Φ N k x, ϕ β,o βk x ) ). We have x ϕ β,o x) = π ξ Φ N k Φβ,k x), ϕ β,o βk Φ β,k x) ))), where π ξ denotes the projection on the ξ variable. Hence, since ϕ β,o βk is bounded in any C l norm independently of β, we only have to prove that Φ β,k is bounded in any C l norm independently of β. 1 On the one hand, we have Φ β,k x ) = π x Φ N k x, ϕ β,o β,k x )) ). Therefore, since Φ N k is a diffeomorphism, the map Φ β,k is bounded in any C l norm independently of β just like ϕ β,o β,k is. On the other hand, if x 1,..., x d ) are local coordinates on X, then for all j = 1...d, and all x O, we claim that j Φ 1 β,k x) ) Suppose for contradiction that we can find x such that Φ 1 j β,k x) < 1. We may then find a sequence x n such that d X x, x n ) = 1/n, and d Φ 1 1 X β,k x), Φ β,k x n) ) =< 1/n for n large enough. 1 Write ρ = Φ β,k x) and ρ 1 n = Φ β,k x n) ). The function d X Φ t ρ), Φ t ρ n )) is bounded for all t 0, thanks to Hypothesis 2.8. Furthermore, we have d X Φ k ρ), Φ k ρ n ) ) < d X ρ, ρ n ). But, if we take n large enough, we can ensure that d X Φ t ρ), Φ t ρ n ) ) < r i for all t, k), thus contradicting Corollary 4.1. This proves 4.9). The theorem then follows from the chain rule, since the derivatives of the inverse map are bounded as long as we apply them to vectors away from zero. 4.4 Distance between Lagrangian manifolds Let us now state a lemma concerning the distance between the Lagrangian manifolds described in Theorem 3.3, which will play a key role in the proof of Corollary 3.2. It is very similar to Proposition 4 in [Ing15]. 26

27 Lemma Let O be a bounded open set in X. There exists a constant C O > 0 such that for any n, n N, for any β B n, β B n, and for any x O, such that x π X Φβ,O L 0 ) ) π X Φβ,OL 0 ) ), we have either ϕ β,o x) = ϕ β,ox) or ϕ β,o x) ϕ β,ox) C 1 e b 0 maxn τβ),n τβ )), 4.10) with τβ) defined as in 3.5), and where b 0 is as in Hypothesis 2.4. Proof. By the remarks at the beginning of the proof of Theorem 3.4, we may restrict ourselves to sequences β, β such that β 2, β 2 0, that is to say, to sequences such that τβ), τβ ) 1, and we may find constants N O and N 1 such that for all N N O + 2N 1 and for all β B N such that β 2 0 and Φ N β L 0) O, we have k = N 1...N N 1 N O, β k B 1. Let n, n N, n, n N O + 2N 1, β B n, β B n, such that β 2, β 2 0, and let x O be such that x π X Φβ,O L 0 ) ) π X Φβ,OL 0 ) ), and ϕ β,o x) ϕ β,ox). Let us write ρ = x, ϕ β,o ) and ρ = x, ϕ β,o). We claim that there exists 0 k maxn, n ) such that for each b B, if Φ k ρ) V b, then Φ k ρ ) / V b. Indeed, if no such k existed, then for each k, there would exist b k B such that Φ k ρ) V bk and Φ k ρ ) V bk for each 0 k maxn, n ). We would then have ρ Φ maxn,n ) β L 0 ), and ρ Φ maxn,n ) β L 0 ) for some sequence β built by possibly adding some 0 s in front of the sequences β and β. This would contradict the statement of Theorem 3.3. Recall that from the elementary) Lemma 22 in [Ing15], we have the existence of a constant c 0 > 0 such that for any ρ, ρ S X 0 such that d ad ρ, ρ ) < c 0, there exists b B such that ρ, ρ V b. Thanks to this, we deduce that there exists 0 k maxn, n ) such that d ad Φ k ρ), Φ k ρ )) c 0. Combining this fact with equation 4.1) and Remark 4.1, we get d ad ρ, ρ ) Ce b 0 maxn,n ) Using the fact that all metrics are equivalent on a compact set, we may compare d ad ρ, ρ ) with ϕ β,o x) ϕ β,ox) and we deduce from this the lemma. 5 Proof of the results concerning the distorted plane waves 5.1 Proof of Theorem 3.4 Proof. We want to write E h as a sum of Lagrangian states associated to Lagrangian manifolds which do all project smoothly on the base manifold. Most of the work was done in [Ing15], and we shall now recall what we need from this paper. Let us write P h := h 2 c 0 h 2 1 and Ut) := e itph/h. Let us fix χ Cc X), and a function ˆχ as in Remark 2.1. Recall that the pseudo-differential operators Π b ) b B were introduced in Theorem 3.1. For each b B, we set U b := e i/h Π b U1). For each n N, for each β = β 0,..., β n 1 B n, we set U β = U βn 1... U β0. Recall that U b : L 2 X) L 2 R d ) is a Fourier integral operator quantizing the symplectic change of local coordinates κ b : x, ξ) y ρ b, η ρ b ), and which is microlocally unitary on the microsupport of Π b. Therefore, Ub : L2 R d ) L 2 X) quantizes the change of coordinates y ρ b, η ρ b ) x, ξ). 27

28 Recall that the time N uns was introduced in Theorem 3.2. Equation 81) in [Ing15] tells us that there exist ˆχ 1 Cc X), and a constant 2 N ˆχ > 0 such that, writing n := n + 2N uns + 1, we have M r log h ) χe h = χˆχe h = χ ˆχU1)) N ˆχ+1 U βn...β n +Nuns+1 Uβ θ n n,β n=1 β B n +Nuns+1 + χ N χ+3n uns+3 l=0 ˆχU1) ) l1 ˆχ)ˆχ1 E 0 h + O L 2h r ). Here the sets B N B N are defined, for N 2N uns + 2 by B N = B 1 B 2 ) Nuns+1 B N 2Nuns 2 1 B 1 B 2 ) Nuns+1, and the θ n,β are Lagrangian states of the form θ n,βy) = a n,βy; h)e iφ n,β y), with y in some bounded open subset of R d. Furthermore, we have estimates analogous to 3.2), that is: for any l N, ɛ > 0, there exists C l,ɛ such that for all n 0, for all h 0, h 0 ], we have β B n a n,β C l C l,ɛ e np1/2)+ɛ). Just as in 3.6), the Lagrangian states θ n,β are associated to the Lagrangian manifolds described in Theorem 3.2. Namely, we have {y β n, φ n,βy β n ))} κ βn Φ n β n L 0) ). Now, ˆχU1)) N ˆχ+1 U βn,...,β n +Nuns+1 Uβ n and ˆχU1)) l 1 ˆχ)ˆχ 1 are Fourier integral operators. We want to apply Lemma A.2 to describe their action on the states e n and Eh 0 respectively. Theorem 3.3 ensures us that A.5) is satisfied. We must check that A.3) is satisfied. For the operators χu1)) Nχ+1 U βn,...,β n +Nuns+1 Uβ n, it follows from the fact that the propagation in the future of small pieces of local unstable manifolds project smoothly on X as explained in Remark 4.3. To apply ˆχU1)) l 1 ˆχ)ˆχ 1 to Eh 0, take ρ L 0 S supp1 ˆχ)ˆχ 1 )) ). We may take coordinates on X near πx) which induces symplectic coordinates ˆx, ˆξ) on a neighbourhood of ρ in T X such that ρ = 0, 0) in these coordinates and such that the manifold L 0 is tangent to {ˆξ = 0}. In this system of coordinates near ρ, and with any system of coordinates on X near πφ t ρ)), we are in the framework of Lemma A.2. Indeed, the condition A.3) is then satisfied thanks to Lemma 4.7. By writing Bn ˆχ := {ββ ; β B n, β B N ˆχ+1 }, if n 2, we obtain that B ˆχ 1 := {β BN ˆχ+3N uns+1 }, 2 The time N ˆχ N is actually chosen so that for any N N, if ρ suppˆχ 1 ) and Φ N ρ) suppˆχ), then for all k with N ˆχ k N N ˆχ, we have Φ k ρ) V b for some b B 1 B 2. 28

29 M r,l log h χe h x) = n=0 β B ˆχ n e iϕ βx)/h a β, ˆχ x; h) + R r, ˆχ, 5.1) where a β, ˆχ S comp X), and each ϕ β is a smooth function defined in a neighbourhood of the support of a β, ˆχ, and where we have R r, ˆχ C l = Oh r ). For any l N, ɛ > 0, there exists C l,ɛ such that β B ˆχ n a β, ˆχ C l C l,ɛ e np1/2)+ɛ). 5.2) Note that the expression 5.1) has interest of its own, because it applies to functions χ Cc with arbitrary support. We shall now show that if the support of χ is small enough, we may regroup the terms of 5.1) in a clever way so that 3.11) is satisfied. This is the content of the next subsection Regrouping the Lagrangian states From now on, we fix a compact set K X, and we suppose that ˆχ = 1 on K. Lemma 5.1. There exists ε K such that, for any open set O K of diameter smaller than ε K, the following holds: n N, β B ˆχ n, t 0 and for all ρ, ρ Φ n β L 0) S O, we have d X Φ t ρ), Φ t ρ )) < r i. Proof. First of all, note that thanks to Hypothesis 2.8, we only have to show the result for t β, since Φ β ρ) and Φ β ρ ) both belong to L 0, and hence can only approach each other in the past. Take ɛ small enough so that for all ρ 1, ρ 2 S K with d ad ρ 1, ρ 2 ) < ɛ, we have d Φ t ρ 1 ), Φ t ρ 2 ) ) < r i for all t [0; N χ + 3N uns + 1]. Thanks to Theorem 3.3, the Lagrangian manifolds Φ n β L 0) S K project smoothly on X, so that we may find a constant C K such that, n N, β B ˆχ n and for all ρ, ρ Φ n β L 0) S O, we have d ad ρ, ρ ) C K d X ρ, ρ ). By taking ε K = ɛ/c K, this gives us the result when n = 1. When n 2, the result follows from the definition of B ˆχ n, from the fact that the sets V b ) b B1 B 2 all have diameter smaller than r i, and from equation 4.2). Without loss of generality, we will always take ε K 1. Let O K be an open set of diameter smaller that ε K. Let us fix L 0 to be a pre-image of L 0 by the projection S X S X. Let us denote by O j ) j J the pre-images of O by the projection X X. If we suppose that the diameter of O is smaller than r i, then the O j are all disjoint. For each b B, we set Ṽb to be the pre-image of V b by the projection S X S X. The truncated propagator Φ k β may then be defined on S X just as in 3.4), but with V b replaced by Ṽb. 29

30 For every n N and every β Bn ˆχ such that Φ β L 0 ) S O, we claim that there exists a unique j β J such that Φ β L 0 ) S O jβ. Indeed, it is by definition, we have Φ β L 0 ) j J S O j. But, by Lemma 5.1, we know that for all t 0, Φ t Φ β L 0 ) S O ) has a diameter smaller than r i, so that it is contained in a single coordinate chart. If β Bn ˆχ and β B ˆχ n are such that Φ βl 0 ) S O =, Φ β L 0 ) S O =, we shall say that β O β if j β = j β. The relation O is clearly an equivalence relation on n N {β Bn; ˆχ Φ β L 0 ) S O }, so we may define ) B O := {β Bn; ˆχ Φ β L 0 ) S O } \ O. 5.3) n N We then define for each β B O : ñ β) := min{n N; β B ˆχ n such that β β }. 5.4) Lemma 5.2. There exists N K such that for all n N, β B ˆχ n, we have β β = n ñ β) + N K. Note in particular that this lemma implies that there are only finitely many elements in the equivalence class β. Proof. First of all, by compactness of X 0, we may find N 0 such that for any ρ L 0, we have Φ N0 ρ) / b B 2 V b. Note that we then have for all t 0: d X Φ N0 t ρ), ) t. 5.5) b B 2 V b Let β, β β, with β B ˆχ n and β B ˆχ n. Suppose for contradiction that n n + N Let ρ Φ β L 0 ) S O jβ and ρ Φ β L 0 ) S O jβ. By assumption, we have d X ρ, ρ ) ε K. Furthermore, by Lemma 4.2, d X Φ t ρ), Φ t ρ )) is non-increasing. Hence d X Φ n ρ), Φ n ρ )) ε K ) On the other hand, since n n + N 0 + 3, we have by 5.5) that d X Φ n ρ ), b B 2 V b ) 3, while d X Φ n ρ ), b B 2 V b ) 1. Therefore, we have dx Φ n ρ), Φ n ρ )) 2, which contradicts 5.6). This concludes the proof by taking for β a sequence which realises the minimum in 5.4). Let us now give a more useful description of the equivalence relation O Lemma 5.3. Let n, n N and let β Bn, ˆχ β B ˆχ n be such that suppϕ β,o) suppϕ β,o). Then we have ) β O β ) x suppϕ β,o ) suppϕ β,o), we have ϕ β,o x) = ϕ β,ox). Proof. Suppose first that β O β, and suppose first for contradiction that there exists x suppϕ β,o ) suppϕ β,o) such that ϕ β,o x) ϕ β,ox). Let us denote by x the unique preimage of x by X X such that x O jβ. In X, the geodesics starting at x with respective speeds ϕ β,o x) and ϕ β,ox) approach each other in the past, since they belong to t 0 Φt L 0 ), which contradicts Lemma

31 Reciprocally, suppose there exists x suppϕ β,o ) suppϕ β,o), such that we have ϕ β,o x) = ϕ β,ox). Consider ρ = Φ n x, ϕ β,o x)) L 0. Let ρ be the pre-image of ρ by S X S X which is in L 0. By definition, we have Φ n ρ) S O jβ, but also Φ n ρ) S O jβ. Therefore, j β = j β, so that β O β. Thanks to this lemma, it is possible for each β B O to build a phase function ϕ β : O R such that for every β β, we have ϕ βx) = ϕ β,o x) for every x suppϕ β,o ). Let χ C c X) be such that suppχ K has a diameter smaller than ε K. Let us write B χ := B suppχ). For every β B χ, we set a β,χ := β β a β,χ e iϕ β ϕ β,suppχ))/h. Defined this way, a β,χ S comp X). Indeed, by Lemma 5.2, the number of terms in the sum is bounded by a constant independent of β, and by Lemma 5.3, the exponentials which appear in the sum are only constants. Therefore, by 5.2), for each l N there exists C l,ɛ > 0 such that for every n N a β,χ C l C l,ɛ e np1/2)+ɛ). 5.7) β B χ ñ β)=n From this construction and from 5.1), we obtain that for any r > 0, l > 0, there exists M r,l > 0 such that we have as h 0: χe h x) = e iϕ β x)/h a βx; h) + R r,l, β B χ ñ β) M r,l log h with R r,l C l = Oh r ). From Lemma 4.10 and from the fact that if β β, we have ϕ βx) ϕ β x) thanks to Lemma 5.3, we deduce that there exists a constant C 1 such that for all β, β B χ, we have ϕ βx) ϕ β x) C 1 e b 0 maxñ β),ñ β )). 5.8) This concludes the proof of Theorem Proof of Corollary 3.2 The main ingredient in the proof of Corollary 3.2 is non-stationary phase. Let us recall the estimate we will use, and which can be proven by integrating by parts. Let a, φ S comp X), with suppa) suppφ). We consider the oscillatory integral: I h a, φ) := X ax)e iφx,h) h dx. The following result is classical, and its proof similar to that of [Zwo12, Lemma 3.12]. 31

32 Proposition 5.1 Non stationary phase). Let ɛ > 0. Suppose that there exists C > 0 such that, x suppa), 0 < h < h 0, φx, h) Ch 1/2 ɛ. Then I h a, φ) = Oh ). Proof of Corollary 3.2. Let a Cc S X), and let us write χ a for a Cc function which is equal to 1 on π X suppa)). We clearly have Op h a)χe h, χe h = Oph a)χ a E h, χ a E h + Oh ). Since the statement of Corollary 3.2 is linear in a Cc S X), it is sufficient to prove it only for a supported in a small open set. In particular, we may suppose that χ a is supported in a small enough set so that Theorem 3.4 applies. We then have with r = 1, l = 0): Oph a)χ a E h, χ a E h = Op h a) = + χa β B ñ β) M log h χa β B ñ β) M log h β, β χa B ñ β ),ñ β) M log h β β e iϕ β x)/h a βx; h), χa β B ñ β) M log h Oph a)e iϕ β x)/h a βx; h), e iϕ β x)/h a βx; h) e iϕ β x)/h a βx; h) + Oh) Oph a)e iϕ β x)/h a βx; h), e iϕ β x)/h a β x; h) + Oh). We want to use Proposition 5.1 to say that the second term corresponding to non-diagonal terms is a Oh ). Thanks to 3.11), we know that if β, β B χa are such that ñ β), ñ β ) 1 2 b 0 ɛ ) log h, then we have ϕ βx) ϕ β x) Ch 1/2 ɛ, so that by Proposition 5.1, we have that Therefore, we have Oph a)χ a E h, χ a E h = Op h a)e iϕ β x)/h a βx; h), e iϕ β x)/h a β x; h) = Oh ). + Oph a)e iϕ βx)/h a βx; h), e iϕ βx)/h a βx; h) β B χa Oph a)e iϕ β x)/h a βx; h), e iϕ β x)/h a β x; h) β, β χa B ñ β) M log h ñ β) or ñ β 1 ) ɛ log h 2 b 0 β β χa β B ñ β) M log h ) Oph a)e iϕ β x)/h a βx; h), e iϕ β x)/h a βx; h) + Oh). We may estimate the second and third terms thanks to 3.10). We obtain a remainder which is a O h min 1, ɛ) ) P1/2) 2 b 0, which gives us 32

33 Oph a)χ a E h, χ a E h = Oph a)e iϕ βx)/h a βx; h), e iϕ βx)/h a βx; h) + O h min β B χa 1, P1/2) 2 b 0 ɛ) ). Each term in the first sum is then the Wigner distribution associated to a Lagrangian state, and the associated semiclassical measure may be computed by stationary phase, just as in [Zwo12, 5.1] and this gives us the first part of the statement of Corollary 3.2. Let us now prove 3.12). Recall that the symbols a β,χ x; h) which appear in 5.1) are built by applying formula A.6) several times to Eh 0 see [Ing15, 5.2] for details). In A.6), the phase θ which appears depends only on the trajectory of the point x 1, φ 1x)), so that these phases are the same for β and β if β suppχ) β. In particular, for all β B χ, for all representative β Bn χ and for all x X, we have that a β x) a β,χ x). Therefore, the result will be proven if for every N N, we can find a constant c N > 0 such that for any x X with χx) = 1, we have n=n β Bn χ σ h aβ,χ 2) x) c N. 5.9) Furthermore, still from A.6), we see that σ h a β,χ x; h))x) > 0 as long as there exists ξ such that x, ξ) Φ n β L 0). But for each x X such that χx) = 1, Corollary 4.2 gives us infinitely many ξ i ) i I Tx X and t i > 0 such that x, ξ) Φ ti L 0 ). In particular, for each of them, there is a n i N and a β i Bn χ i such that x, ξ) Φ ni β i L 0 ). Now, using Corollary 4.1, we see that if i i, we have β i β i : otherwise, we could build two distinct geodesics staying at a distance less than r i for all negative times, and whose distance is 0 at time zero, thus contradicting convexity. Hence, since each Bn χ is finite, there exists a β Bn χ for some n N and a i I such that β = β i Therefore, σ h a βi,χ x; h))x) > 0. By continuity, this is true uniformly in a neighbourhood of x. By compactness of suppχ, we obtain 5.9). 6 Small-scale equidistribution Thanks to Corollary 3.2 along with Corollary 3.1, we know that there exist constants C 1, C 2 > 0 such that for any x 0 X such that χx 0 ) = 1, for any r > 0 small enough, we have C 1 r d E h 2 x)dx C 2 r d, 6.1) Bx 0,r) where Bx 0, r). The goal of this section is to show that 6.1) still holds if we replace E h by RE h, and if r depends on h, as long as r > Ch for some C > 0 large enough. Let us start by showing small-scale equidistribution for E h. 33

34 6.1 Small-scale equidistribution for E h The aim of this section is to show the following proposition. Proposition 6.1. Let X be a manifold which satisfies Hypothesis 2.1 near infinity, and which satisfies Hypothesis 2.4. Suppose that the geodesic flow Φ t ) satisfies Hypothesis 2.2 on hyperbolicity, Hypothesis 2.3 concerning the topological pressure. Let E h be a generalized eigenfunction of the form described in Hypothesis 2.7, where Eh 0 is associated to a Lagrangian manifold L 0 which satisfies the invariance Hypothesis 2.5, part iii) of Hypothesis 2.4 and Hypotheses 2.8 and 2.9. Let χ Cc X). Then there exist constants C, C 1, C 2 > 0 such that the following holds. For any x 0 X such that χx 0 ) = 1, for any sequence r h such that 1 >> r h > Ch, we have for h small enough: C 1 rh d E h 2 x)dx C 2 rh. d 6.2) Bx 0,r h ) Proof. The upper bound is a direct consequence of Corollary 3.1. Let us explain why the lower bound holds. As in the proof of Corollary 3.2, E h 2 x) can be written, up to a remainder of order Oɛ) as a sum of terms of the form a β,β x; h)e iφ β,β x)/h, such that β,β a β,β x) C 0 for some C 0 > 0. Furthermore, there exists cε) > 0 such that for all β, β satisfying φ β,β x 0 ) 0, we have φ β,β x 0 ) > cε). Consider a term where φ β,β x 0 ) 0. By a change of variables and by Stokes theorem, we have Bx 0,r h ) a β,β x; h)e iφ β,β x)/h dx = rh d = rh d B0,1) B0,1) a β,β r h x + x 0 ; h)e iφ β,β r hx+x 0)/h dx [ h r h a β,β r h x + x 0 ; h) φ β,β r h x + x 0 ) 2 φ β,β r h x + x 0 ) ) e iφ β,β r hx+x )] 0)/h dx [ h = rh d a β,β r h x + x 0 ; h) r h φ β,β r h x + x 0 ) 2 B0,1) e iφ β,β r hx+x 0)/h φ β,β r h x + x 0 ) )] d n [ h rh d aβ,β r h x + x 0 ; h) φβ,β r r h φ β,β r h x + x 0 ) 2 h x + x 0 ) )) B0,1) e iφ β,β r hx+x 0)/h ] dx c ε)hr d 1 h a β,β ; h) C 1, where c ε) is a constant depending on ε, but not on h. By summing over β and β, we obtain that the sum of non-diagonal terms is bounded by c ε)c 0 hr d 1 h + rh dr ε, where R ε = Oε). On the other hand, thanks to 3.12), the diagonal terms will give a contribution larger than c 0 rh d for some c 0 > 0 independent of h and of x. By taking ε small enough so that R ε < c 0 /3 and C sufficiently small so that c ε)c 0 /C < ε/3, we obtain the result. 34

35 Remark 6.1. Recall that theorem 3.4 tells us that χe h x) = e iϕ β x)/h a β,χ x; h) + R r,l, β B χ ñ β) M r,l log h Since equation 3.12) is still true if we limit ourselves to β such that ñ β) > N for any N 0, the proof above can be easily adapted to give the following result. For any N > 0, there exist C 0, C 1, C 2 > 0 such that for any x 0 X with χx 0 ) = 1, for any sequence r h such that 1 >> r h > C 0h, we have for h sufficiently small: C 1r h d Bx 0,r h ) β B χ N ñ β) M r,l log h e iϕ β x)/h a β,χ x; h) + R r,l x) 2 dx C 2r h. d 6.3) 6.2 Small-scale equidistribution for RE h The aim of this section is to prove Theorem 3.5, which we now recall. Theorem. Let X be a manifold which satisfies Hypothesis 2.1 near infinity, and which satisfies Hypothesis 2.4. Suppose that the geodesic flow Φ t ) satisfies Hypothesis 2.2 on hyperbolicity, Hypothesis 2.3 concerning the topological pressure. Let E h be a generalized eigenfunction of the form described in Hypothesis 2.7, where Eh 0 is associated to a Lagrangian manifold L 0 which satisfies the invariance Hypothesis 2.5, part iii) of Hypothesis 2.4 and Hypotheses 2.8 and 2.9. Let χ Cc X). Then there exist constants C, C 1, C 2 > 0 such that the following result holds. For all x 0 X such that χx 0 ) = 1, for any sequence r h such that 1 >> r h > Ch, we have for h small enough: C 1 rh d RE h 2 x)dx C 2 rh. d 6.4) Bx 0,r h ) In particular, for any bounded open set U X, there exists cu) > 0 and h U > 0 such that for all 0 < h < h U, we have RE h 2 cu). 6.5) U To prove Theorem 3.5, we first need to prove the following lemma, which says that few trajectories go backwards. Lemma 6.1. There exists N N such that for any n N, for any n N and for any β Bn, χ β B χ n, the following holds. Suppose x suppχ) is such that ϕ β x) = ϕ β x). Then there exists a small neighbourhood V of x such that for all y V, we have ϕβ y) = ϕ β y) ) = y, φ β y) ) = Φ t x, φ β x) ) for some t small enough. In the proof, we shall use the following notation. If ρ = x, ξ) E, we shall write ρ t = x, ξ). 35

36 Proof. Let us consider integers n, n such that the statement above is false, and show that there can be only finitely many values for n. For such n, n, there exists a sequence y k converging to some x suppχ) such that ϕ β y k ) = ϕ β y k ), and such that ρ k and ρ do not belong to the same trajectory, where we write ρ k := y k, ϕ β y k )) and ρ = x, ϕ β x)). Write N = 1 + maxn, n ). Thanks to 4.1), by taking k large enough, we may assume that d X Φ t ρ k ), Φ t ρ) ) < r i for t N. On the other hand, since Φ N ρ k ), Φ N ρ) L 0, the point iii) in Hypothesis 2.4 ensures us that for t N, we also have d X Φ t ρ k ), Φ t ρ) ) < r i. We also have Φ N ρ k ) ) t, Φ N ρ) ) t L0, so that d X Φ t ρ k ), Φ t ρ) ) < r i for all t N. Therefore, by Corollary 4.1, we see that d X Φ t ρ k ), Φ t ρ) ) must be constant with respect to time. If n is large enough, then there will be a t > 0 such that Φ t ρ) and Φ t ρ k ) are in a small neighbourhood of the trapped set K. But then, because of hyperbolicity, the two trajectories Φ t ρ k ) and Φ t ρ) cannot remain at a constant distance from each other, unless they belong to the same trajectory. This proves the lemma. Proof of Theorem 3.5. Let us apply Theorem 3.4 for r = 1, l = 0) with χ 1 on U. We have, for x U: M log h E h x) = e iϕ β x)/h a β,u x; h) + O L h) =: S 1 + S 2 + O L h), 6.6) n=0 β B ˆχ n where S 1 denotes the sum for ñ β) N, and S 2 the sum for ñ β) > N, where N is as in Lemma 6.1. We may write RE h 2 = RS 1 ) + S 2 + S 2 )/2) 2 + O L h) = RS 1 ) 2 + S S S RS 1 )RS 2 ) + O L h) ) When we compute Bx 0,r h ) RE h 2, the term Bx 0,r h ) RS 1) 2 + S S2 2 4 is of course non-negative. As for the term Bx 0,r h ) S 2 2, we may use 6.3) to find a constant c > 0 independent of h such that S 2 2 crh d + o h 0 rh). d Let us show that Bx 0,r h ) Bx 0,r h ) 2RS 2 )RS 1 ) crd h ) The function 2RS 2 )RS 1 ) can be written, up to a remainder of size ε, like a finite sum of oscillating terms with phases ±ϕ βx) ± ϕ β x) with N < ñ β) N ε) and ñ β ) N ε). Thanks to Lemma 6.1, we know that for each β, the phase can only cancel on a finite number of curves. Just as in the proof of proposition 6.1, we rewrite 1 rh d e iϕ β x)/h a β,χ x; h)e i±ϕ β x)/h a β,χx; h)dx Bx 0,r h ) = e iϕ β r hx+x 0)/h a β,χ r h x + x 0 ; h)e i±ϕ β r h x+x 0)/h a β,χ r hx + x 0 ; h)dx B0,1) 36

37 By the method of stationary phase, we obtain that each of these integrals is bounded by ) d 1)/2 h c ε o h 0 r h ). By taking ε = c/4, and then C large enough, we obtain 6.8). The lemma follows. 6.3 Lower bound on the nodal volume The aim of this section is to prove the following corollary of Theorem 3.5. Corollary 6.1. Let X be a manifold which satisfies Hypothesis 2.1 near infinity, and which satisfies Hypothesis 2.4. Suppose that the geodesic flow Φ t ) satisfies Hypothesis 2.2 on hyperbolicity, Hypothesis 2.3 concerning the topological pressure. Let E h be a generalized eigenfunction of the form described in Hypothesis 2.7, where E 0 h is associated to a Lagrangian manifold L 0 which satisfies the invariance Hypothesis 2.5, part iii) of Hypothesis 2.4 and Hypotheses 2.8 and 2.9. Fix K a compact subset of X. There exists C K such that Haus d 1 {x K; RE h )x) = 0}) C K h. Proof. Let us fix U X a bounded open set which contains K. The proof relies on the so-called Dong-Sogge-Zelditch formula, which we recall. Let f C0 X), and let g h C X) be a family of smooth functions such that h 2 + 1)g h = 0. Let us write Z h := {x X; g h x) = 0}. Then, as proven in [Don92], [SZ11] 3 we have h 2 + 1)f ) g h dx = h 2 f g h ds, 6.9) X Z h where ds denotes the d 1)-dimensional Hausdorff measure. Let f Cc X; [0, 1]) be such that suppf) K, and f is not identically zero. Applying 6.9) with g h = RE h ), we have h 2 + 1)f ) RE h ) dx = h 2 f RE h ) ds X Z h 6.10) Ch fds Z h ChHaus d 1 {x K; RE h )x) = 0}), where we used Theorem 1.1 to go from the first to the second line. Therefore, the theorem will be proven if we can find a lower bound on the left-hand side in 6.10) uniform in h. Since f is not identically zero, we may find a constant c > 0 and an open set U X such that 3 This formula was proven on compact manifolds, but the proof works exactly the same on noncompact manifolds, as long as the function f is compactly supported. 37

38 f c on U. We have h 2 + 1)f ) RE h ) = f RE h ) + Oh 2 ) X X c RE h ) + Oh 2 ) U c RE h ) 2 E h C 0 U) U C RE h ) 2 U thanks to Theorem 1.1. Therefore, Theorem 6.1 follows from 6.5). Appendix A Reminder of semiclassical analysis A.1 Pseudodifferential calculus Let Y be a Riemannian manifold. We will say that a function ax, ξ; h) C T Y 0, 1]) is in the class S comp T Y ) if it can be written as h ) ) ax, ξ; h) = ã h x, ξ) + O, ξ where the functions ã h Cc T Y ) have all their semi-norms and supports bounded independently of h. If U is an open set of T Y, we will sometimes write S comp U) for the set of funtions a in S comp T Y ) such that for any h ]0, 1], ã h has its support in U. Definition A.1. Let a S comp T Y ). We will say that a is a classical symbol if there exists a sequence of symbols a k S comp T Y ) such that for any n N, a n h k a k h n+1 S comp T Y ). k=0 We will then write a 0 x, ξ) := lim h 0 ax, ξ; h) for the principal symbol of a. We will sometimes write that a S comp Y ) if it can be written as ax; h) = ã h x) + O h ), where the functions ã h Cc Y ) have all their semi-norms and supports bounded independently of h. We associate to S comp T X) the class of pseudodifferential operators Ψ comp h X), through a surjective quantization map Op h : S comp T X) Ψ comp h X). 38

39 This quantization map is defined using coordinate charts, and the standard Weyl quantization on R d. It is therefore not intrinsic. However, the principal symbol map is intrinsic, and we have and σ h : Ψ comp h X) S comp T X)/hS comp T X) σ h A B) = σ h A)σ h B) σ h Op : S comp T X) S comp T X)/hS comp T X) is the natural projection map. For more details on all these maps and their construction, we refer the reader to [Zwo12, Chapter 14]. For a S comp T X), we say that its essential support is equal to a given compact K T X, if and only if, for all χ ST X), ess supp h a = K T X, suppχ T X\K) χa h ST X). For A Ψ comp h X), A = Op h a), we define the wave front set of A as: W F h A) = ess supp h a, noting that this definition does not depend on the choice of the quantization. When K is a compact subset of T X and W F h A) K, we will sometimes say that A is microsupported inside K. Let us now state a lemma which is a consequence of Egorov theorem [Zwo12, Theorem 11.1]. Recall that Ut) is the Schrödinger propagator Ut) = e itp h/h. Lemma A.1. Let A, B Ψ comp h X), and suppose that Φ t W F h A)) W F h B) =. Then we have AUt)B = O L 2 L 2h ). If U, V are bounded open subsets of T X, and if T, T : L 2 X) L 2 X) are bounded operators, we shall say that T T microlocally near U V if there exist bounded open sets Ũ U and Ṽ V such that for any A, B Ψ comp h X) with W F A) Ũ and W F B) Ṽ, we have AT T )B = O L2 L 2h ) Tempered distributions Let u = uh)) be an h-dependent family of distributions in D X). We say it is h-tempered if for any bounded open set U X, there exists C > 0 and N N such that uh) H N h U) Ch N, where H N h U) is the semiclassical Sobolev norm. For a tempered distribution u = uh)), we say that a point ρ T X does not lie in the wave front set W F u) if there exists a neighbourhood V of ρ in T X such that for any A Ψ comp h X) with W F a) V, we have Au = Oh ). 39

40 A.2 Lagrangian distributions and Fourier Integral Operators Phase functions Let φx, θ) be a smooth real-valued function on some open subset U φ of X R L, for some L N. We call x the base variables and θ the oscillatory variables. We say that φ is a nondegenerate phase function if the differentials d θ1 φ)...d θl φ) are linearly independent on the critical set C φ := {x, θ); θ φ = 0} U φ. In this case Λ φ := {x, x φx, θ)); x, θ) C φ } T X is an immersed Lagrangian manifold. By shrinking the domain of φ, we can make it an embedded Lagrangian manifold. We say that φ generates Λ φ. Lagrangian distributions Given a phase function φ and a symbol a S comp U φ ), consider the h-dependent family of functions ux; h) = h L/2 R L e iφx,θ)/h ax, θ; h)dθ. A.1) We call u = uh)) a Lagrangian distribution, or a Lagrangian state) generated by φ. By the method of non-stationary phase, if suppa) is contained in some h-independent compact set K U φ, then W F h u) {x, x φx, θ)); x, θ) C φ K} Λ φ. Definition A.2. Let Λ T X be an embedded Lagrangian submanifold. We say that an h- dependent family of functions ux; h) Cc X) is a compactly supported and compactly microlocalized) Lagrangian distribution associated to Λ, if it can be written as a sum of finitely many functions of the form A.1), for different phase functions φ parametrizing open subsets of Λ, plus an Oh ) remainder. We will denote by I comp Λ) the space of all such functions. Fourier integral operators Let X, X be two manifolds of the same dimension d, and let κ be a symplectomorphism from an open subset of T X to an open subset of T X. Consider the Lagrangian Λ κ = {x, ν; x, ν ); κx, ν) = x, ν )} T X T X = T X X ). A compactly supported operator U : D X) Cc X ) is called a semiclassical) Fourier integral operator associated to κ if its Schwartz kernel K U x, x ) lies in h d/2 I comp Λ κ ). We write U I comp κ). The h d/2 factor is explained as follows: the normalization for Lagrangian distributions is chosen so that u L 2 1, while the normalization for Fourier integral operators is chosen so that U L2 X) L 2 X ) 1. Note that if κ κ is well defined, and if U I comp κ) and U I comp κ ), then U U I comp κ κ ). If U I comp κ) and O T X is an open bounded set, we shall say that U is microlocally unitary near O if U U I L 2 X) L 2 X) microlocally near O κo). 40

41 A.3 Local properties of Fourier integral operators In this section we shall see that, if we work locally, we may describe many Fourier integral operators without the help of oscillatory coordinates. In particular, following [NZ09, 4.1], we will recall the effect of a Fourier integral operator on a Lagrangian distribution which has no caustics. Let κ : T R d T R d be a local symplectic diffeomorphism. By performing phase-space translations, we may assume that κ is defined in a neighbourhood of 0, 0) and that κ0, 0) = 0, 0). We furthermore assume that κ is such that the projection from the graph of κ T R d T R d x 1, ξ 1 ; x 0, ξ 0 ) x 1, ξ 0 ) R d R d, x 1, ξ 1 ) = κx 0, ξ 0 ), A.2) is a diffeomorphism near the origin. Note that this is equivalent to asking that the n n block x 1 / x 0 ) in the tangent map dκ0, 0) is invertible. A.3) It then follows that there exists a unique function ψ C R d R d ) such that for x 1, ξ 0 ) near 0, 0), κ ξ ψx 1, ξ 0 ), ξ 0 ) = x 1, x ψx 1, ξ 0 )), det 2 x,ξψ 0 and ψ0, 0) = 0. The function ψ is said to generate the transformation κ near 0, 0). Thanks to assumption A.2), a Fourier integral operator T I comp κ) may then be written in the form T ux 1 1 ) := 1,ξ 0 ) x 0,ξ 0 /h 2πh) d αx 1, ξ 0 ; h)ux 0 )dxdx 0 dξ 0, A.4) R 2n e iψx with α S comp R 2d ). Now, let us state a lemma which was proven in [NZ09, Lemma 4.1], and which describes the effect of a Fourier integral operator of the form A.4) on a Lagrangian distribution which projects on the base manifold without caustics. Lemma A.2. Consider a Lagrangian Λ 0 = {x 0, φ 0x 0 )); x Ω 0 }, φ 0 Cb Ω 0), contained in a small neighbourhood V T R d such that κ is generated by ψ near V. We assume that κλ 0 ) = Λ 1 = {x, φ 1x)); x Ω 1 }, φ 1 C b Ω 1 ). A.5) Then, for any symbol a S comp Ω 0 ), the application of a Fourier integral operator T of the form A.4) to the Lagrangian state ax)e iφ0x)/h associated with Λ 0 can be expanded, for any L > 0, into T ae iφ0/h )x) = e iφ1x)/h L 1 ) b j x)h j + h L r L x, h), where b j S comp, and for any l N, we have j=0 b j Cl Ω 1) C l,j a C l+2j Ω 0), 0 j L 1, r L, h) Cl Ω 1) C l,l a C l+2l+n Ω 0). 41

42 The constants C l,j depend only on κ, α and sup Ω0 β φ 0 for 0 < β 2l + j. Furthermore, if we write g : Ω 1 x gx) := π κ 1 x, φ 1x)) Ω 0, the principal symbol b 0 satisfies where ξ 0 = φ 0 gx 1 ) and where θ R. b 0 x 1 ) = e iθ/h α 0 x 1, ξ 0 ) det ψ x,ξ x 1, ξ 0 ) 1/2 det dgx1 ) 1/2 a gx 1 ), A.6) Appendix B Hyperbolic near infinity manifolds In this appendix, we will explain how the results of [Ing15] and of the present paper apply in the case of hyperbolic near infinity manifolds. Namely, we have to check that the hypotheses of Section 2 which depend on the structure of the manifold at infinity are satisfied in the case of Hyperbolic near infinity manifolds. More precisely, we will have to build distorted plane waves E h so that Hypothesis 2.7 is satisfied. This section partly follows [DG14, 7]. Definition B.1. We say that X is hyperbolic near infinity if it fulfils the Hypothesis 2.1, and if furthermore, in a collar neighbourhood of X, the metric g has sectional curvature 1 and can be put in the form g = db2 + hb) b 2, where b is the boundary defining function introduced in section 2.1.1, and where hb) is a smooth 1-parameter family of metrics on X for b [0, ɛ). Construction of Eh 0 Let us fix a ξ X. By definition of a hyperbolic near infinity manifold, there exists a neighbourhood V ξ of ξ in X and an isometric diffeomorphism ψ ξ from V ξ X into a neighbourhood V q0,δ of the north pole q 0 in the unit ball B := {q R d ; q < 1} equipped with the metric g 0 : 4dq 2 V q0,δ := {q B; q q 0 < δ}, g 0 = 1 q 2 ) 2, where ψ ξ ξ) = q 0, and denotes the Euclidean length. We shall choose the boundary defining function on the ball B to be b 0 = 2 1 q 1 + q, B.1) and the induced metric b 2 0g 0 S d on S d = B is the usual one with curvature +1. The function b ξ := b 0 ψ 1 ξ can be viewed locally as a boundary defining function on X. For each p S d, we define the Busemann function on B Note that we have lim p q φ B pq) = +. There exists an ɛ > 0 such that the set 1 q φ B 2 ) pq) = log q p 2. U ξ := {x X; d gx,ξ)<ɛ } 42

43 Figure B4: The Lagrangian manifolds L 0 and L 0 when X, g) is hyperbolic near infinity. lies inside V ξ, where g = b 2 ξg is the compactified metric. We define the function φ ξ x) := φ B q 0 ψξ x) ), for x U ξ, 0 otherwise. Let ˆχ : X [0, 1] be a smooth function which vanishes outside of U ξ, which is equal to one in a neighbourhood of ξ. The incoming wave is then defined as E 0 hx; ξ) := χx)e d 1)/2+i/h)φ ξx) if x U ξ, 0 otherwise. E 0 h is then a Lagrangian state associated to the Lagrangian manifold L 0 := {x, x φ ξ x)), x U ξ }, represented on figure B4. L 0 satisfies the invariance hypothesis 2.7), as can be easily seen by working in B, but it will not satisfy hypothesis 2.8). To obtain a manifold L 0 which satisfies hypothesis 2.8) from L 0, we just have to continue propagating the points of L 0 which are already in DE +, that is to say, which go directly to infinity in the future, as follows : L 0 := L 0 Φ t ρ). t 0 ρ L 0 DE+ If U ξ has been chosen small enough, then L 0 will be included in V ξ, and by working in B, we can check that the hypotheses 2.8 and 2.9 are satisfied. 43