Yannick Bonthonneau

RESONANCE-FREE REGIONS FOR NEGATIVELY CURVED MANIFOLDS WITH CUSPS

Keywords: Finite volume manifolds with cusps, scattering determinant, resonances, semiclassical parametrix

The Laplace-Beltrami operator on cusp manifolds has continuous spectrum. The resonances are complex numbers that replace the discrete spectrum of the compact case. They are the poles of a meromorphic function ϕpsq, s P C, the scattering determinant. We construct a semi-classical parametrix for this function in a half plane of C when the curvature of the manifold is negative. We deduce that for manifolds with one cusp, there is a zone without resonances at high frequency. This is true more generally for manifolds with several cusps and generic metrics.

We also study some exceptional examples with almost explicit sequences of resonances away from the spectrum.

The object of our study are complete connected d `1-dimensional negatively curved manifolds of finite volume pM, gq with a finite number κ of real hyperbolic cusp ends. Such a manifold can be decomposed as follows:

M " M 0 \ Z 1 \ ¨¨¨\ Z κ ,
where M 0 is a compact manifold with smooth boundary and negative curvature, and Z i are hyperbolic cusps [START_REF] Belabas | Counting horoballs and rational geodesics[END_REF] pa i , `8q ˆTd i » Z i Q x " py, θq, θ " pθ 1 , . . . , θ d q, i " 1 . . . κ, where a i ą 0, and T d i " T d Λ i " R d {Λ i are d-dimensional flat tori, and the metric on Z i in coordinates py, θq P pa i , `8q ˆTd i is ds 2 " dy 2 `dθ 2 y 2 , which has constant ´1 sectional curvature. Notice that the manifold has finite volume when equipped with this metric. The choice of the coordinate y on a cusp is unique up to a scaling factor, and we choose it so that all T d i 's have volume 1. Such a manifold will be referred to as a cusp-manifold. Mind that we require that they have negative curvature.

The Laplace operator on M is denoted ∆ in the analyst's convention that ´∆ ě 0. The resolvent Rpsq " p´∆ ´spd ´sqq ´1 is a priori defined on L 2 pM q for s ą d{2. Thanks to the analytic structure at infinity, one shows that R can be continued to C as a meromorphic family of operators C 8 c Ñ C 8 whose set of poles is called the resonant set RespM, gq. The original proof is due to Selberg in constant ´1 curvature, to Lax and Phillips [START_REF] Lax | Scattering theory for automorphic functions[END_REF] for surfaces, and this subject was studied by both Yves Colin de Verdière [START_REF] De Verdière | Une nouvelle démonstration du prolongement méromorphe des séries d'Eisenstein[END_REF][START_REF] De Verdière | Pseudo-laplaciens[END_REF] and Werner Müller [START_REF] Müller | Spectral theory for Riemannian manifolds with cusps and a related trace formula[END_REF][START_REF] Müller | The point spectrum and spectral geometry for Riemannian manifolds with cusps[END_REF][START_REF] Müller | Spectral geometry and scattering theory for certain complete surfaces of finite volume[END_REF]. It fits in the general theory of spectral analysis on geometrically finite manifolds with constant curvature ends, see [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF][START_REF] Guillopé | Scattering asymptotics for Riemann surfaces[END_REF].

The spectrum of ´∆ divides into both discrete L 2 spectrum, that may be finite, infinite or reduced to t0u, and continuous spectrum rd 2 {4, `8q with multiplicity κ. We can find in [START_REF] Müller | Spectral theory for Riemannian manifolds with cusps and a related trace formula[END_REF] a precise description of the structure of its spectral decomposition given by the Spectral Theorem. For each cusp Z i , i " 1 . . . κ, there is a meromorphic family of Eisenstein functions tE i psqu sPC on M such that [START_REF] Bonthonneau | Weyl laws for manifolds with hyperbolic cusps[END_REF] ´∆E i psq " spd ´sqE i psq.

The line t s " d{2u corresponds to the continuous spectrum and is called the unitary axis.

The poles of the family are contained in t s ă d{2u Y pd{2, ds, and are called resonances. We also consider the vector E " pE 1 , . . . , E κ q. Let tu u be the discrete L 2 eigenvalues. Then, any f P C 8 c pM q expands as:

f " ÿ xu , f yu `1 4π κ ÿ j"1 ż `8 ´8 E j ˆd 2 `it ˙BE j ˆd 2 `it ˙, f F dt [18, eq. 7.36],
where x¨, ¨y is the L 2 duality product. An important feature of the Eisenstein functions is the following: in cusp Z j , the zeroth Fourier mode in θ of E i writes as

(3) δ ij y s `φij psqy d´s , where φ ij is a meromorphic function. Combining this with (2), we deduce that the family E i psq is unique. If we take the determinant of the scattering matrix φ " tφ ij u, we obtain the scattering determinant ϕpsq. It is known that the set RpM, gq of poles of ϕ is the same as that of tEpsqu s -again, see [18, theorem 7.24]. It also coincides with the poles of the meromorphic continuation of the kernel of the resolvent of the Laplacian that are not on t s " d{2u, [START_REF] Müller | Spectral theory for Riemannian manifolds with cusps and a related trace formula[END_REF].

The uniqueness property of E i psq gives a relation between E i psq and E i pd ´sq, which implies ϕpsqϕpd ´sq " 1. Hence, studying the poles of ϕ in t s ă d{2u is equivalent to studying the zeroes in t s ą d{2u. In this article, we will be giving information on the zeroes of ϕ, keeping in mind that the really important objects are the poles.

The first examples of cusp manifolds to be studied had constant curvature, and were arithmetic quotients of the hyperbolic plane. Let Γ 0 pN q be the congruence subgroup of order N , that is, the kernel of the morphism π : SL 2 pZq Ñ SL 2 pZ N q. Then, H{Γ 0 pN q is a cusp surface -or orbifold but we will ignore this technicality here. For such examples, and more generally, for all constant curvature cusp surfaces H{Γ, if cusp Z i is associated with the point 8 in the half plane model, then the associated Eisenstein functions can be written as a series [START_REF] Cheeger | Comparison theorems in Riemannian geometry[END_REF] E i psqpzq "

ÿ rγsPΓ i zΓ r pγzqs s
where Γ i is the maximal parabolic subgroup of Γ associated with Z i . Recall a Dirichlet series is a function of the form

f psq " ÿ kě0 a k λ s k
, where pλ k q is a strictly increasing sequence of real numbers.

In the case of constant curvature cusp surfaces, Selberg proved -see [START_REF] Selberg | Remarks on the distribution of poles Eisenstein series[END_REF] -that there is a non-zero Dirichlet series L converging absolutely for t s ą du so that [START_REF] De Verdière | Une nouvelle démonstration du prolongement méromorphe des séries d'Eisenstein[END_REF] ϕpsq " ˆπΓps ´1{2q Γpsq ˙κ{2 Lpsq, κ being the number of cusps. This implies:

Theorem (Selberg). Let pM, gq be a constant curvature cusp surface. There may be a finite set of resonances in p1{2, 1s. The other resonances are contained in a vertical strip of the form t1{2 ´δ ď s ă 1{2u, for some δ ą 0.

While conducting his systematic study of the spectral theory of the Laplacian on cusp surfaces, Müller wondered whether Selberg's theorem still holds in variable curvature -see [20, page 274]. Froese and Zworski [START_REF] Froese | Finite volume surfaces with resonances far from the unitarity axis[END_REF] gave a counter-example, that had positive curvature. The following theorem gives a partial answer in negative curvature.

Theorem 1. For M a cusp manifold, let GpM q be the set of C 8 metrics g on M such that pM, gq is a cusp manifold with negative sectional curvature. If U ĂĂ M is open, let G U pM q be the set of metrics in GpM q that have constant curvature outside of U . Endow GpM q and G U pM q with the C 2 topology on metrics. Then (I) There exist hyperbolic cusp surfaces pM, g 0 q and non-empty open sets U ĂĂ M such that for all g P G U pM q, RespM, gq is still contained in a (possibly different) vertical strip. (II) Given any cusp manifold M , for an open and dense set of g P GpM q, or all of GpM q when there is only one cusp, there is a δ ą d{2 such that for any constant C ą 0, ts P RespM, gq, s ă d ´δ, s ą ´C log | s|u is finite.

(III) There is a 2-cusped surface pM, gq with the following properties. The resonant set RespM, gq is the union of Res strip , Res f ar and an exceptional set Res exc , so that

Res strip " ts P RespM, gq, s ą d ´δu

Res exc " ! s i , s i , i P N, s i " si `Op|s i | ´β q
) for some β ą 0, Res f ar X t s ą ´C log | s|u is finite for any C ą 0.

where δ ą 0, and the si 's and si 's are the zeroes of se ´sT ´C0 for some constants T ą 0 and C 0 ‰ 0 -they are related to the several branches of the Lambert W function. (IV) For a bigger open and dense set of metrics g P GpM q, containing the example in (III), there are constants δ ą 0, and C 0 ą 0 such that ts P RespM, gq, s ă d ´δ, s ą ´C0 log | s|u is finite. Our theorem does not solve completely the problem. First, there may exist some metrics for which the theorem does not say anything. We conjecture that this set is empty, that is to say: Conjecture 1. The set of metrics in (IV) is actually GpM q.

Not being able to prove this, part 4 is dedicated to showing that the complement is contained in a C 8 infinite codimensional submanifold of GpM q. Except for some special cases, our theorem does not give much insight on the presence or the absence of resonances far from the spectrum -i.e in the region s ąą log | s|. It seem that one would have to invoke different techniques to make progress in the direction of Conjecture 2. For an open and dense set of g P GpM q, there is an infinite number of resonances outside of any strip d{2 ą s ą d ´δ.

Our reason for conjecturing this is that the existence of such resonances seems to be more stable than their absence.

The main tool to prove theorem 1 is a parametrix for the scattering determinant ϕ in a half plane t s ą δ g u. Thanks to the form of that parametrix -sums of Dirichlet serieswe will be able to determine zones where ϕ does not vanish.

Theorem 2. Let pM, gq be a negatively curved cusp manifold with κ cusps. There is a constant δ g ą d{2 and Dirichlet series L 0 , . . . , L n , . . . with abscissa of absolute convergence δ g such that if at least one of the L n 's does not identically vanish, for s ą δ g , as s Ñ ˘8, ϕpsq " s ´κd{2 L 0 psq `s´κd{2´1 L 1 psq `. . . . Actually, the constant δ g is the pressure of the potential pF su `dq{2, where F su is the unstable jacobian. In constant curvature, F su " ´d and δ g " d.

This theorem is a consequence of a more precise estimate -see Theorem 5. The L n 's are defined by dynamical quantities related to scattered geodesics. Those are geodesics that come from one cusp and escape also in a cusp -maybe the same -spending only a finite time in the compact part of M , called the Sojourn Time. This terminology was introduced by Victor Guillemin [START_REF] Guillemin | Sojourn times and asymptotic properties of the scattering matrix[END_REF]. In that article, for the case of constant curvature, he gave a version similar to ours of [START_REF] De Verdière | Une nouvelle démonstration du prolongement méromorphe des séries d'Eisenstein[END_REF]. He also conjectured that his formula could be generalized to variable curvature, which is the point of the present article, some 40 years later -see the concluding remarks pp. 79 in [START_REF] Guillemin | Sojourn times and asymptotic properties of the scattering matrix[END_REF]. Lizhen Ji and Maciej Zworski gave a related result in the case of locally symmetric spaces [START_REF] Ji | Scattering matrices and scattering geodesics of locally symmetric spaces[END_REF].

Sojourn Times are objects in the general theory of classical scattering -see [START_REF] Petkov | Sojourn times, singularities of the scattering kernel and inverse problems[END_REF]. Maybe ideas from different scattering situations may help to prove Conjecture 1, that may be reformulated as Conjecture 1'. Given g P GpM q, at least one L i is not identically zero.

The structure of the article is the following. In section 1 we recall some definitions and results on cusp manifolds, and prove the convergence of a modified Poincaré series. Section 2 is devoted to building a parametrix for the Eisenstein functions, via a WKB argument, using the modified Poincaré series. In section 3, we turn to a parametrix for the scattering determinant. To use the Stationary Phase method, most of the effort goes into proving the non-degeneracy of a phase function. The purpose of section 4 is to study the behaviour of the series L i when we vary the metric. Finally, we prove theorem 1 in section 5. In appendix A, for lack of a reference, we give a proof of a regularity result on horocycles. This result may be of interest for the study of negatively curved geometrically finite manifolds in general.

This work is part of the author's PhD thesis. In a forthcoming article [START_REF] Bonthonneau | Weyl laws for manifolds with hyperbolic cusps[END_REF], we will deduce precise spectral counting results from Theorem 2.

Scattered geodesics and some potential theory on cusp manifolds

Recall that a manifold N is said to have bounded geometry when its injectivity radius is strictly positive, and when ∇ k R is bounded for all k " 0, 1, . . . -R being the Riemann curvature tensor of N . Since the injectivity radius goes to zero in a cusp, a cusp manifold cannot have bounded geometry. However, its universal cover Ă M does. Since the curvature of M is negative, Ă M is also a Hadamard space -diffeomorphic to R d`1 -and we can define its visual boundary B 8 Ă M homeomorphic to S d , and visual compactification

M " Ă M Y B 8 Ă M .
In all the article, unless stated otherwise, we will refer to the projection T ˚Ă M Ñ Ă M as π; when we say geodesic, we always mean unit speed geodesic.

The results given without proof are from the book [START_REF] Paulin | Equilibrium states in negative curvature[END_REF].

1.1. Hadamard spaces with bounded geometry and negative curvature. Let us define the Busemann cocycle in the following way. For p P B 8 Ă M , Let β p px, x 1 q :" lim wÑp dpx, wq ´dpx 1 , wq.

For each p P B 8 Ă M , we pick m p P Ă M -we will specify this choice later, see remark 2. Then, we define the horosphere Hpp, rq (resp. the horoball Bpp, rq) of radius r P R `based at p as Hpp, rq :"

!

x P Ă M ˇˇβ p px, m p q " ´log r

) and Bpp, rq :"

! x P Ă M ˇˇβ p px, m p q ď ´log r ) . (6)
We also define [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF] G p pxq :" β p px, m p q.

Beware that with these notations, horoballs Bpp, rq increase in size as r decreases. The number r will correspond to a height y in the coming developments.

Since the curvature of Ă M is pinched-negative ´k2 max ď K ď ´k2 min , Ă M has the Anosov property. That is, at every point of S ˚Ă M , there are subbundles such that

T pS ˚M q " RX ' E s ' E u ,
where X is the vector field of the geodesic flow ϕ t . This decomposition is invariant under ϕ t , and there are constants C ą 0, λ ą 0 such that for t ą 0

}dϕ t | E s } ď Ce ´λt and }dϕ ´t| E u } ď Ce ´λt .
The subbundle E s (resp. E u ) is tangent to the strong stable (resp. unstable) foliation W s (resp. W u ). The subbundles E s , E u are only Hölder -see [23, theorem 7.3

] -but each leaf of W s , W u is a C 8 submanifold of Ă M -see lemma A.1.
Remark 1. We have to say how we measure regularity on Ă M and T Ă M . In T T Ă M , we have the vertical subbundle V " ker T π : T T Ă M Ñ T Ă M . Since Ă M is riemannian, we also have a horizontal subbundle H given by the connection ∇. Both V and H can be identified with T Ă M , and the Sasaki metric is the one metric on T Ă M so that V K H and those identifications are isometries.

We endow T Ă

M with the Sasaki metric, and then also T ˚Ă M by requesting that v Þ Ñ xv, ¨y is an isometry. For a detailed account on the Sasaki metric, see [START_REF] Gudmundsson | On the geometry of tangent bundles[END_REF]. On all the manifolds that appear, when they have a metric, we define their C k spaces, k P N, using the norm of their covariant derivatives:

}f } C n :" sup k"0,...,n }∇ k f } 8 .
Then, C 8 " X ně0 C n . For a more detailed account of C k spaces on a riemannian manifold, see for example the appendix "functionnal spaces in a cusp" in [START_REF] Bonthonneau | Long time quantum evolution of observables on cusp manifolds[END_REF].

There are useful coordinates for describing the geodesic flow ϕ t on S ˚Ă M . We associate with a geodesic its endpoints p ´, p `. Then we have the identification

S ˚Ă M » B 2 8 Ă M ˆR given by ξ Þ Ñ pp ´, p `, t " β p ´pπξ, m p ´qq. Here B 2 8 Ă M is obtained by removing the diagonal from B 8 Ă M ˆB8 Ă M .
In those coordinates, ϕ t is just the translation by t in the last variable. Moreover, the strong unstable manifold of ξ is the set tp ´" p ´pξq, t " tpξqu. For the strong stable manifold, it is a bit more complicated in this choice of coordinates.

We deduce that W u pξq is the set of outer normals to the horosphere based at p ´pξq, through πξ. The horospheres Hpp, rq are C 8 submanifolds, and each G p is a smooth function so that dG p P C 8 p Ă M q. The proof uses the fact that the unstable manifolds W u are C 8 (lemma A.1), and the fact that there can be no conjugate points in negative curvature.

For p P B 8 Ă M , we introduce W u0 ppq as the set of ξ P S ˚Ă M such that p ´pξq " p. It is the set of outer normals to horospheres based at p. It is the graph of dG p , and

G p pπϕ t px, dG p qq " G p `t.
We will refer to W u0 ppq as the incoming Lagrangian from p.

1.2. Parabolic points and scattered geodesics. Now, let Γ " π 1 pM q. It is a discrete group acting freely on Ă M by isometries. The elements of Γ can be seen to act by homeomorphisms on M . We can define the limit set ΛpΓq as Γ ¨x0 X B 8 Ă M , where the closure was taken in M , and x 0 is an arbitrary point in Ă M . This does not depend on x 0 .

If γ P Γ is not the identity, one can prove that it has either. (1) Exactly one fixed point in Ă M , (2) Exactly two fixed points on B 8 Ă M , (3) Exactly one fixed point in B 8 Ă M . Then we say that it is (1) elliptic, (2) loxodromic, or (3) parabolic. Here there are no elliptic elements in Γ, since Γ acts freely on Ă M . Our study will be focused of the parabolic elements of Γ.

All the parabolic elements γ of Γ are regular, in the following sense: there is r γ P R ˚so that if p γ is the fixed point of γ, Bpp γ , r γ q has constant curvature ´1. We denote by Γ par the set of parabolic elements in Γ. The set Λ par of p γ 's is the set of parabolic points of B 8 Ă M .

Let p P Λ par . Then, horoballs centered at p will project down to M as neighbourhoods of some cusp Z i , and we say that p is a parabolic point that represents Z i . When γ.p " p, we also say that γ represents Z i . Objects (points in the boundary, or elements of Γ par ) representing the same cusp will be called equivalent. Γ acts on Γ par by conjugation, and elements of the same orbit under Γ are equivalent -however observe that the equivalence classes gather many different orbits under Γ.

If p is a parabolic point representing Z i , write p P Λ i par . Let Γ p ă Γ be its stabilizer. It is a maximal parabolic subgroup. We always have Γ p » π 1 pT d i q » Z d . The set of parabolic points equivalent to p is in bijection with Γ p zΓ " tΓ p γ, γ P Γu.

Remark 2. We will not use the functions G p when p is not a parabolic point. When p P Λ i par , one can choose the point m p so that G p coincides with ´log ỹp on the horoball Hpp, r p q, where ỹp is obtained on Hpp, r p q by lifting the height function y on the cusp Z i . With this choice, for p P Λ par and γ P Γ, we have the equivariance relation [START_REF] Froese | Finite volume surfaces with resonances far from the unitarity axis[END_REF] G γ ´1p " G p ˝γ.

The following lemma seems to be well known in the literature. However, since we cannot give a reference for a proof, we have written one down. Proof. Let us pick a cusp Z i , and a point p P Λ i par . Then, we consider x P BZ i in the boundary of Z i in M . We can lift x to x P Hpp, a i q. The orbit under Γ of any x1 P Hpp, a i q will remain at bounded distance of the orbit of x under Γ. We deduce that ΛpΓq is the intersection of the closure of Y γ γHpp, a i q with the boundary B 8 Ă

M . This implies in particular that Λ i par Ă ΛpΓq. Now, we can find a distance d on M that is compatible with its topology. Indeed, take a point m P Ă M , and consider the distance d obtained on M by requesting that

v P Bp0, 1q Ă T m Ă M Þ Ñ exp m tv ˆargth|v|u is an isometry.
Then, for that distance, the sequence of images γHpp, a i q have shrinking radii. Now, take a sequence of points xj P γ j Hpp, a i q, so that xj Ñ q P ΛpΓq. We have γ j Hpp, a i q " Hpγ j p, a i q, and so dpx j , γ j pq Ñ 0. This proves that ΛpΓq " Λ i par . Next, consider the open set U in Ă M obtained by taking only points of Ă M that project to points in the compact part M0 ĂĂ M . There is C ą 0 such that given x1 P U , for any x2 P U , there is a γ P Γ such that dpγ x1 , x2 q ď C.

Let U be the closure of U in M . Since U is at distance at most C of the orbit of any of its points under Γ, we deduce that the limit set is U X B 8 Ă M .

Then, we find that ΛpΓq " Y γ γHpp, a i q X B 8 Ă M " U X B 8 Ă M . But, we also have Y γ γHpp, a i q X B 8 Ă M " Y γ γBpp, a i q X B 8 Ă M . We deduce that [START_REF] Gudmundsson | On the geometry of tangent bundles[END_REF] ΛpΓq "

! Y γ γBpp, a i q Y U ) X B 8 Ă M " U Y γ γBpp, a i q X B 8 Ă M " B 8 Ă M .
Geodesics that enter a cusp eventually come back to M 0 when they are not vertical, that is, when they are not directed along ˘By . A geodesic that is vertical in a cusp is said to escape in that cusp.

Definition 1.2. The scattered geodesics are geodesics on M that escape in a cusp for both t Ñ `8 and t Ñ ´8.

The set of scattered geodesics is denoted by SG. Such a geodesic, when lifted to Ă M , goes from one parabolic point to another, and hence is entirely determined by its endpoints. Take p, q representing Z i , Z j . For γ, γ 1 P Γ, the pair of endpoints pp, γqq and pγ 1 p, γ 1 γqq represent the same geodesic on M . We let SG ij be the set of geodesics scattered from Z i to Z j . From the above, we deduce that when i ‰ j, [START_REF] Guillemin | Sojourn times and asymptotic properties of the scattering matrix[END_REF] SG ij » Γ i zΓ{Γ j and SG ii » Γ i zpΓ ´Γi q{Γ i , where Γ i (resp. Γ j ) is any maximal parabolic subgroup representing Z i (resp Z j ).

On the other hand, we can consider the set of C 1 curves that start in Z i above the torus ty " a i u and end in Z j , above the torus ty " a j u. Among those curves, we can consider the classes of equivalence under free homotopy. Let π ij 1 pM q be the set of such classes. One can prove that in each class rcs P π ij 1 pM q, there is exactly one element c of SG ij -this is implied by the fact that the riemannian distance on Ă M is uniquely geodesic. In particular, this proves that SG is countable. Hence, we have an identification SG ij » π ij 1 pM q. In what follows, when there is no ambiguity on the metric, we will write directly c P π ij 1 pM q. In section 4, we will study variations of the metric, and will come back to the notation rcs P π ij 1 pM q. For a scattered geodesic c ij , we define its Sojourn Time in the following way. Take one of its lifts cij to Ă M , with endpoints p, q. Let T be the (algebraic) time that elapses between the time cij hits tỹ p " a i u, and the time it crosses tỹ q " a j u. Then, let [START_REF] Guillopé | Scattering asymptotics for Riemann surfaces[END_REF] T pc ij q :" T ´log a i ´log a j .

This does not depend on the choice of a i and a j (as defined in (1)), nor on the choice of the lift cij . We say that T pc ij q is the Sojourn Time of c ij , and we can see T as a function on π ij 1 pM q. Given T ą 0, there is a finite number of c P SG ij with sojourn time less than T (otherwise, we would have two such curves that would be so close from one another that they would be homotopic).

We denote by ST (resp. ST ij ) the set of T pcq for scattered geodesics (resp. between Z i and Z j ). We also call the Sojourn Cycles and denote by SC the set of sums [START_REF] Heintze | Geometry of horospheres[END_REF] T 1 `¨¨¨`T κ where σ is a permutation of t1, . . . , κu, and T i P ST i,σpiq . A set of scattered geodesics tc 1 , . . . , c κ u such that c i P SG iσpiq will be called a geodesic cycle. where ş γx x V ´s is the integral of V ´s along the geodesic from x to γx. The convergence of both series does not depend on x, only on s.

We will write ş V ´s instead of ş pV ´sq to reduce the size of the expressions. We will assume that the integrand is all that is written after the sign ş , until we encounter another ş sign.

When p is a point on the boundary, x and x 1 in Ă M , ş p x ´şp x 1 V will refer to the limit of

ş p x V ´şp x 1 V as Ă M Q p Ñ p.
When V is Hölder, this limit exists because the geodesics rx, ps and rx 1 , ps are exponentially close.

When we sum over trγs P ΞzΓu we mean that we sum over a set of representatives for ΞzΓ (Ξ being assumed to be a subgroup of Γ).

We only work with reversible potentials V . That means that ı ˚V is cohomologous to V (following [START_REF] Paulin | Equilibrium states in negative curvature[END_REF], ı is the antipodal map in S Ă M ). In other words, we require that (13)

ż y x V ´ı˚V " Apyq ´Apxq
where A is a bounded Hölder function on S Ă M , invariant by Γ. In particular when this is the case, we can replace V by ı ˚V in the integrals, losing a Op1q remainder. It is then harmless to integrate along a geodesic in a direction or the other.

In our case where Γ is the π 1 of M , it is a general fact that there is a finite δpΓ, V q P R such that P Γ,V converges for s ą δpΓ, V q and diverges for s ă δpΓ, V q. This number is called the critical exponent of pΓ, V q. We also call δ Γ " δpΓ, 0q the critical exponent of Γ.

The exponent of convergence of a maximal parabolic subgroup Γ p is always δ Γp " d{2. Additionally, the Poincaré series for Γ p diverges at d{2 (Γ p is divergent). This can be seen computing explicitely with the formula for the distance between two points py, θq and py, θ 1 q in the half-space model of the real hyperbolic space H d`1 [START_REF] Ji | Scattering matrices and scattering geodesics of locally symmetric spaces[END_REF] dppy, θq, py, θqq " 2argsh |θ ´θ1 | 2y .

Definition 1. [START_REF] Bonthonneau | Long time quantum evolution of observables on cusp manifolds[END_REF]. In what follows, we say that a potential V is admissible if the following holds. First, V is a Hölder function on S Ă M , invariant by Γ and reversible. Second, there are positive constants C, λ, and a constant V 8 P R such that whenever T ą 0, if πϕ t pξq stays in an open set of constant curvature ´1 for t P r0, T s, then for t P r0, T s, [START_REF] Kobayashi | Foundations of differential geometry[END_REF] |V pϕ t pξqq ´V8 | ď Ce ´λt .

Observe that an admissible potential has to be bounded. We will mostly use the potential V 0 " pF su `dq{2 where F su is the unstable jacobian (see (31) and (32)). We start with the following lemma: Lemma 1.4. Let V be an admissible potential. Then δpΓ, V q ą δ Γp `V8 .

If V " 0, this is the consequence of [7, Proposition 2]. We will actually follow their proof closely, but before, we need two observations on triangles in Ă M .

Remark 3. 1. Consider a triangle with sides a, b, c and angles α, β, γ in a complete Hadamard space Ă M k of curvature ´k2 . We have cosh kc " cosh ka cosh kb ´sinh ka sinh kb cos γ.

Assume that γ ą π{2 (the triangle is obtuse). Then we find that there is a constant

C k ą 0 -smooth in k ‰ 0 and k ‰ 8 -such that (16) |c ´pa `bq| ď C k .
Since the curvature of M is pinched, by the Topogonov comparison theorem for triangles, the same is true for obtuse triangles in Ă M , with a constant C controlled by k min and k max .

2. Now, we consider a triangle with sides c 0 , c 1 , c 2 in Ă M , and V an admissible potential on Ă M . Take C ą 0. Among those triangles, we restrict ourselves to the ones such that the length of c 0 is at most C. Then (17)

ż c 1 ´żc 2 V " Op1q.
this is still valid if the vertex at c 1 X c 2 is at infinity. Actually, to prove this, first observe that it suffices to make computations for that case when c 1 X c 2 is at infinity. Then it follows directly from the fact that the two curves are exponentially close in that case.

Proof of lemma 1.4. The limit set of Γ p is reduced to tpu. In Hpp, a i q, Γ p has a Borelian fundamental B domain whose closure is compact. We can obtain a fundamental domain G for Γ p on B 8 Ă M ztpu by taking the positive endpoints of geodesics from p through B. From [23, Proposition 3.9], which is due to Patterson, there exists a Patterson density µ of dimension δpΓ, V q on Ă M , i.e, a family of finite non-zero borelian measures pµ x q xP Ă M on B 8 Ă M , so that for any x,

x 1 P Ă M , γ P Γ, (18)
γ ˚µx " µ γx , dµ x dµ x 1 pqq " exp "ż q x ´ż q x 1 V ´δpΓ, V q * , q P B 8 Ă M .
Additionally, the µ x 's are exactly supported on ΛpΓq " B 8 Ă M , so µ x pG q ą 0. Take x P B. We have

8 ą µ x pB 8 Ă M q " ÿ γPΓp µ x pγG q `µx ptpuq But, µ x pγ ´1G q " γ ˚µx pG q " ż G exp "ż q γx ´ż q x V ´δpΓ, V q * dµ x pqq So we find ż G ÿ γPΓp exp "ż q γx ´ż q x V ´δpΓ, V q * dµ x pqq " ÿ γPΓp µ x pγG q ă 8.
For q P G , let x q P B be its projection on Hpp, a i q. Since we have dpx, x q q " Op1q -from the choice of B -we use [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF] and uniformly in γ P Γ p ,

ż q xq ´ż q x V ´δpΓ, V q " Op1q ; ż γx xq ´ż γx x V ´δpΓ, V q " Op1q
Take zpx 1 q the intersection of the geodesic rq, x 1 s and the horosphere H q based at q through x q . The set of zpx 1 q, x 1 P Hpp, a i q has to be bounded. Indeed, Hpp, a i q is not compact, but the only way to go to infinity in Hpp, a i q is to tend to p, and we find that as x 1 Ñ p, zpx 1 q Ñ x q . The geometry is described in figure 2.

Using again [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF],

ż q xq ´ż q zpx 1 q V ´δpΓ, V q " Op1q ; ż zpx 1 q x 1 ´ż xq x 1 V ´δpΓ, V q " Op1q, q p
Hpp, a i q

H q x q G B x 1 zpx 1 q Figure 2.
and sum everything up (with x 1 " γx)

ż q γx ´ż q x V ´δpΓ, V q " Op1q `ż zpγxq γx `ż q zpγxq ´ż q xq V ´δpΓ, V q " Op1q `ż xq γx V ´δpΓ, V q " Op1q `ż x γx V ´δpΓ, V q. ( 19
)
As a consequence, P Γp,V px, δpΓ, V qqµ x pG q ă 8, and since µ x pG q ą 0, (20) P Γp,V px, δpΓ, V qq ă 8.

Since V is an admissible potential, for each x P Ă M , there is C x ą 1 such that for all s P R,

1 C x P Γp,V px, sq ď P Γp px, s ´V8 q ď C x P Γp,V px, sq.
Since Γ p is divergent, taking s " δpΓ, V q, we deduce that δpΓ, V q ´V8 ą d{2.

In the following developments, we will need the convergence of a modified Poincaré series. Take V an admissible potential. For a cusp Z i , take a point p P Λ i par , and let π a i p pxq be the intersection of the geodesic through p and x with Hpp, a i q. The horoballs Bpp, a i q, p P Λ i par are all pairwise disjoint. Indeed, the restriction of the projection Ă M Ñ M to any such horoball is a universal cover of Z i . This implies that for x P Bpp, a i q, the part of the orbit of x under Γ that stays in Bpp, a i q has to be its orbit under Γ p .

For x P M , take x P Ă M a lift of x, and define

P Z i ,V px, sq :" ÿ rγsPΓpzΓ,γ xRBpp,a i q exp # ż γ x π a i p pγ xq V ´s+ .
This does not depend on the choice of x. Given a point x P Ă M , among a family tγx, rγs P Γ p zΓu, there is at most one point in Bpp, a i q, and such a point has to be one that minimizes G p . So, for a point x P M , take x p to be a lift minimizing G p among the lifts of x, and let G i pxq :" G p px p q. For q P Λ j par , also let ( 21)

P ij V psq :" ÿ ΓpγΓq‰Γp exp " pV 8 ´sqT pp, γqq `ż γq p V ´V8 * ,
where T pp, γqq is the sojourn time of the geodesic on M that lifts to rp, γqs. Observe that the set tΓ p γΓ q ‰ Γ p u can be identified with SG ij , from equation [START_REF] Guillemin | Sojourn times and asymptotic properties of the scattering matrix[END_REF]. The main result of this section is Lemma 1.5. The series P Z i ,V px, sq and P ij V psq converge if and only if s ą δpΓ, V q. Additionally, when ą 0, there is a constant C ą 0 such that for s ą δpΓ, V q ` , ( 22)

}P Z i ,V px, sq} L 2 pM q ď C
Our proof is inspired by [START_REF] Belabas | Counting horoballs and rational geodesics[END_REF], and we generalize their Theorem 1.1. One can also see the article [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF], or the proposition 3 in [START_REF] Paulin | Regards croisés sur les séries de Poincaré et leurs applications[END_REF]. For two real valued functions f and g, we write fg when there is a constant C ą 0 with Cg ď f ď g{C. In the following, when we use that notation, we let the constant C depend on s, but not on x, γ, p. We fix a cusp Z i , a representing parabolic point p P Λ i par .

Proof. The proof is divided into 3 parts. First, we compare the values of terms of the sum for different x's, to check that the convergence does not depend on x indeed. Then, we study the sum for some well chosen x, to find the convergence exponent. At last, we turn to asymptotics in cusps. We let P ˚be the series where we have not excluded γx q P Bpp, a i q from the sum.

1. Take x, x 1 two points in M , at distance D ą 0, and two lifts x and x1 such that dpx, x1 q " D.

Take γ P Γ. Assume that G p pγ x1 q ě G p pγ xq. Then the projection x 1 γ of γ x1 on the horoball Bpp, G p pγ xqq is at distance OpD `1q from γ x. This is a simple consequence of equation ( 16) for the triangle with vertices γ x, γ x1 , x 1 γ . Write

ż γ x1 π a i p pγ x1 q V ´s ´ż γ x π a i p pγ xq V ´s " ż γ x1 x 1 γ V ´s `ż x 1 γ π a i p pγ x1 q V ´ż γ x π a i p pγ xq V.
Since V is Hölder, and bounded, we deduce that

ż γ x1 π a i p pγ x1 q V ´s ´ż γ x π a i p pγ xq V ´s " OpD `1q " p1 `|s|q `ż 8 0 pe ´kmin t q µ dt *
where µ is the Hölder exponent of V . The constants in the estimates do not depend on x and x 1 . We have used that the geodesics joining γ x, π a i p γ x and x 1 γ , π a i p γ x1 are on the same strong stable manifold. We deduce that for some constant C ą 0, [START_REF] Paulin | Equilibrium states in negative curvature[END_REF] e ´CpD|s|`1q ď P ˚px 1 , sq P ˚px, sq ď e CpD|s|`1q x, x 1 P M, dpx, x 1 q " D.

2. Take now a point x P M so that x p P Hpp, a i q Ă Bpp, a i q. We claim that for all x 1 P Hpp, a i q, (24)

ż γxp x 1 V ´s " pV 8 ´sqdpx 1 , π a i p pγx p qq `Op1q `ż γxp π a i p pγxpq V ´s.
The remainder being bounded independently from x p and γ. Let us assume that this holds for now. Then, we write

P Γ,V px p , sq ´PΓp,V px p , sq " ÿ Γpγ‰Γp ÿ αPΓp exp # ż γxp αxp V ´s+ , - ÿ Γpγ‰Γp ÿ αPΓp exp # pV 8 ´sqdpαx p , π a i p pγx p qq `ż γxp π a i p pγxpq V ´s+ -P Z i ,V px p , sqP Γp px p , s ´V8 q. Hence P Z i ,V px p , sq - P Γ,V px p , sq ´PΓp,V px p , sq P Γp px p , s ´V8 q .
But from lemma 1.4, we know that δpΓ, V q ą δpΓ p , V q.

For the proof of ( 24), we will just say that it is based on the fact that the triangle with vertices x 1 , π a i p pγx p q, and γx p is obtuse at π a i p pγx p q. The estimate follows from remark 3 and the way to obtain it was exemplified in the proof of [START_REF] Müller | The point spectrum and spectral geometry for Riemannian manifolds with cusps[END_REF].

3.

We turn to asymptotics in the cusps. Take x P Z j and q P Λ j par (if i " j, take p " q). Let x q minimize G q among the lifts of x. Observe that the map pΓ p γΓ q ‰ Γ p , α P Γ q q Þ Ñ Γ p γα is a bijection onto Γ p zΓ if i ‰ j, and Γ p zpΓ ´Γp q if i " j. We hence rewrite

P Z i ,V px, sq " ÿ ΓpγΓq‰Γp ÿ αPΓq exp # ż γαxq π a i p pγαxqq V ´s+ .
Consider H q the horosphere based at q, through x q . Let z γ (resp. z 1 γ ) be the point of intersection of the geodesic rp, γqs with γH q (resp. Hpp, a i q). From ( 19), we have

ż γαxq π a i p pγαxqq V ´s " Op1q `ż zγ p `ż γαxq zγ ´ż π a i p pγαxqq p V ´s.
However, the distance between z 1 γ and π a i p pγαx q q is uniformly bounded. This is a direct consequence of lemma 3.

*

`pV 8 ´sq `T pp, γqq ´Gq px q q `dpγ ´1z γ , αx q q ˘`Op1q.

where T pp, γqq is the sojourn time for the geodesic rp, γqs. It follows that

P Z i ,V px, sq - ÿ ΓpγΓq‰Γp exp " pV 8 ´sqT pp, γqq `ż γq p V ´V8
* ˆÿ αPΓq exp pV 8 ´sqp´G q px q q `dpγ ´1z γ , αx q qq ( In the RHS, the first term does not depend on x; we recognize P ij V psq. The second is related to P Γq px q q. We can see it as a Riemann sum as x q Ñ q. Indeed, Γ q » Z d , and we can write explicitely the second term as

*

As x q Ñ q, y " e ´Gqpxqq Ñ `8, and we can see this as a Riemann sum for the function f " expt2pV 8 ´sq argshu for the parameter 2y. It should be equivalent to p2yq d ş R d f . However f is integrable if and only if s ´V8 ą d{2. As a result, we find that ÿ αPΓq exp pV 8 ´sqp´G q px q q `dpγ ´1z γ , αx q qq ( -e ps´V8´dqGqpxqq , s ą V 8 `d{2.

It is easy to check that the L 2 norm of this is finite whenever s ě V 8 `d{2 ` . The proof of the lemma is complete when we observe that the L 2 norm decreases when s increases.

Parametrix for the Eisenstein functions

In the case of constant curvature, the universal cover Ă M is the real hyperbolic space H d`1 . On it, there is the Poisson kernel P px, p, sq that associates a function f of p P B 8 Ă M » S d on the boundary with a function on H d`1 , upxq such that p´∆ ´spd ´sqqupxq " 0 upxq " ż S d P px, p, sqf ppqdp.

We say that u corresponds to the superposition of outgoing stationary plane waves at frequency s, with weight f ppq in the direction p. When the curvature is variable, one cannot build such a kernel anymore, because the geometry of the space Ă M near the boundary is quite singular. In other words, the metric structure on the boundary is not differentiable, only Hölder. Hence, no satisfactory theory of distributions is available. However, in the special case of parabolic points that correspond to hyperbolic cusps, the fact that small enough horoballs have constant curvature enables us to construct an approximate Poisson kernel for p P Λ par .

Taking the half space model for H d`1 , the Poisson kernel for the point p " 8 is P " y s , so one can rewrite formula (4) as

E i ps, xq " ÿ rγsPΓpzΓ P pγx, p, sq.
This is exactly the type of expression we are looking for. In the first subsection, we introduce some notations. In the second we recall some facts on Jacobi fields that we will need. Then we build the approximate Poisson kernel, and later, we prove that summing over p P Λ i par gives a good approximation of E i .

2.1. Some more notations. Fix some Z i and let p P Λ i par be a parabolic point. We denote by ϕ p t the flow on Ă M generated by ∇G p . It is conjugated to the geodesic flow on W u0 ppq by the projection π : Thanks to the rigid description in the cusps, we have G p ď ´log a i ô we are above cusp Z i and G p " ´log y i , for all p P Λ i par .

T ˚Ă M Ñ Ă M .
In that case, we can compute ∆G p " d, and it makes sense to define a twisted Jacobian:

(27) Jp pxq :" lim tÑ`8 b Jac ϕ p ´tpxqe td " b Jac ϕ p ´tpxqe td těGppxq`log a i , for p P Λ i par .
This Jp is constant equal to 1 in the horoball Bpp, a i q. It is useful to define (28) b i :" infty ą 0, Bpp, yq has constant curvatureu.

We have b i ď a i , and J equals 1 on Bpp, b i q. We also let (29) F p pxq :" log Jp pxq.

Recall the curvature of M is pinched between ´k2 max ď ´1 ď ´k2 min ă 0. Then by Rauch's comparison theorem, [4, Theorem 1.28], (30)

dp1 ´kmax q ď 2F p pG p `log b i q `ď dp1 ´kmin q. What is more, by B.1, ∇ n F p is bounded for n ě 1, because ∇G p is in C 8 p Ă M q.
On the other hand, the Unstable Jacobian F su is the Hölder function on SM defined by (31)

F su px, vq :" ´d dt |t"0 det " pdϕ t q |E u px,vq ‰ ă 0.
The fact that it is Hölder is a consequence of the Hölder regularity of E u -see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theorem 7.1]. In what follows, we will be interested by the potential

(32) V 0 " 1 2 F su `d 2 .
We let δ g " δpΓ, V 0 q. This is the relevant abscissa of convergence of theorem 2 in the introduction, as we will see.

Unstable Jacobi fields.

We want to relate V 0 and F p . We have to make a digression, and recall some facts on Jacobi fields. Take a geodesic xptq, and a Jacobi field J along xptq, orthogonal to x 1 ptq. By parallel transport, one can reduce J to some function of time valued in T xp0q M . If one also uses parallel transport for the curvature tensor, we get the equation (33) J 2 ptq `KptqJptq " 0.

If xptq lives in constant curvature ´1, K is the constant matrix ´1. If Jp0q " J 1 p0q, then

Jptq " e t Jp0q, and conversely, if Jp0q " ´J1 p0q, Jptq " e ´tJ p0q.

For v P T x M , denote by v K the space of vectors in T x M orthogonal to v. Recall that H and V are the horizontal and vertical subspaces introduced in remark 1. Then we can identify

T v SM » pRv ' v K q ' v K . In this identification, the first term Rv ' v K is H v . The second term v K is V v X T v SM .
In this notation, Rv is the direction of the geodesic flow, and v its vector. This identification is consistent with Jacobi fields in the sense that if dϕ t .pl, v 1 , v 2 q " plptq, v 1 ptq, v 2 ptqq, then lptq " l for all t, v 1 ptq is a Jacobi field orthogonal to vptq " x 1 ptq, and v 2 ptq is its covariant derivative (also orthogonal to vptq).

An unstable Jacobi field J u ptq along xptq is a d ˆd matrix-valued solution of (33) along xptq that is invertible for all time, and that goes to 0 as t Ñ ´8 -it just gathers a basis of solutions. Similarly, one can define the stable Jacobi fields. Such fields always exist; they never vanish, nor does their covariant derivative -see [START_REF] Ruggiero | Dynamics and global geometry of manifolds without conjugate points, volume 12 of Ensaios Matemáticos[END_REF]. Given a geodesic xptq, we denote by J u s ptq the unstable Jacobi field that equals 1 for t " s -. Actually, the fields t Þ Ñ J u s ps`tq are all equal and only depend on v " pxpsq, x 1 psqq P SM . We will write it t Þ Ñ J u v ptq. Identifying with T SM , we find that vectors in E u take the form pJ u ptqw, J u1 ptqwq, whence we deduce that (34)

E u v " tpw, J u v 1 p0qwq|w K vu.
The matrix J u v 1 p0q only depends on v, we denote it by U v . Similarly, we define S v for the stable Jacobi fields. They satisfy the Ricatti equation (along a geodesic vptq):

U 1 `U2 `K " 0.
They take values in symmetric matrices (with respect to the metric), which is equivalent to saying that the stable and unstable directions are Lagrangians. Given a geodesic curve xptq, J u ptq and J s ptq two Jacobi fields along it, we can write U " ppJ u q ´1q T ppJ u q 1 q T , and find that (35) d dt pJ u q T pU ´SqJ s ( " 0. This is a Wronskian identity. We can also compute

(36) det dϕ t|E u pvq " det J u v ptq g f f e det ´1 `U2 ϕtpvq det p1 `U2 v q .
We have a map i u : w P Hpvq Þ Ñ pw, U v wq P E u pvq from the horizontal subspace to the unstable one. If one considers the metric ds 2 u obtained on E u by restriction of the Sasaki metric in T SM , this gives a structure of Euclidean bundle to E u over SM .

Lemma 2.1. The matrix 1 `U2 v is the matrix of the metric pi u q ˚ds 2 u on H. This is bounded uniformly on SM .

Proof. This metric is always ě 1 -here, 1 refers to the metric on H, i.e, the metric on T M . The only way it can blow up would be that for a sequence of v, ṽ K v, U v ṽ Ñ 8. If v 8 was a point of accumulation of v in Ă M , that implies that E u and H are not transverse at v 8 . That is not possible since there are no conjugate points in strictly negative curvature. We deduce that πv P M has to escape in a cusp.

However, in the cusp, the curvature K is constant with value ´1. Hence, unstable Jacobi fields in the cusp write as Ae t `Be ´t, where A and B are constant matrices along the orbit. Then U v " 1 `Ope ´tq as the point v travels along a trajectory ϕ t that remains in a cusp. In particular, pi u q ˚ds 2 u " 2.1 `Op1{yq for points of height y in a cusp.

In this context, from the definition, we find that for x P Ă M , (37) J2 p pxq " e td det J u px,∇Gppxqq p´tq, for t ě G p pxq `log a i .

As a consequence, Lemma 2.2. For x P Ă M , and t P R,

ż ϕ p t pxq x V 0 " F p pϕ p t pxqq ´Fp pxq `Op1q.
What is more, V 0 is an admissible potential.

Proof. The first part of the lemma comes directly from equations (37) and (36), and the observation just afterward.

To prove the second part, it suffices to prove that F su is an admissible potential. Consider a point v P T SM so that ϕ t pvq remains in a cusp for times t P r0, T s. Taking the Jacobi fields starting from v along its orbit, for t P r0, T s, we find (38)

U ϕtv " pAe t ´Be ´tqpAe t `Be ´tq ´1 " 1 `Ope ´tq, and

F su " ´d ds |s"0 # det J u ϕtpvq psq d det 1 `Ope ´tq det 1 `Ope ´t´s q + " ´d `Ope ´tq.
The last thing we have to check is that F su is reversible. However, ı ˚F su is the strong Stable Jacobian F ss (39)

F ss " d dt |t"0 log det dϕ t|E s px,vq .

From equation (36), and the Wronskian identity (35), we find that (40) det dϕ t|E s px,vq det dϕ t|E u px,vq " detpU v ´Sv q detpU ϕtpvq ´Sϕtpvq q

g f f e det ´1 `U2 ϕtpvq ¯´1 `S2 ϕtpvq det p1 `U2 v q p1 `S2 v q
.

Since the function (41) a detp1 `U2 qp1 `S2 q detpU ´Sq is well defined on S Ă M , Hölder continuous, and bounded, F su is reversible.

2.3. On the universal cover. In this section, we fix p P Λ par , and we omit the dependency on p; it shall be restored afterwards. We use notations introduced in section 2.1. We will use the WKB Ansatz to find our approximate Poisson kernel. Consider a formal series of functions on Ă M , f pxq "

ÿ ně0 s ´nf n pxq,
with s P C and f 0 " 1, and compute

p´∆ ´spd ´sqqre ´sG Jf s " e ´sG " s ´2∇G.∇p Jf q `Jf ∆G ´Jf d ¯´∆p Jf q ı ,
where we have used that G satisfies the eikonal equation |∇G| 2 " 1. If we expand the formal series, we find that this expression (formally) vanishes if for all n ą 0, 2 J∇G.∇f n " ∆p Jf n´1 q.

Indeed, 2∇G.∇ J " Jpd ´∆Gq.

We can rewrite those equations in terms of F " log J :

(42) 2∇G.∇f n " Qf n´1 where Qf pxq " ∆f `2∇F.∇f `p|∇F | 2 `∆F qf.

These are transport equations, with solutions :

(43) f n " 1 2 ż 0 ´8pQf n´1 q ˝ϕp τ dτ
Remark that on tG ď ´log bu, from the definition (28) F vanishes, and so does Qf 0 . Hence all f n 's but f 0 vanish, and the formula above makes sense. We prove :

Lemma 2.3. There are constants C n,N ą 0 for n ą 1, N P N, such that for all τ P R }f n } C N ptGpďτ ´log buq ď C n,N τ n .
Proof. We use lemma B.1 again, and proceed by induction. The result is obvious for n " 0. Now assume it holds for some n ě 0. Taking g 0 " f n , g 1 " f n`1 , " ´log b, the lemma enables us to conclude directly if we can prove that

}Qf n } C k pGďτ ´log bq ď C n,k τ n .
But this is a simple consequence of the induction hypothesis and the fact that ∇F P C 8 p Ă M q.

All the functions defined above depended on choosing a parabolic point p, and now we make it appear in the notations: 44) and ( 45) in Ă M are already invariant under the action of Γ p , so to define a function on M , we only have to sum over Γ p zΓ. As in section 1.3, take a cusp Z i , a parabolic point p P Λ i par . For x P M , let x p P Ă M be a point minimizing G p amongst the lifts of x. Then Lemma 2.4. For ą 0, and N P N, there is a constant C N, ą 0 such that for all x P M , and all s ą δpΓ, V 0 q ` ,

(44) f N p psq :" N ÿ n"0 s ´nf n,
› › › › › › ÿ rγsPΓpzΓ,rγs‰r0s
|P N pγx p , p, sq|

› › › › › › L 2 x pM q ă C N, a s i .
Further, with the same condition on s, the remainder satisfies

› › › › › › ÿ rγsPΓpzΓ |R N pγx p , p, sq| › › › › › › L 2 x pM q ă C N, b s i .
with b i as defined in (28).

Proof. First, we give a proof for N " 0. write

ÿ rγsPΓpzΓ,rγs‰r0s |P 0 pγx p , p, sq| " ÿ rγsPΓpzΓ,rγs‰r0s exp # s log a i `Fp pγx p q ´ż γxp π a i p pγxpq s + .
Recall ´Fp pπ a i p pγx p qq " 0. By lemma 2.2, losing constants not depending on s, the RHS is comparable with

a s i ÿ rγsPΓpzΓ,rγs‰r0s exp ˜ż γxp π a i p pγxpq V 0 ´ s ¸.
Lemma 1.5 states that the term in the right part of the product is bounded uniformly in L 2 norm. Now, we deal with the higher order of approximation. Let n ą 0 and consider the sum

ÿ rγsPΓpzΓ pe ´sGp Jp f n,p q ˝γ.
By lemma 2.3, this is bounded term by term by

C k ÿ rγsPΓpzΓ pe ´sGp Jp ppG p `log b i q `qn q ˝γ. Inserting 1 " b s i b ´s i , this is (47) C k b s i ÿ rγsPΓqzΓ,Gqě´log b i B k s pe ´spGq`log b i q Jq q ˝γ. Let L b i 0 :" ÿ rγsPΓqzΓ,Gqě´log b i pe ´spGq`log b i q Jq q ˝γ.
By the argument above, for s ą δpΓ, V 0 q ` , this sum converges and the value is bounded in L 2 norm by some function of s. What is more, since all the exponents are nonpositive, this a decreasing function of s P R. We deduce that when ą 0, there is C ą 0 such that for x P M , s ą δpΓ, V 0 q ` , we have

}L b i 0 } L 2 ď C . Consider L " ř a k λ s k a Dirichlet series, with a k P R `, λ k ě 1, converging for s ą s 0 . Then, if s ´ ą s 0 , we find |L 1 psq| ď Lp s ´ q sup n | log λ n |λ ´ n .
Since L b i 0 has this Dirichlet series structure in the s variable, it implies that for some constants C ,k ą 0,

}B k s L b i 0 psq} L 2 ď C ,k , x P M, s ą δpΓ, V 0 q ` . Observe that C ,k may depend on b i . Hence › › › › › › ÿ rγsPΓpzΓ |pe ´sGp Jp f n,p q ˝γ| › › › › › › L 2 pM q ď C ,n b s i , x P M, s ą δpΓ, V 0 q ` .
Moreover, this also holds if we replace f n,p by Q p f N,p , and this observation concludes the proof.

Now, we can state and prove our first theorem. Recall E i ps, xq is the Eisenstein function incoming from cusp Z i , defined in (2) and (3).

Theorem 3 (Parametrix for the Eisenstein functions). For N P N, let Z i be some cusp, and p P Λ i par a representing point. For s ą δpΓ, V 0 q, let P N px, p, sq be defined by (44) and E i,N ps, xq :"

ÿ rγsPΓpzΓ P N pγx p , p, sq.
this function is defined on Ă M , but invariant by Γ, so it descends to M ; it does not depend on the choice of p P Λ i par . Then, uniformly in s when s stays away from δpΓ, V 0 q, and s R rd{2, ds,

(48) }B m s pE i ´Ei,N q} H k pM q " O ´sk´N b s i ¯ Proof.
From lemma 2.4, we deduce that E i,N is well defined; it does not depend on p thanks to the equivariance relation (46). Additionally, for a cutoff χ that equals 1 sufficiently high in Z i and vanishes outside of Z i , E i,N ´χy s i is in L 2 , uniformly bounded in sets t s ą δpΓ, V 0 q` u. The sum ÿ rγsPΓpzΓ R N pγ¨, p, sq converges normally on compact sets, and in L 2 pM q also, so we find

E i ´Ei,N " s ´N p´∆ ´spd ´sqq ´1 ÿ rγsPΓpzΓ R N pγ¨, p, sq
Since B m s p´∆ ´spd ´sqq ´1 is bounded on H n pM q with norm Op1q when s stays in sets t s ą d{2 ` , s R rd{2, dsu, it suffices to prove that when s ą δpΓ, V 0 q ` ,

› › › › › › B m s ÿ rγsPΓpzΓ R N pγ¨, p, sq › › › › › › H k pM q " Ops k b s i q
Actually, since the sum has a Dirichlet series structure, we see that this is true for all m ě 0 as long as it is true for m " 0. From the bounds in lemma 2.3, and the bounds on ∇G p P C 8 , we see that for

x 1 P Ă M , }∇ k R N p¨, p, sq}px 1 q ď Ce ´sGppx 1 q Jp ppG p `log b i q `qN
We conclude the proof using the arguments of the proof of lemma 2.4 again -from equation (47) and below.

Remark 4. We have given estimates for the convergence in H k , k ě 0. However, the sum also converges normally in C k topology on compact sets.

Parametrix for the scattering matrix

Let us recall that the zero-Fourier mode of E i at cusp Z j is

y s δ ij `φij psqy d´s .
This formula is valid a priori for y ě a j . However, if we integrate E i along a projected horosphere of height b i ď y ď a i , we still obtain the same expression, even though the projected horosphere may have self-intersection -recall they are the projection in M of horospheres in Ă M . This is true because following those projected horospheres, we do not leave an open set of constant curvature ´1 -see (28) -and we can apply a unique continuation argument.

The smaller the b i 's are, the better the remainder is. In constant curvature, there is no remainder -the remainder in (48) goes to zero as N Ñ 8, with fixed s. Observe that the parameters b i are only related to the support of the variations of the curvature, and not to their size.

3.1.

Reformulating the problem. In this section, let p P Λ i par , q P Λ j par . Recall from (10) that when i ‰ j, SG ij » Γ p zΓ{Γ q , and SG ii » Γ p zpΓ ´Γp q{Γ p . We prove Lemma 3.1. When s ą δpΓ, V 0 q, integrating on horospheres in Ă M , (49)

φ ij psq " b s j ÿ rγsPSG ij ż Hpγq,b j q P N p¨, p, sqdµpθq `Ops 1{2´N b s i b s j q.
The constants are uniform in sets t s ą δpΓ, V 0 q ` u, ą 0. What is more, this expansion can be differentiated, differentiating the remainder.

Proof. First, for Hj pb j q » Γ q zHpq, b j q the projected horosphere from cusp Z j at height b j , integrating in M , we claim (50)

φ ij psq " ´b2s´d j δ ij `żxP Hj pb j q b s´d j E i,N dθ d `Ops 1{2´N b s i b s j q,
where the remainder can be differentiated. Considering zero Fourier modes of E i in the cylinder Γ q z Ă M we see that the formula holds if we replace E i,N by E i , without remainder. Hence, we only have to estimate

ˇˇˇB m s ˆbs´d j ż E i ´Ei,N dθ d ˙ˇˇˇ.
The surface measure obtained by disintegrating the riemannian volume on ty " b j u is dµpθq " dθ d {b d j . According to the Sobolev trace theorem, the L 2 norm of a restriction to Hj pb j qit is an immersed hypersurface -is controlled by the H 1{2 norm on M . Using this and theorem 3, we obtain that the remainder is bounded up to a constant by b s´d{2 j sup k"0,...m }B k s pE i ´Ei,N q} H 1{2 pM q " Ops 1{2´N b s i b s j q.

Now, to go from (50) to (49), we just have to use the description given by theorem 3. Indeed, consider a cube C q in Hpq, b j q that is a fundamental domain for the action of Γ q » Z d . Then We want to give an asymptotic expansion for each term in (49). To be able to use stationary phase, the next section is devoted to giving sufficient geometric bounds on the position of Hpγq, b j q with respect to W u0 ppq.

3.2. Preparation lemmas and main asymptotics. Take p P Λ i par , q P Λ j par , q ‰ p. We will work in Hpq, b j q Ă Ă M . As an embedded Riemannian submanifold, it is isometric to R d , and the isometry is given by the θ coordinate; we use this to measure distances on Hpq, b j q unless mentioned otherwise. We are considering

P N p¨, p, sqdµpθq " ż R d e ´spGp´log b j q Jp f N p ps, θqdθ d
At all the points where ∇G p is not orthogonal to the horosphere, this integral is nonstationary as |s| Ñ `8. There is only one point in Hpq, b j q where ∇G p is orthogonal to Hpq, b j q; it is exactly the point where the geodesic c p,q from p to q intersects Hpq, b j q for the first time -the second is q. It is reasonable to expect that the behaviour of the approximate Poisson kernel around this point will determine the asymptotics of the integral.

It is indeed the case, as we will show that G p , Jp and f N p satisfy appropriate symbol estimates on Hpq, b j q. If a P C 8 pR d q, we say that a is a symbol of order n P Z if for all k P N, (52) |xxy ´n`k B k apxq| L 8 pR d q ă 8 where xxy 2 " 1 `x2 .

For a geodesic coming from p intersecting Hpq, b j q, call the first intersection the point of entry and the second one the exit point -they may be the same. We can assume that the point of entry of c p,q is 0 in the θ coordinate -denoted 0 θ . It is also the point where G p attains its minimum on Hpq, b j q, and this is T pc p,q q `log b j , with T pc p,q q the sojourn time as defined in [START_REF] Guillopé | Scattering asymptotics for Riemann surfaces[END_REF]. We start with a lemma : Lemma 3.2. Let I Ă Hpq, b j q be the set of entry points. It is compact. Its radius is bounded independently from p, q, for the distance on Hpq, b j q given by Hpq, b j q » R d .

Proof. First, we prove it is compact. By continuity, I contains a small neighbourhood U of 0 θ . Let U 1 be the set of exit points of geodesics whose entry point is in U . It is a neighbourhood of q in Hpq, b j q -by definition of the visual topology on M . The complement of U 1 has to contain I, and it is compact, so I is relatively compact. The claim follows because I is closed. Now, since G p is C 8 , and the horosphere is smooth, the boundary of I only contains points where ∇G p is tangent to Hpq, b j q. Take such a point θ, and consider the triangle in Ă M with vertices p, 0 θ , and θ. Let α be the angle at 0 θ , and L the distance between 0 θ and θ in Ă M . Since the horoball Bpq, b j q is convex, α ą π{2. From the remark on obtuse triangles (16) for rp, θ, 0 θ s, if l " G p pθq ´Gp p0 θ q, we have L ´l " Op1q.

We want to prove that L is bounded independently from p, q. To see that, consider p 1 the other endpoint of the geodesic through p and θ, and θ 1 its projection on Hpq, b j q. Let l 1 " G p 1 pθq´G p 1 pθ 1 q and let L 1 be the distance in Ă M between θ and θ 1 . By the same argument,

q p p 1 θ 0 θ θ 1 Hpq, b j q L 1 L l l 1 c p,q L 2 α Figure 3
. Geometric presentation of the proof.

L 1 " l 1 `Op1q. Moreover, l `l1 is the distance between Hpp, G p p0 θ qq and Hpp 1 , G p 1 pθ 1 qq. So that if L 2 is the distance between 0 θ P Hpp, G p p0 θ qq and θ 1 P Hpp 1 , G p 1 pθ 1 qq, L 2 ą l `l1 . However, by the triangle inequality, L 2 ď L `L1 . We deduce that L 2 " L `L1 `Op1q. By theorem 4.9 and 4.6 of [START_REF] Heintze | Geometry of horospheres[END_REF], L 2 is bounded by constants depending only on the pinching of M , and so is L.

Our second lemma is the following: Lemma 3.3. In I, G p is convex. That is, on I, if α is the angle ∇G p makes with the horosphere Hpq, b j q, we have d 2 θ G p ě psin α `sin 2 αK min q{b 2 j .

Proof. Let θ P I. Take u P R d with |u| " 1 and θ 1 " θ ` u for ą 0 small. We apply Topogonov's theorem to the triangle with vertices θ, θ 1 and p, where p is a point that will tend to p. Let αp q be the angle at θ. Then by comparison, we have coshpK min dpθ 1 , pqq ě coshpK min dpθ, pqq coshpK min dpθ, θ 1 qq ´sinhpK min dpθ, pqq sinhpK min dpθ, θ 1 qq cos αp q.

As we let p Ñ p, coshpK min dpθ 1 , pqq coshpK min dpθ, pqq and coshpK min dpθ 1 , pqq sinhpK min dpθ, pqq Ñ exppK min pG p pθ 1 q ´Gp pθqqq and K min pG p pθ 1 q ´Gp pθqq ě log " coshpK min dpθ, θ 1 qq ´sinhpK min dpθ, θ 1 qq cos αp q ‰ . Now, we let go to 0. We have G p pθ 1 q ´Gp pθq " ∇G p pθq.u ` 2 d 2 θ G p pθq.u b2 {2 `op 2 q. Additionally, ∇G p pθq.u " ´cos αp0q{b j and dpθ, θ 1 q " {b j by ( 14), so the RHS becomes log

« 1 ´Kmin cos αp q b j `K2 min 2 2b 2 j `op 2 q ff " ´ K min b j cos αp q` 2 K 2 min 2b 2 j p1´cos αp0q 2 `op1qq.
And we deduce

d 2 θ G p pθq.u b2 ě K min b 2 j sin 2 αp0q `2 pcos αq 1 p0q b j .
Now, computing in the hyperbolic space, we find that the angle β at which the geodesic between θ and θ 1 intersects Hpq, b j q satisfies β " {2b j . If α is the angle between ∇G p pθq and Hpq, b j q, we find cos α " cos α cos >pu, B θ G p q `sin α sin β and pcos αq 1 p0q " sin α 2b j .

Finally, we can observe that αp0q ě α and

d 2 θ G p pθq ě sin α `Kmin sin 2 α b 2 j .
We have to separate the integral (51) into two parts, let us explain how we choose them. The stable and the unstable distributions of the flow ϕ t are always transverse. Since they are continuous, the angle between them is uniformly bounded by below by some α ą 0 in any given compact set of M -we say that they are uniformly transverse. Lifting this to Ă M , the angle is uniformly bounded by below on sets that project to compact sets in M . In particular, this is true on the union of the Hpq, b j q for q P Λ j par . Now, we can also consider the geodesic flow in the hyperbolic space of dimension d `1. It has stable and unstable distributions. The cusp Z j is the quotient of an open set of that space by a group of automorphisms, so that those stable and unstable distributions project down to subbundles E s hyp , E u hyp of T S ˚Zi , invariant by the geodesic flow. We call them the ˚-stable and ˚-unstable manifolds of Z i . The angle between them is constant equal to π{2, and they are smooth -even analytic.

By definition of the stable and ˚-stable manifolds, if the trajectory of a point ξ P S ˚Zi stays in Z i for all times positive, its stable and ˚-stable manifolds coincide. This is the case of p0, dG p q. As a consequence, there is a small neighbourhood V 1 of 0 in the θ plane, whose size can be taken independent from p, q, where the unstable manifold of pθ, dG p pθqq and its ˚-stable manifolds are uniformly transverse.

By the arguments in the proof of lemma 3.2, we see that the set of points of Hpq, b j q that are not exit points of geodesics whose entry points are in V 1 , is a compact set. Denote it by V 2 . Its radius is also bounded independently from p and q. Now, let χ P C 8 c pR d q take value 1 on V 2 , and introduce 1 " χpθq `p1 ´χpθqq in (51), to separate it into pIq and pIIq.

From theorem 7.7.5 (p.220) in Hörmander [START_REF] Hörmander | The analysis of linear partial differential operators. I. Classics in Mathematics[END_REF] -the stationnary phase -we deduce Lemma 3.4. For each p, q, there are coefficients A n pp, qq so that for every N ě 1

pIq " ´π s ¯d{2 exp # ż cp,q
V 0 ´sT pc p,q q + « ÿ nďN ´1 A n pp, qq s n `1 s N O `p1 `pT pc p,g q `log b j q `qN ˘ff .

We have A 0 pp, qq " 1, and A n pp, qq " Op1 `pT pc p,q q `log b j q `qn . What is more, the A n do not depend on N for n ď N ´1.

Proof. From lemma 2.3, we already know that the functions under the integral are smooth, uniformly in p, q. From lemma 3.3, we know that the phase is non-degenerate at 0. To apply Hörmander's theorem, we need to check that the derivative |B θ G p | is uniformly bounded from below in V 2 ´V1 .

The general observation is that |B θ G p | remains bounded from below if ∇G p stays away from being the outer normal to Hpq, b j q.

Start with θ P V 2 an exit point. Consider c the geodesic along ∇G p , going out at θ. The closer to the outer normal ∇G p pθq is, the longer the time c had to spend in the horoball. Since the set of entry points is uniformly compact, this implies that points where ∇G p is almost vertical -i.e along B y -have to be far from 0. But V 2 is uniformly bounded, so |B θ G p | is bounded by below on the exit points in V 2 .

For the entry points that are not exit points, we use the uniform convexity from lemma 3.3. By that lemma, B θ G p is a local diffeomorphism in I 1 " tθ, ∇G q pθq.∇G p pθq ă 0u. On the boundary of

I 1 , |B θ G p | " b 2 j .
By continuity, there is ą 0 such that |B θ G p pθq| ă b 2 j {2 implies dpθ, BI 1 q ą . As a consequence, from the local inversion theorem, there is 0

ă 1 ă and 2 ą 0 such that if |B θ G p pθq| ă b 2 j {2, BpB θ G p pθq, 2 q Ă B θ G p pBpθ, 1 qq.
Then, when |B θ G p pθq| ă 2 , θ has to be at most at distance 1 from a zero of B θ G p , i.e 0 θ .

The constants 1 and 2 can be estimated independently from p and q.

Now, we have an expansion (53)

pIq " ˆ2π s ˙d{2 exp t´sT pc p,q qu ˆC0 `1 s C 1 `. . .

˙.

We have (54) C 0 " Jp p0 θ q pdet d 2 θ G p p0 θ qq 1{2 . We factor out C 0 from the sum, and define A n pp, qq " C n {C 0 . From lemma 2.3 and the fact that ∇F p is C 8 p Ă M q, it is quite straightforward to prove the estimates on the A n 's. Now, we have to compute C 0 . From [12, proposition 3.1], we see that

(55) ∇ 2 G p pxq " U x,∇Gppxq ,
where U was introduced after equation (34).

We use a simple trick. Along the geodesic c p,q , ∇pG p `Gq q " 0, so that the Hessian d 2 pG p `Gq q is well defined along c p,q . This implies that d 2 θ pG p `Gq q " ∇ 2 pG p `Gq q. But on the horosphere Hpq, b j q, G q is constant, and we find d 2 θ G p p0 θ q " ∇ 2 G p p0 θ q `∇2 G q p0 θ q. The unstable Jacobi fields along c q,p are the stable Jacobi fields along c p,q so U x,∇Gqpxq " ´Sx,∇Gppxq . Hence, (56) d 2 θ G p p0 θ q " U x,∇Gppxq ´Sx,∇Gppxq . In constant curvature, this is the constant matrix 2 ˆ1.

Now, we give another expression for J2

p p0 θ q. Let x P Ă M . Consider J u the unstable Jacobi field along pϕ p t pxqq, that equals p1{yq1 for a point along the orbit that is close enough to p -where y is the height coordinate exp ´Gp . Then (57) J2 p pxq " e d.Gppxq det J u pxq .

When x " 0 θ , for t ą 0, we can write J u pϕ p t p0 θ qq " Ae t `Be ´t. We can also define J s the stable Jacobi field along ϕ p t p0 θ q that equals 1 at 0 θ . From the equation ( 35), we find that (58) W :" J u ptq T pUptq ´SptqqJ s ptq is constant.

Hence

C 2 0 " e d.Gpp0 θ q det W det J s p0q.
The limit for t Ñ `8 gives W " 2A. Whence (59)

C 2 0 " e d.Gpp0 θ q 2 d det A .
On the other hand, exp

"ż q p ´2V 0 * " lim tÑ`8
det ´dϕ t|E u pvq ¯e´td for v P rp, qs sufficiently close to p.

" det ! Ae ´Gpp0θq
) from formulae (36) and (38).

We conclude that (60) C 0 " 2 ´d{2 exp # ż cp,q V 0 + .

3.3.

Estimating the remainder terms. Now, we consider (61)

pIIq :" ż R d e ´spGp´log b j q Jp f N p ps, θqp1 ´χpθqqdθ " 1 s k ż R d e ´sGp L k q p Jp f N p ps, θqp1 ´χpθqqqdθ where L p f " div " B θ Gp }B θ Gp} 2 f
ı . This holds for any k P N; if we get symbolic estimates on the integrand, we will find that pIIq " Ops ´8qpIq. Our next step is to study the growth of G p as θ Ñ 8.

Lemma 3.5. The function

B θ Gp }B θ Gp} is a symbol of order 1 in θ in R d ´V2 , bounded indepen- dently from p, q.
Additionally, expp´spG p ´log b j qq is integrable, and for any ą 0, there is a constant C ą 0 such that whenever s ą d{2 ` , ż

R d ´V2
e ´spGp´log b j q dθ ď C e ´ sT pcp,qq .

Proof. With each θ P R d ´V2 we associate the point of entry θ 0 -θ 0 P V 1 by definition. Consider the geodesic coming from p, entering the horoball at θ 0 and going out at θ. Then, if is the angle of this geodesic with the normal to the horosphere,

|θ ´θ0 | " 2b tan .
but, we also have that

|B θ G p | " sin b . Hence 1 |B θ G p | " 1 2 |θ ´θ0 | d 1 `4b 2 |θ ´θ0 | 2 ,
and

B θ G p }B θ G p } 2 " 1 2 pθ ´θ0 q d 1 `4b 2 pθ ´θ0 q 2 .
It suffices to see that θ Þ Ñ θ 0 is a symbol of order ´1 to obtain the first part of the lemma. But θ Þ Ñ θ 0 is a one-to-one map, and by means of an inversion in the hyperbolic space, we see that θ 0 Ñ θ{}θ} 2 is a smooth map. Its derivatives are controlled by the angle that ∇G p makes with the vertical (and its derivatives). As a consequence θ Ñ θ 0 is a symbol of order ´1, uniformly in p and q.

Then, using formula ( 14), as θ Ñ 8, G p " 2 log |θ| 2b `Gp pθ 0 q " 2 log |θ| 2b `T pc p,q q `log b j `op1q

where the remainder is a symbol of order ´1. We deduce that exp ´sG p is integrable ( s ą d{2), and ż

R d
dθ d e ´ spGp´log b j q ď Ce ´ sT pcp,qq .

Lemma 3.6. On the horosphere Hpq, b j q, Jp is a symbol of order 0 with respect to θ. In symbol norm, it is Op Jp p0 θ qq.

Proof. We use Jacobi fields and notations introduced in section 2.2. We also use the uniform transversality condition in the definition of V 1 -see page 26. In the neighbourhood V 1 , since E u is transverse to the constant curvature stable direction, there exists a smooth matrix Apθq such that E u pθq " tX ``X ´| X `P E s hyp , X ´P E u hyp , X `" ApθqX ´u. When we transcribe this to Jacobi field coordinates,

E u pθq " tpp1 `Aqξ K , p1 ´Aqξ K q|ξ K PKu.
Remark here that p1 `Aq is invertible ; indeed, if it were not, there would be an unstable Jacobi field on M that would vanish at some point. But a Jacobi field that vanishes at some point cannot go to 0 as t Ñ ´8, it has to grow. Now, we consider a trajectory entering the horoball at θ 0 . We use the coordinates pθ 0 , tq to refer to ϕ p t pθ 0 q. We use parallel transport to work with vectors in T S ˚M|V 1 . We have E u pθ 0 , tq " tX ``X ´| X `P E s hyp , X ´P E u hyp , X `" e ´2t Apθ 0 qX ´u and in the horizontal-vertical coordinates E u pθ 0 , tq " tpp1 `e´2t Aqξ, p1 ´e´2t Aqξq|ξ K d{dtu.

Actually, for t P r0, T s, the jacobian Jac ϕ p t pθ 0 q is the determinant of Jp0q Þ Ñ Jptq where Jptq are the unstable Jacobi fields along the trajectory ϕ p t pθ 0 q. From the description with the matrix A above, we deduce that this is

e t.d det " 1 `e´2t Apθ 0 q 1 `Apθ 0 q  .
As a consequence, Jp pϕ p t pθ 0 qq Jp pθ 0 q "

d det " 1 `e´2t Apθ 0 q 1 `Apθ 0 q  .
Recall that t " 2 log |θ| when the trajectory reaches the horosphere again, and that θ Ñ θ 0 is a symbol of order ´1. We deduce that Jp pθq is a symbol of order 0. Lemma 3.7. For all n ě 0, in the region of the horoball corresponding to trajectories entering in V 1 , we can write f n,p pθ 0 , tq " fn,p pθ 0 , e ´2t q.

We have for all k ě 0, } fn,p } C k ď C n,k ppT pc p,q q `log b j q `qn with C n,k not depending on p nor on q.

Proof. We start by considering two functions a 1 and a 2 of θ 0 and e ´2t . Then e 2t ∇a 1 .∇a 2 and e 2t ∆a 1 are still smooth functions of θ 0 and e ´2t . Consider a trajectory xptq " pθ 0 , tq. We can take normal coordinates along this geodesic pt, x 1 q. We then only need to prove that e t Bθ 0 {Bx 1 |x 1 "0 is a smooth function of θ 0 and t. First, we observe that Bθ 0 {Bx 1 |x 1 "0,t"0 is only controlled by the angle between the geodesic and the horosphere Hpq, b j q, and this angle we have shown to be smooth. We only have to consider Bx 1 ptq{Bx 1 p0q |x 1 "0 , that is, the differential of the flow ϕ p t transversally to ∇G p . We have computed it in the previous proof; it is e t p1 `e´2t Apθ 0 qqp1 `Apθ 0 qq ´1. Now, we proceed by induction on n. First, f 0,p " 1 so it obviously satisfies the assumptions; it is also the case of F p " log Jp . Assume that the hypothesis has been verified for some n ě 0. Then by the above and (42), 2.3, e 2t Q p f n,p is a smooth function of θ 0 and e ´2t , with the same control as for f n,p , and

f n`1,p pθ 0 , tq " f n`1,p pθ 0 , 0q `1 2 ż t 0 Q p f n,p ds.
we can write the integral as ż t 0 e ´2s apθ 0 , e ´2s qds " "ż apθ 0 , ρqdρ

 1 e ´2t
, for some smooth function a. This ends the proof. Now, recall that e ´2t " |θ ´θ0 | ´4, so this proves that f n,p pθq is a symbol of order 0 as θ Ñ 8.

Putting lemmas 3.5, 3.6, 3.7 together, we deduce from equation ( 61) that for all N, k ą 0 and ą 0, there is a constant C N,k, ą 0 such that, when s ą d{2 ` , (62) |pIIq| ď C N,k e ´T pcp,qq s p1 `pT pc p,q q `log b j q `qN s ´k.

Main result.

With the notations of lemma 3.4, for c P π ij 1 pM q with endpoints p, q in Ă M , we define (63)

a n pcq :" exp

"ż c V 0 * A n pp, qq.
We also define

(64) T 0 ij :" inftT pcq, c P π ij 1 pM qu and T # ij " minp´log b i b j , T 0 ij q.
Putting together lemmas 3.1, 3.4, and equation (62), we get Theorem 4. For two cusps Z i and Z j not necessarily different, and for every N ą 0, when s ą δpΓ, V 0 q, the element of scattering matrix defined in (3)

(65) φ ij psq " ´π s ¯d{2 ÿ rcsPπ ij 1 pM q N ´1 ÿ n"0 1 s n a n pcq e sT pcq `Op1q s N e sT # ij .
We proceed to give a parametrix for ϕ. When taking the determinant of the scattering matrix φpsq, we use the Leibniz formula ϕpsq "

ÿ σ εpσq κ ź i"1 φ i,σpiq psq.
The sum is over the permutations σ of v1, κw, and εpσq is the signature of σ. The remainder will be bounded by terms of the form

s κ{2´N exp ! ´s´T # 1σp1q `¨¨¨`T # κσpκq ¯
) . This one corresponds to the error of approximation for the product φ 1σp1q psq . . . φ κσpκq psq where σ is a permutation of v1, κw. Hence, we define (66)

T # " min σ ÿ T # iσpiq . It corresponds to the slowest decreasing remainder term as s Ñ `8. Recall the definition in [START_REF] Heintze | Geometry of horospheres[END_REF]: the scattering cycles (SC) are numbers of the form T 1 `¨¨¨`T κ , where T i P ST iσpiq . We define T 0 to be the smallest scattering cycle. It corresponds to the slowest decreasing term in the parametrix. By definition, T # ď T 0 . Remark 5. There are two cases. When T # ă T 0 , the error is bigger than the main term in the parametrix, for s too big with respect to s. That occurs when the incoming plane waves from the cusps encounter variable curvature before they have travelled the shortest scattered geodesics.

When T # " T 0 , the error term is always smaller than the main term in the parametrix. This means that the variations of the curvature happen not too close to the cusps. It is in particular the case when the curvature is constant.

In any case, let λ # " exp T # . Also let λ 0 ă λ 1 ă ¨¨¨ă λ k ă . . . be the ordered elements of texp T , T P SCu. We can now state the conclusion of this section: Theorem 5. There exist real coefficients ta n k u k,ně0 such that if

L n :" ÿ kě0 a n k λ s k ,
all the L n 's converge in the half plane t s ą δpΓ, V 0 qu. In that half plane, for all N ě 0,

ϕpsq " s ´κd{2 « N ÿ n"0 s ´nL n psq `Op1q s N `1λ s # ff .

Dependence of the parametrix on the metric

This section is devoted to studying the regularity of the coefficients a n pcq with respect to the metric. We prove that they are continuous in the appropriate spaces in sections 4.1 and 4.2. Then, we prove an openness property in C 8 topology on metrics. While essential to the proof of theorem 1, this part is quite technical, and the impatient reader may skip directly to section 5.

4.1.

The marked Sojourn Spectrum. As announced in section 1.2, we emphasize the dependence of objects on the metric from now on. In particular, when we write rc g s P π ij 1 pM q, we mean that we take some class in π ij 1 pM q, and consider c g , the unique scattered geodesic for g in that class.

We denote by T g the application T g : rc g s P π ij 1 pM q Þ Ñ T pc g q. We also write a n pg, rcsq instead of just a n pcq.

In what follows, we are interested in the regularity and openness properties of ϕ. It is obtained as the determinant of φpsq. Since the determinant is a polynomial expression, it is certainly smooth and open with respect to φ. As a consequence, it suffices to study the regularity of each φ ij independently, and the openess properties of φpsq instead of ϕ. Definition 4.1. Let pg q PR be a family of C 8 cusp metrics on M , so that their curvature varies in a compact set independent from . Also assume that they coincide in some cusp neighbourhood ty ě y 0 u. Suppose additionally that g is C 2`k on R ˆM for k ě 0. Then we say that g is a C 2`k family of cusp metrics.

The following lemma is classical: Lemma 4.2. Assume that pg q is C 2`k family of cusp metrics. Then the geodesic flow

t; , x, ξ Þ Ñ ϕ g t px, ξq is C 1`k on R ˆR ˆS˚M for k ě 0.
This is the direct consequence of

Lemma 4.3. Let f be a C 1`k map from R ˆpU Ă R m q to R m
where k ě 0 and U is an open set. Then the flow Ψ : s, t, xpsq Þ Ñ xptq associated to

9 x " f pt, xq. is C 1`k on its maximal domain of definition in R ˆR ˆU .
This is a basic lemma in dynamical systems -for example see proposition 6.2 in [START_REF] Taylor | Partial differential equations I. Basic theory[END_REF]. Now, we can prove that both the marked set of scattered geodesics and the marked Sojourn Spectrum are continuous along a perturbation of the metric that is at least C 1 in the C 2 topology on metrics. Lemma 4.4. Let k ě 0 and let g be a C 2`k family of cusp metrics on M . Let c be a scattered geodesic for g " g 0 . Then there is a C 1`k family of curves c on M such that c is a scattered geodesic for g . In particular, this proves that g Þ Ñ c g (given a class in π ij 1 pM q) and g Þ Ñ T g are C 1`k in C 2`k topology on g.

Proof.

Let us assume that c enters M in Z i and escapes in Z j . We can assume that the variations of g always take place below y " y 0 . Let x 0 (resp. x 1 ) be the point where c intersects the projected horosphere H i (resp. H j ) at height y 0 in Z i (resp. Z j ), entering (resp. leaving) the compact part. For x P H i and close to 0, we can consider the following curve: c x, is the geodesic for g , that passes through x, and is directed by ´By at x. We have c " c x 0 ,0 . For px, q close enough to px 0 , 0q, c x, intersects the projected horosphere H j , for a time close to T pcq `2 log y 0 . We let x 1 px, q be that point of intersection, and vpx, q the vector c 1

x, at x 1 px, q. Now, by the lemma above, vpx, q is C 1`k , and by the Inverse Function Theorem, there is a unique Þ Ñ xp q, C 1`k , such that vpxp q, q is the vertical for all sufficiently close to 0, as soon as B x vp0, 0q is invertible. But the fact that it is invertible is a direct consequence of the non-degeneracy of the phase function shown in lemma 3.3.

Let

L ij n psq " ÿ rcsPπ ij 1 pM q
a n pg, rcsq e sTgprcsq Lemma 4.5. Let g be a C 2`k family of metrics, k ě 0. Then, as a formal series, L ij 0 depends on in a C k fashion. In particular, the series L 0 giving the first asymptotics for ϕ at high frequencies, also depends in a C k fashion on .

Proof. We only have to prove that a 0 pg , rcsq depends on in a C k fashion. Since V 0 is only a Hölder function, it is easier to study the regularity of a 0 with the original expression (54).

That is, we have to study d 2 θ G p p0 θ q and Jp p0 θ q.

First, consider Jp . It is a function of the jacobian of the flow ϕ p t along c. Since ϕ p t is just some restriction of ϕ t , ϕ p t is C 1`k on R ˆM . We also have that c is C 1`k , so that Jp is C k on .

For d 2 θ G p p0 θ q, consider that it is obtained as the first variation of ∇G p along the horocycle Hpq, b j q. But this means that d 2 θ G p p0 θ q is again obtained directly in terms of dϕ p t along c and this ends the proof.

Here already, we see that the first order behaviour of the scattering determinant at high frequency ( s bounded and s Ñ ˘8) depends continuously on g in C 2 topology. The next section is devoted to studying this regularity for other terms. In other words, the coefficient of π d{2 s ´d{2´n is (67

) n ÿ "0 1 !4 ∆ pσ n´ ˝Ψ. JacpΨqq
It is a well known fact that the Morse chart Ψ is not uniquely defined. However, from the computations above, the operators σ Þ Ñ ∆ j pσ ˝Ψ. Jac Ψqp0q do not depend on the choice of Ψ, but only on G. By writing the condition G ˝Ψ " x 2 , one can see that dΨp0q.dΨ T p0q " 2pd 2 Gp0qq ´1. This determines dΨp0q up to isometries of R d . The higher order derivatives of Ψ are undetermined, but one can see that they can be chosen recursively, so that d k Ψp0q only depends on the pk `1q-jet of G at 0.

Remark. One can check that the numbers ∆ j Jac Ψp0q are, up to universal constants, the Taylor coefficients in the expansion of the function volpG ď r 2 q. Now comes our lemma:

Lemma 4.6. Let g be a C 2`k family of cusp metrics on M . Then the coefficients a n pg, rcsq are C k´2n functions of , as soon as k ě 2n.

In the following proof, we fix two points p, q on the boundary. Most of the functions that appear depend on p and q, but to simplify notations, we omit that dependence. We do not fix n, but k will always be assumed to be greater or equal to 2n.

Proof. We apply the discussion above to a n pg, rcsq. The notations are coherent with section 3.2 and equation (51) if one set G " G p and σ " Jp f N p . We decompose a n , following equation (67). We find that a n depends on derivatives of J, f and G. We expand each summand in the decomposition, using the Leibniz rule. Then we gather the terms involving the highest order derivatives of the metric. They are (68)

1 n!4 n ´Jac Ψp0q∆ n θ Jp0q `Jp0q∆ n θ Jac Ψp0q ¯`J p0q Jac Ψp0q n ÿ "1 1 pn ´ q!4 n´ ∆ n´ θ pf ˝Ψq.
In the proof of 4.5, we saw that along a C 2`k perturbation, J is C k , so that ∆ n θ p J ˝Ψq is C k´2n . By the same argument, we find that the p2n `2q-jet of G at 0 θ is a C k´2n function of , so that ∆ n θ Jac Ψ is also C k´2n . Now, we deal with the f n 's. From the definition of Q in (42) and f n in (43), we can prove by induction that for n ě 1, (69)

f n " 1 2 n ż 0 ´8 dt n "ż 0 tn dt n´1 ¨¨¨ż 0 t 2 dt 1 Q t n´1 ¨¨¨Q t 1 Q 0 f 0  ˝ϕp tn
where Q t is defined by Qpf ˝ϕp t q " pQ t f q ˝ϕp t . Since F is essentially a jacobian of ϕ p t , it is C k on R ˆM along a C 2`k perturbation. From the formula (69), we deduce that f n is C k´2n along a C 2`k perturbation when k ě 2n, and this ends the proof.

4.3.

Openness in smooth topology. To find that the coefficients of the parametrix are open, we are going to adopt a different point of view from the previous section. We let a ´1pg, rcsq " T g prcsq. We aim to prove the following: Lemma 4.7. Let rc 1 s, . . . , rc N s be distinct elements of π i 1 j 1 1 pM q, . . . , π i N j N 1 pM q, and take indices n 1 , . . . , n N . Then the application a N : g Þ Ñ `pa ´1, a 0 , . . . , a n 1 qpg, rc 1 sq, . . . , pa ´1, a 0 , . . . , a n N qpg, rc N sq ˘P R

ř n i `2
is open in C 8 topology on g.

Proof.

First, observe that it suffices to prove that the differential of a N is surjective. Indeed, we can then use the inverse function theorem to prove the openness property.

For each class rc i s, we will compute the variation of pa 0 , . . . , a n i q along a well chosen smooth family of cusp metrics g . We will find that this variation is a linear form in a jet of the variation B g along c i . From the properties of this linear form, we will find that there are functions with arbitrary compact support on which it does not vanish. This will prove the lemma for N " 1.

For the case when N ą 1, observe that since the rc i s are distinct, the c i are also distinct. Then, it suffices to observe that we can take a finite number of small open sets U i such that U i X U j " H when i ‰ j, and U i X c i ‰ H. Then we can perturb in each open set independently, and in this way, we see that the differential of a N is surjective, and this ends the proof. Remark 6. There might seem to be a difficulty when the geodesic c has a self intersection, because at the point of intersection, we have less liberty on the perturbations we can make. However, we will always choose to perturb away from those intersection points.

As we have reduced the proof to the case N " 1, let rcs P π ij 1 pM q. First case, n " 0. In particular, if U is an open set that intersects c 0 , one can find a perturbation of the metric, supported in U , along which B T prcsq ‰ 0.

Proof. From the arguments above, we can construct a variation c of c 0 such that each γ is an unparametrized geodesic for g . We can assume that for t negative (resp. positive) enough, y i pc q " y i pc 0 q (resp. y j pc q " y j pc 0 q). Then in local coordinates B T prcsq " 1 2

ż " pB gq 0 pc 1 0 ptq, c 1 0 ptqq `2g 0 pc 1 0 ptq, B c 1 0 ptqq

‰ dt

In the RHS, the second term, we can interpret as the 1st order variation of the length of the curve c for g 0 . Since c 0 is a geodesic, this has to be zero Lemma 4.9. The logarithmic differential d g log a 0 pg, rcsq is non-degenerate on the set of symmetric 2-tensors h on M such that h and dh vanish at c. This proves the property for n " 0, because if h is such a 2-tensor, along the perturbation g ` h, the curve c is always a scattered geodesic of constant sojourn time, and d g a ´1pg, rcsq.h " 0.

Proof. since the curve c does not depend on the metric in our context, it is reasonable to use the method of variation of parameters. Let g " g ` h, and consider J u, " J u ` Ju `op q the unstable Jacobi field for g along c defined in page 28. We also use J s as defined in the same page. We can write Ju " J u r Aptq `Js r Bptq and we find the equations for r A and B:

J u r A 1 `Js r B 1 " 0 J 1 u r A 1 `J1 s r B 1 " ´pd g K.hqJ u .
Since Hpp, b i q » R d , we are now working in R d`1 . In the coordinates ς g , the flow has a very simple expression: ϕ p t px, sq " px, s `tq. The metric also:

(72) ς g g " gpx, t; dxq `dt We can find that gpx, 0q actually does not depend on x. We also have that G p px, sq " s´log b i . We will refer to horospheres ϕ p t pHpp, b i qq as slices. Idea of proof. If we perturb g by a symmetric 2-tensor h on the slices, we obtain a new metric gh on R d`1 . We can obtain a metric g 1 h on Ă M , pushing forward by ς g . If the support Ω of this perturbation g 1 h ´g is small enough so that γΩ X Ω " H for all γ ‰ 1, we can periodize the perturbation to obtain a metric on M , or equivalently, a metric g h invariant by Γ on Ă M .

The metric g h , seen in the chart ς g , does not have the nice decomposition (72) anymore. However, that decomposition still holds in the complement of ς ´1 g pΥq, where Υ :" Ť γRΓp γΩ. To apply our perturbative argument, we need to be able to choose Ω so that there is a neighbourhood Ω 1 of the geodesic c that we wanted to perturb with Ω 1 X Υ " H. The condition for Ω to be appropriate is that the projection Ă M Ñ M is injective on Ω, and that γΩ does intersect the lift rp, qs of c. For this, it suffices that Ω is not too close to the points I in rp, qs that project to self-intersection points of c. See figure 4.

There one more difficulty. The point of intersection of c with Hpq, b j q, 0 θ , is represented by p0, t 1 q with t 1 " T g pg, rcsq `log b i b j -see the paragraph after equation (52). It is possible that t 1 ă 0. In that case, the geodesic c only encounters constant curvature. To perturb the coefficients, we will need to create variable curvature along the geodesic. In particular, that will change the values of b i and b j .

To overcome this problem, we proceed in the following way. Instead of integrating along the projected horosphere at height b j in the cusp Z j , we integrate on the projected horosphere at height b ‹ j ě b j in the proof of theorem 4. We do it so that for all rcs P π ij 1 pM q, T pcq`log b ‹ i b ‹ j ą 0 (for all i, j. . . ). Since the marked sojourn time function is proper, only a finite number of scattered geodesics intervene here. All quantities that depended on b i , b j before will now receive a ‹ when we replace b i by b

‹ i . Hpp, b ‹ i q Hpq, b ‹ j q 0 ‹ θ t x Support of h Figure 5. A close up.
Coming back to perturbing coefficients, the point 0 ‹ θ is represented by p0, t ‹ 1 q with t ‹ 1 " T g pg, rcsq `log b ‹ i b ‹ j . If the perturbation h is compactly supported in t0 ă t ă t ‹ 1 u, the expression of g h , Hpq, b ‹ j q, ∆ ‹ θ , J, G p will not depend on h in the chart ς ‹ g , around 0 ‹ θ . In particular, a ´1 and a 0 are always constant along such perturbation. Now, we assume that the change in the slices is h where h is a 2-symmetric tensor such that h, dh, . . . , d 2n´1 h vanish along tx " 0u in the chart ς ‹ g , and h is supported for 0 ă T ă s ă T 1 ă t ‹ 1 . Let g :" g h .

Lemma 4.10. Under such a perturbation, a ´1, a 0 , . . . , a n´1 are constant.

Proof. As we saw in section 4.2, the coefficient a k is computed from the 2k ´2 jet of f , " 1, . . . , k, at 0 θ , and also the 2k jet of J and G. Those computations are done with ∆ θ , which in our chart ς g has a complicated expression. However, since we are not perturbing the metric around 0 θ , the coefficients of ∆ θ do not change under the perturbation. From equation (73), and the expression for G in this chart, we see that the contribution of J and G to a k will not change under perturbation (independently from the order of cancellation of h).

We are left to prove that the 2k ´2 jet of f at 0 θ does not change for 0 ď ď k ď n ´1. From formula (74), we see that the 2n ´1 jet of F along c will not change along the perturbation. From equation (69), we see that the m jet of f at 0 θ depends on the m `2 jet of F , and the m `2 ´1 jet of g -recall that the coefficients in the Laplacian ∆ depend on dg, and the coefficients in ∇ depend on g. Taking this for m " 2k ´2 and ď k ď n ´1, we find that the 2k ´2 jet of f can be computed with only the 2n ´2 jet of g at c, and this proves the lemma.

From the proof of the lemma above, we see that in a n , the only change will come from the change in the derivatives of order 2n ´2k of f k , and more precisely, the parts of these variations that come from the change in 2n derivatives of F , in the x direction. As a consequence, we can do all the forecoming computations as if the differential operators appearing had constant coefficients, and replace ∆ (resp. ∆ θ p¨˝Ψq ˝Ψ´1 ) by ∆ :" gij psqB i B j presp. ∆θ :" c ij B i B j q where the matrices pg ij qptq and pc ij q are symmetric, positive matrices. Recall the metric g has the expression g px,tq pdx, dtq " gx,t pdxq `dt 2

and pg ij qptq is the value of g´1 0,t , but this fact will not be used later. Recall that the operator Q t was defined by pQ t f q ˝ϕq t " Qpf ˝ϕq t q. We define ∆t in the same way. An easy computation shows that ∆t " ř g ij ps ´tqB i B j . Now, we use formula (68). We only keep the terms that vary under the perturbation g . This yields a n pg , rcsq ´an pg, rcsq " a 0 pg, rcsq

n ÿ "1 1 pn ´ q!4 n´ ∆n´ θ tpf q ´f u .
Next we use equation (69), leaving out the constant terms again. We find: a n pg , rcsq ´an pg, rcsq a 0 pg, rcsq "

n ÿ "1 1 pn ´ q!4 n 2 ´ ż S dt . . . dt 1 ∆n´ θ ∆t ´1 . . . ∆t 1 ∆ tF ´F u p0, t ‹ 1 `t q
Here, S is the simplex t´8 ă t ď t ´1 ď ¨¨¨ď t 1 ď 0u. Let t 0 " 0. Now, since 4d x F " Tr g ´1dg, each integrand in the above formula reduces to (75)

4 Tr $ & % g ´1pt ‹ 1 `t q ÿ tpim,jmqu n´1 ź m" c imjm ´1 ź m"0 gimjm pt ‹ 1 `t ´tm q ˜n ź m"1 B im B jm ¸hpt ‹ 1 `t q , .
-.

It is still not clear why such a formula would lead to a non-degenerate differential. However, let us assume that h has the following form in a neighbourhood of tx " 0u hpx, s, dxq " λpsqdx 2 ÿ |α|"2n u α x α `op|x| 2n q where u α " u α 1 . . . u α 2n , and likewise for x α . We take u a constant vector in R d , and λpsq a smooth function, supported in s0, t ‹ 1 r. Formula (75) becomes pcpu, uqq n´ Tr g ´1pt ‹ 1 `t q

# λpt ‹ 1 `t q ´1 ź m"0 gpt ‹ 1 `t ´tm qpu, uq + .
Observe that these are nonnegative numbers. From those computations, we see that a n pg , rcsq ´an pg, rcsq a 0 pg, rcsq "

ż t ‹ 1 0 λptqHptqdt
where Hptq is a function that does not vanish. This ends the proof for n ě 1.

Applications

We use simple Complex Analysis to locate zones without zeroes for ϕ. We also give some explicit examples corresponding to part (I) and (III) of the main theorem.

Complex Analysis and Dirichlet

Series. Let λ 0 ă λ 1 ă ¨¨¨ă λ k ă . . . be positive real numbers. For δ ą 0, we let Dpδ, λq be the set of Dirichlet series Lpsq whose abscissa of absolute convergence is ď δ, and

Lpsq " 8 ÿ k"0 a k λ s k .
We let D k pδ, λq be the set of L P Dpδ, λq such that a 0 " ¨¨¨" a k´1 " 0 and a k ‰ 0. For 0 ă λ # ď λ 0 , also consider Dpδ, λ, λ # q the set of holomorphic functions f on t s ą δu such that there are L n P Dpδ, λq with, for all n ě 0 f psq " L 0 psq `1 s L 1 psq `¨¨¨`1 s n L n psq `O ˜1 s n`1 λ s # ¸.

We will denote pa n k q the coefficients of L n . By taking notations coherent with the rest of the article, we have s κd{2 ϕ P Dpδ g , λ, λ # q. For δ 1 ą δ and C ą 0, let Ω δ 1 ,C :" s P C s ą δ 1 s ď C log | s| ( .

Lemma 5.1. Let f P Dpδ, λ, λ # q such that L 0 P D 0 pδ, λq. Then there is a δ 1 ą δ such that for any constant C ą 0, f has a finite number of zeroes in Ω δ 1 ,C .

In the special case where λ # " λ 0 , we can take δ 1 ą 0 such that f has no zeroes in When λ 0 " λ # , the condition on s is void. When λ 0 ą λ # , by taking N " C logpλ 0 {λ # q, we find that the zeroes of f in the region described in the lemma are actually in a bounded region of the plane. Since f is holomorphic, they have to be in finite number.

We give another lemma:

Lemma 5.2. Let f P Dpδ, λ, λ # q be such that L 0 P D 1 pδ, λq and L 1 P D 0 pδ, λq. Let There is a δ 1 ą δ such that for any constant C ą 0, there is a mapping W from the zeroes of f in Ω δ 1 ,C to the zeroes of f in Ω δ 1 ,C , that only misses a finite number of zeroes of f , and such that W psq ´s " Op|s| ´β q, for some β ą 0.

A picture gives a better idea of the content of this abstract lemma. Using usual asymptotic expansion techniques, we observe that the zeroes of f are asymptotically distributed along a vertical log line s " a log | s| `b, at intervals of lengths " 2πplogpλ 1 {λ 0 qq ´1. d{2 Figure 6. The zeroes of f . Remark 8. Instead of assuming a 0 0 " 0 and a 1 0 , a 0 1 ‰ 0, we could have assumed a finite number of explicit cancellations and non-cancellations. In that case, it is likely that one could prove a similar lemma, with f having zeroes close to a finite (arbitrary) number of log lines instead of only one line. However, this leads to tedious computations that we did not carry out entirely.

Proof. This is an application of Rouché's Theorem. We aim to give a good bound for |f ´f | on appropriate contours. To this end, we decompose |f ´f | ď ˇˇˇL 0 ´a0 For some δ 1 ą δ, and for s ą δ 1 ,this gives

ď C ˜1 λ s 2 `1 |s|λ s 1 `1 |s| 2 λ s 0 `1 |s| n`1 λ s # ( 77
)
where C ą 0 is a constant. We can always choose δ 1 big enough so that Observe that in the region Ω " tδ 1 ď s ď n log | s|plog λ 0 {λ # q ´1u, arg s " π{2 Òplog |s|{|s|q. In particular, there is a (possibly large) constant C ą 0 such that, on each line s " C `2πkplog λ 1 λ 0 q ´1, k P N, in Ω, 2| f | ě ˇˇˇa In a Op1q-sized neighbourhood of that curve, there is a α ą 0 such that the RHS in (77) is bounded by

Op|s| ´αq ˇˇˇa 0 1 λ s 1 ˇˇˇ
(5) Finally, we can adapt the proof of lemma 5.1 to show that whenever at least one L i is not the zero function, the conclusions of lemma 5.1 apply, if we replace "for all constant C ą 0" by "for some constant C ą 0". This proves part (IV).

Two examples.

In this last section, we construct explicit hyperbolic examples that satisfy the conditions given in lemma 5.3.

An example with one cusp.

Here, we construct a surface with one cusp, such that there are parts of the surface that are far from the cusp, in the appropriate sense. Topologically, we are looking at the most simple cusp surface: a punctured torus. It can be obtained explicitely by glueing two hyperbolic pentagons. Consider two copies of the pentagon in figure 7 -the Euclidean distance between sides A and B has to be strictly larger than 2 to build such a pentagon, because we require right angles. Glueing sides A Ø B 1 , B Ø A 1 , D Ø D 1 , C Ø E and C 1 Ø E 1 , we obtain a punctured torus.

The scattered geodesic c 0 with the smallest sojourn time corresponds to the sides AB 1 and BA 1 . Its sojourn time is 0, i.e T 0 " 0. However, the set U of points that are strictly below the line ty " 1u is non empty (and open). This is the example in 4) in lemma 5.3 5.2.2. An example with 2 cusps. Now, we aim to construct an example of surface with two cusps pM, gq such that L 0 P D 1 . We consider a two-punctured torus. As in the previous example, we will glue pentagons. Only this time we glue 4 identical pentagons a, b, c, d, and they will not be symmetrical -see figure 8.

The cusp corresponding to pentagons a and b will be called cusp Z 1 , and the other one, corresponding to pentagons c and d will be cusp Z 2 . We obtain a surface pM, g q with two cusps, depending on the hyperbolic length . To ensure the normalization condition that the volume of a projected horosphere at height y is y ´1, we have to take (78) y 0 " 1 2 1 ?

2 `?1 `e´2 .

We number the geodesics c ij n from i to j (i " 1, 2, and j " 1, 2) by their sojourn time with T pc ij 0 q ď T pc ij 1 q ď . . . . Since a geodesic coming from a cusp has to go under the y " y 0 line to exit a pentagon, we see that actually, c 11 0 and c 12 0 are the geodesics designated in figure 8. Actually, we also get that :

´2 log y 0 " T pc 11 0 q " T pc 12 0 q ă T pc ij 1 q, i, j " 1, 2.

This proves that a 0 0 " 0. Now, to obtain that L 0 P D 1 , we need to show that the second shortest sojourn time is 2 ´2 log y 0 , and that it is simple. That is to say, c 12 1 really is the curve drawn in figure 8, and the only other curves with sojourn time ď 2 ´2 log y 0 are c 11 0 and c 12 0 . In order to prove this, draw a line at height y 0 e ´2 . A scattered geodesic coming from cusp Z 1 can be lifted to H 2 as a curve coming from 8 in the pentagon a, that stays in the same pentagon as y ď y 0 e ´2 . When is small enough, there are only 3 geodesics that satisfy such a property, and they are drawn on figure 8.

The T i ps, tq's are tensors with a particular structure. Either they are of order 1 and T i ps, tqpXq " X, or they are of higher order and T i ps, tq " ż t s pϕ G u q ˚Ri " pϕ G u q ˚Ti,1 pu, tq, . . . , pϕ G u q ˚Ti,k i pu, tq ‰ du where R i is a bounded tensor with all derivatives bounded, and the T i,j have the same structure. Observe that for X P T N , }pϕ G t q ˚X } ď C}X} when t ą 0, and }pϕ G t q ˚X } ď Ce ´λt }X} when X K V . By induction, we deduce that for t ą 0, }∇ X 1 ,...,Xn pg 0 ˝ϕG ´tq}pxq ď C n e ´λnt }g 0 } C n pGďGpxqq . We just have to integrate this for t P r0, `8r, and the exponential decay ensures the convergence.

2 Figure 1 .

 21 Figure 1. The resonances: 4 cases in theorem 1.

Lemma 1 . 1 .

 11 Since M has finite volume, ΛpΓq " B 8 Ă M , and the parabolic points are dense in B 8 Ă M .

żxPP

  Hj pb j q b s´d j E i,N dθ d " N p¨, p, sqdθ d It suffices to observe now that ż Cp b s´d i P N p¨, p, sqdθ d " b 2s´d i .

Lemma 4 . 8 . 2 ż

 482 Let g be a C 8 family of cusp metrics. Then B T prcsq "1 pB gq 0 pc 1 0 ptq, c 1 0 ptqqdt.

Figure 4 .

 4 Figure 4. Global situation.

There is a δ 1 ą δ such that whenever s ą δ 1 , 1 N

 11 Take N ą 0. Then for |s| big enough -say |s| ą C N -and for s ą δ 1 ,1 |s| |L 1 psq| `¨¨¨`1 |s| N ´1 |L N ´1psq| ď # q log | s| `Op1q and |s| ą C N ùñ ˇˇˇˇC

  for s " δ 1 and | s| big enough. Then, on the vertical line s " δ 1 , for | s| big enough,|f ´f | ă | f |.Now, on the line s " n log | s|plog λ 0 {λ # q ´1, the 3 first terms of the RHS of equation (77) are very small in comparison to f . We can check that the last one is Op1{sq| f | to see that on that curve also, |f ´f | ă | f |.Now, we observe that| f | " ˇˇˇa

  Since on Ω, we also have for | s| large enough, on each of those horizontal lines, |f ´f | ă | f |. Now, the zeroes of f are located on the curve

1 Figure 7 .

 17 Figure 7. symmetric pentagon with an ideal vertex.

Figure 8 .

 8 Figure 8. A tiling of the hyperbolic plane with pentagons.

  p and P N p¨, p, sq :" e ´sGp Jp f N p psq. This is the approximate Poisson kernel. Then for all N ą 0, r´∆ ´spd ´sqse ´sGp Jp f N p psq " ´s´N e ´sGp Jp Q p f N,p psq.

	so we let
	(45)	R N p., p, sq :" ´e´sGp	Jp Q p f N,p psq
	This will be the remainder term. Now, as the last point in this section, observe the equiv-
	ariance relation
	(46)	P

N pγx, p, sq " P N px, γ ´1p, sq.

2.4. Poincaré series and convergence. The functions defined by (

  4.2. Higher order coefficients of the parametrix. Now, we are interested in the regularity of a n pg, rcsq for n ě 1. Before stating a lemma and its proof, let us start by a discussion of classical stationary phase in R d . Let σ be a smooth compactly supported function on R d . Then, as |s| Ñ `8 with s ą 0, " ∆ . . . ∆σ. If G is a non-degenerate phase function around 0, we find Ψ smooth around 0 such that G ˝Ψpxq " x 2 , by Morse theory. Then, if σ is still compactly supported but has an expansion σps, xq " σ 0 pxq `σ1 pxq{s `¨¨¨`σ n pxq{s n `. . . , we find

	R d ż	e ´sx 2 σpxqdx "	´π s	¯d{2	"	σp0q	`1 4s	∆σp0q `¨¨¨`1 !4 s	∆ σp0q `Ops ´ ´1q 
	where ∆ σ ż R d	e ´sGpxq σps, xqdx "	´π s	¯d{2	« n"0 8 ÿ	8 ÿ "0	s ´n´ 1 !4	∆ pσ n ˝Ψ. JacpΨqqp0q	ff	.

Acknowledgment We thank Colin Guillarmou and Stéphane Nonnenmacher for suggesting the idea that led to this article. We also thank Colin Guillarmou, Nalini Anantharaman and Maciev Zworski for their very helpful suggestions.

Recall from the arguments in page 28 that a 0 pg , rcsq 2 " lim tÑ`8 e d.Gpp0 θ q det J u J s " a 0 pg 0 , rcsq 2 1 det 1 ` r Ap`8q `op q .

where r Ap`8q is the limit of r A when t Ñ 8. Hence When the curvature of g is constant along c, one may observe that this gives a particularly simple expression -a quarter of the integral of the variation of the curvature along the geodesic. Now we prove that the differential h Þ Ñ d g K.h is surjective on the set of symmetric matrices along the geodesic c. We consider Fermi coordinates along c. That is, the coordinate chart in a neighbourhood of c given by

Remark 7. When c has self-intersection, this chart is not injective. However, we can assume that h vanishes around such points of intersection, and the computations below remain valid.

In those coordinates, g ´1 and dg vanish along the geodesic, which is c » tx 1 " 0u. We deduce that the Christoffel coefficients Γ k ij also vanish to second order on c. Now we recall from [START_REF] Kobayashi | Foundations of differential geometry[END_REF] two useful formulae.

We see that 2Kptq " ´d2 g 11 ptq, so that d g K.h " ´1{2d 2 h 11 , and this is certainly surjective onto the set of smooth functions along the geodesic valued in symmetric matrices. In particular, the RHS of (70) defines a non-degenerate linear functional on the set of compactly supported 2-symmetric tensors along c.

This ends the case n " 0.

General case, n ě 1. We introduce a special coordinate chart on Ă M :

The proof of the lemma will be complete if we can find some circles C n around the zeroes s n of f , whose radii r n shrink like a power of s n , but such that on C n ,

Actually, this kind of estimate is true on the circles C n centered at s n of radius r n , as long r n Ñ 0 with r n ąą |s n | ´α. Thus, we can take any β ă α.

Now,

Lemma 5.3. There are different situations.

(1) When there is only 1 cusp, we always have L 0 P D 0 pδ, λq.

(2) In general, the set of g P GpM q such that L 0 P D 0 pδ, λq is open and dense in C 2 topology.

(3) There are examples of hyperbolic cusp surfaces with L 0 P D 1 pδ, λq. (4) There are examples of hyperbolic cusp surfaces M that satisfy the following. First, L 0 P D 0 pδ, λq. Second, there is an open set U ĂĂ M such that for any cusps Z i , Z j , dpU, Z i q `dpU, Z j q ě T 0 ij `log a i `log a j . Then λ # " λ for all the metrics g P G U pM q (the metrics with variable curvature supported in U ). Proof. When κ " 1, ϕ " φ 11 . From lemma 3.4, we see that a 0 0 is a sum of positive terms over the set of scattered geodesics whose sojourn time is T 0 11 , hence it cannot vanish. In the general case, the openness property of lemma 4.7 shows that for an open and dense set of g P GpM q for the C 2 topology, the smallest element T 0 of the set of sojourn cycles is simple. That implies that a 0 0 ‰ 0. For the third part of the lemma, an example will be constructed in section 5.2.2.

For the last part, an example will be given in section 5.2.1. The conclusion λ # " λ 0 is a consequence of the discussion just before theorem 5

Proof of theorem 1. We can list the cases (1) Consider the hyperbolic surface described in lemma 5.3 [START_REF] Cheeger | Comparison theorems in Riemannian geometry[END_REF]. For such a surface, for all g P G U pM q, we have L 0 P D 0 pδ, λq, and λ # " λ 0 . We can apply the special case of lemma 5.1, to prove part (I). ( 2) For all manifolds with one cusp only, L 0 P D 0 pδ, λq so we can apply the general case of lemma 5.1. (3) When there is more than one cusp, case (2) of lemma 5.3 and lemma 5.1 lead to part (II) of theorem 1. (4) The example in case (3) of lemma 5.3 can be perturbed, preserving the condition L 0 P D 1 pδ, λq, and with L 1 P D 0 pδ, λq, according to lemma 4.7. We can then apply lemma 5.2 to prove part (III).

Appendix A. Regularity of Horospheres for some Hadamard manifolds

In this appendix, we recall some results on the regularity of stable and unstable foliations.

Lemma A.1. Let N be a simply connected manifold of dimension d `1, with sectio nal curvature ´|K max | ă K ă ´|K min | ă 0. Assume additionally that all the covariant derivatives of the curvature tensor R of N are bounded. For a point ξ P S ˚N , we define W s pξq as tξ 1 P S ˚M, dpπϕ t ξ 1 , πϕ t ξq Ñ 0 as t Ñ `8u, and similarly W u pξq. Those are C 8 submanifolds of T S ˚N , uniformly in ξ; they form a continuous foliation of T S ˚N , tangent respectively to E s and E u .

Proof. We just check that the proof of the compact case also works for us.

Let ξ P S ˚N . Take ν ą 0, t ą 0 and 0 ă ă t. For k ě 0, let ξ k " ϕ kt pξq. Using the exponential charts for the Sasaki metric on S ˚N , we can conjugate ϕ t to diffeomorphisms from T ξ k S ˚N to T ξ k`1 S ˚N that map 0 to 0. We still refer to those as ϕ t . Let

This is a Banach space when endowed with }pz k q} ν :" sup }z k }e νk .

On H ν , we can define Ψpz k q :" p0, ϕ t pz 0 q ´z1 , ϕ t pz 1 q ´z2 , . . . q. This is a C 8 function on H ν . We want to solve Ψ " 0 in H ν . As the stable manifold ξ should be a graph over E s pξq, for pz k q P H ν , we decompose pz k q " pz s 0 , rq. We need to show that B r Ψ is injective and surjective on a closed subspace, to use the implicit function theorem for Banach spaces. Let V " pv u0 0 , v 1 , . . . q P H ν where u0 refers to the weak unstable direction E u ' RX. We have

First, we prove this is injective. Assume B r Ψp0qV " 0. Then, we have d ξ ϕ t ¨vu0 0 " v 1 . Since the weak unstable direction is stable by the flow, v 1 P E u0 pξ 1 q. By induction, v k P E u0 pξ k q. However, V has to be in H ν , so that there is a constant C ą 0 such that for all k ą 0, }v u0 0 } " }pd ξ ϕ kt q ´1v k } ď Ce ´νk . This implies that v u0 0 " 0, and V " 0. Now, we prove that B r Ψ is surjective on the space of sequences whose first term vanishes. Let W " pw u0 0 , w 1 , . . . q P H ν . We decompose each w k " pw s k , w u0 k q, and we try to solve B r Ψ.V " W , with v i " pv s i , v u0 i q. If V is a solution, then, for all k ą 0,

That is why we let

This sum converges because }pdϕ kt q ´1| E u0 } is bounded independently from k, and we assumed w is in H ν . Then, we have to check that the equations

define a sequence in H ν . By the Anosov property, there are constants λ ą 0 and C ą 0 not depending on w, nor on ξ such that }dϕ kt w s l } ď e ´λkt }w s l }. So

Ce ´λpk´lqt e ´lν }w} ν `8 ÿ l"1 e ´pk`lqν }w} ν .

It suffices to choose ν ă λt, and we find that v P H ν .

By the Implicit Function Theorem, in a small enough neighbourhood of ξ, the strong stable manifold of ξ is a graph of a C 8 function from E s pξq Ñ E u0 pξq. Additionally, the derivatives of this function are controlled by the C k norms of Ψ. These norms are the C k norms of ϕ t . They can be bounded independently from ξ -recall Ψ depends on ξ -according to Lemma B.1 and Proposition C.1 of [START_REF] Bonthonneau | Long time quantum evolution of observables on cusp manifolds[END_REF]. Hence, the stable manifolds are uniformly smooth in the manifold.

Moreover, the tangent space of W s pξq at ξ has to be E s pξq, according to the dynamical definition of W s pξq. We deduce that the regularity of the lamination formed by the collection of W s pξq, ξ P S ˚M has the regularity of the splitting E s ' E u . Using the description as Green's fiber bundles for E s and E u , one can prove that they are Hölder, and the lamination is actually a foliation.

The case of unstable manifolds is similar.

Appendix B. Estimating the regularity of solutions for transport equations Lemma B.1. Let N be a riemannian manifold such that all the derivatives of its curvature are bounded. Let G be a C 8 function on N , such that }∇G} " 1, and }∇G} C n pN q is bounded for all n. Let ϕ G t be the flow generated by V " ∇G. Assume that ϕ G is expanding, that is, there is a λ ą 0 such that if u K V , }dϕ G t .u} ě Ce λt }u} for t ą 0. Let g 0 be a C 8 function on N , supported in G ě . Let

Then if L pτ q " supt|g 0 pxq|, Gpxq " τ u, for all n there is a constant C n ą 0 only depending on G such that

Proof. The first part of the statement is obvious. We concentrate on the second part. The basic idea is that when differentiating in the direction of the flow, one obtains g 0 , and when differentiating in other directions, one can use the contracting properties of ϕ G t in negative time. Let x P N , and X 1 , . . . , X n vectors at x. We want to evaluate ∇ X 1 ,...,Xn g 1 pxq. We can decompose the X i 's according to

By linearity, we can assume that either X i 9V or X i K V . Additionally, we assume }X i } " 1. By taking symmetric parts, and antisymmetric parts of ∇, we see that it suffices to evaluate ∇ X 1 ,...,Xn g 1 when the X i 's colinear to V are the last in the list. That corresponds to differentiating g 1 first along V . Now, there are two cases. First, assume that one of the X i 's is colinear to V . Then ∇ X 1 ,...,V g 1 " ∇ X 1 ,...,X n´1 g 0

We are left to consider the case when all the X i 's are orthogonal to V . For this, we use the proof from [3, appendix B]. From therein, we know that for t ą 0, ∇ X 1 ,...,Xn pg 0 ˝ϕG ´tq " W n t g 0 ppϕ G t q ˚X1 , . . . , pϕ G t q ˚Xn q Where -lemma B.2 -W n t g 0 is a sum of tensors of the form ∇ k T 1 p0,tq,...,T k p0,tq g 0 .