Colin Guillarmou

Frederic Naud
email: frederic.naud@univ-avignon.fr

F R Éd

Éric Naud

U M R Dma

EQUIDISTRIBUTION OF EISENSTEIN SERIES ON CONVEX CO-COMPACT HYPERBOLIC MANIFOLDS

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Since the early work of Schnirelman [START_REF] Schnirelman | Ergodic properties of eigenfunctions[END_REF], Colin de Verdière [START_REF] De Verdière | Ergodicité et fonctions propres du Laplacien[END_REF] and Zelditch [START_REF] Zelditch | Uniform distribution of eigenfunctions on compact hyperbolic surfaces[END_REF], it is a well known fact that on any compact Riemannian manifold X whose geodesic flow is ergodic, one can find a full density sequence λ j → +∞ of eigenvalues of the Laplacian ∆ X such that the corresponding normalized eigenfunctions ψ j are equidistributed i.e. for all f ∈ L 2 (X), we have

lim j→+∞ X f (z)|ψ j (z)| 2 dv(z) = X f (z)dv(z),
where dv is the normalized volume measure. Much less is known for non-compact manifolds and the closest analogs of the above celebrated theorem are due to Zelditch [START_REF] Zelditch | Mean lindelöf hypothesis and Equidistribution of Cusps forms and Eisenstein series[END_REF], Luo-Sarnak [START_REF] Luo | Quantum ergodicity of eigenfunctions on P SL2(Z)/H 2[END_REF] and Jakobson [START_REF] Jakobson | Quantum unique ergodicity for Eisenstein series on P SL2(Z)\P SL2(R)[END_REF] in the case of non-compact finite area hyperbolic surfaces. Let us recall their results. By H 2 we denote the usual hyperbolic plane. Let X = Γ\H 2 be a finite area surface where Γ is a non co-compact co-finite Fuchsian group. The non compact ends of X are cusps which can be viewed on the universal cover as fixed points c j in ∂H 2 of parabolic elements in Γ. The number of non-equivalent cusps is finite. The spectrum of the Laplacian ∆ X consists of two types. The discrete part which corresponds to L 2 (X)-eigenfunctions and may be not finite as in the arithmetic case, but is conjectured by Philips-Sarnak [START_REF] Phillips | On cusps forms for co-finite of PSL(2, R)[END_REF] to be finite in the generic case. The absolutely continuous part [1/4, +∞) which is parametrized (t ∈ R) by the finite set of Eisenstein series E X (1/2 + it; z, j) related to each cusp c j . The Eisenstein series E X (1/2 + it; z, j) are smooth (non L 2 (X)) eigenfunctions ∆ X E X (1/2 + it; z, j) = (1/4 + t 2 )E X (1/2 + it; z, j).

In the Poincaré upper half-plane model, Eisenstein series are usually defined through the series (absolutely convergent for Re(s) > 1)

E X (s; z, j) = γ∈Γ j \Γ Im(σ -1 j γz) s ,
which enjoys a meromorphic continuation to the whole complex plane and is analytic on the critical line {Re(s) = 1/2}. In the above formula, Γ j is the stabilizer of the cusp c j and Date: July 13, 2011.

1 σ j ∈ PSL 2 (R) is such that σ j (∞) = c j . In the case of the modular group PSL 2 (Z), there is only one Eisenstein series given by the formula (valid for Re(s) > 1) E X (s; x + iy) = 1 2

gcd(c,d)=1

y s |cz + d| 2s . For all t ∈ R, define the density µ t by X a(z)dµ t (z) := j X a(z)|E X (1/2 + it; z, j)| 2 dv(z),

where a ∈ C ∞ 0 (X). If we assume that we have only finitely many eigenvalues (believed to be the generic case) then Zelditch's equidistribution result 1 is as follows: for a ∈ C ∞ 0 (X),

1 s(T ) T -T X adµ t -∂ t s(t) X a dv dt → 0 as T → ∞
where s(t) is the scattering phase, this coefficient appearing as a sort of regularization of Eisenstein series due to the fact that the Weyl law involves the continuous spectrum.

On the other hand, for the modular surface X = PSL 2 (Z)\H 2 , Luo and Sarnak [START_REF] Luo | Quantum ergodicity of eigenfunctions on P SL2(Z)/H 2[END_REF] showed that as t → +∞,

X adµ t = 48 π log(t) X adv + o(log(t)),
which is a much stronger statement obtained via sharp estimates on certain L-functions. A microlocal version was later proved by Jakobson [START_REF] Jakobson | Quantum unique ergodicity for Eisenstein series on P SL2(Z)\P SL2(R)[END_REF].

In the present paper, we focus on the case of infinite volume hyperbolic manifolds without cusps, more precisely convex co-compacts quotients X = Γ\H n+1 of the hyperbolic space. Let us recall some basic notations. A discrete group of orientation preserving isometries of H n+1 is said to be convex co-compact if it admits a polygonal, finitely sided fundamental domain whose closure does not intersect the limit set of Γ. The limit set Λ Γ and the set of discontinuity Ω Γ are usually defined by

Λ Γ := Γ.o ∩ S n , Ω Γ := S n \ Λ Γ ,
where o ∈ H n+1 is a point in H n+1 . By a result of Patterson and Sullivan [START_REF] Patterson | The limit set of a Fuchsian group[END_REF][START_REF] Sullivan | The density at infinity of a discrete group of hyperbolic motions[END_REF], the Hausdorff dimension of Λ Γ δ Γ := dim Haus (Λ Γ ) is also the exponent of convergence of the Poincaré series, i.e. for all m, m ′ ∈ H n+1 and s > 0, [START_REF] Beardon | The Geometry of discrete groups[END_REF] γ∈Γ e -sd(γm,m ′ ) < ∞ ⇐⇒ s > δ Γ , where d(m, m ′ ) denotes the hyperbolic distance.

In that case the spectrum of ∆ X has been completely described by Lax-Philips and consists of the absolutely continuous spectrum [n 2 /4, +∞) and a (possibly empty) finite set of eigenvalues in (0, n 2 /4). Just like in the finite volume case, there is a way (see next § for full details) to define Eisenstein functions which parametrize the continuous spectrum. These functions replace in this setting the plane wave solutions e ±isz.ω of (∆-s 2 )u = 0 in R n z (here ω ∈ S n-1 ). Let g denote the usual hyperbolic metric. Convex co-compact quotients X = Γ\H n+1 fall in the class of conformally compact manifolds, which means that X compactifies smoothly to a 1 More details about the appropriate normalization can be found in his paper [START_REF] Zelditch | Mean lindelöf hypothesis and Equidistribution of Cusps forms and Eisenstein series[END_REF] where a microlocal statement is actually proved. manifold X with boundary ∂X such that there exists a smooth boundary defining function x with the property that x 2 g extends to a smooth metric on X and |dx| x 2 g = 1 on ∂X. The boundary ∂X is called the conformal boundary and can be identified with Γ\Ω Γ . The Eisenstein series 2 are smooth functions E X (s; m, ξ) where s ∈ C is the spectral parameter as above and (m, ξ) ∈ X ×∂X. They are generalized non-L 2 (X) eigenfunctions of the Laplacian. Using the ball model and an appropriate boundary defining function, Eisenstein series lift to

H n+1 to E X (s; m, ξ) = γ∈Γ 1 -|γm| 2 4|γm -ξ| 2 s ,
which is absolutely convergent for Re(s) > δ Γ . Our first result is the following.

Theorem 1. Let X = Γ\H n+1 be a convex co-compact quotient with δ Γ < n/2. Let a ∈ C ∞ 0 (X)
and let E X (s; •, ξ) be an Eisenstein series as above with a given point ξ ∈ ∂X at infinity. Then we have as t → +∞,

X a(m) E X ( n 2 + it; m, ξ) 2 dv(m) = X a(m)E X (n; m, ξ)dv(m) + O(t 2δ Γ -n ).
This result shows individual equidistribution of high energy Eisenstein series. The limit measure on X is given by the harmonic density E X (n; m, ξ) whose boundary limit is the Dirac mass at ξ ∈ ∂X.

A microlocal extension of this theorem is actually proved in §3. We first need to introduce some adequate notations. Fix any ξ ∈ ∂X. Let L Γ ξ defined by

L Γ ξ := ∪ γ∈Γ L γξ ⊂ S * X,
where L γξ are stable Lagrangian submanifolds of the unit cotangent bundle S * X: the Lagrangian manifold L γξ is defined to be the projection on Γ\S * H n+1 of

{(m, ν γξ (m)) ∈ S * H n+1 ; m ∈ H n+1 },
where ν γξ (m) is the unit (co)vector tangent to the geodesic starting at m and pointing toward γξ ∈ S n . The set L Γ ξ "fibers" over X, and the fiber over a point m ∈ X corresponds to the closure of the set of directions v ∈ S * X such that the geodesic starting at m with directions v converges to ξ ∈ ∂X as t → +∞. Since the closure of the orbit Γ.ξ satisfies Γ.ξ ⊃ Λ Γ , we actually have

L Γ ξ ⊃ T + := {(m, ν) ∈ S * X : g t (m, ν) remains bounded as t → +∞},
where g t : S * X → S * X is the geodesic flow. The set T + is often refered as the forward trapped set. The Hausdorff dimension of

L Γ ξ is n + δ Γ + 1 and satisfies n + 1 < δ Γ + n + 1 < 2n + 1 if Γ is non elementary.
Our phase-space statement is the following, we refer the reader to §3.2 for the required background on pseudo-differential operators.

2 They depend on the boundary defining function x; in the boundary variable ξ they should be thought of as section of a complex power of the conormal bundle to the boundary Theorem 2. Let A be a compactly supported 0-th order pseudodifferential operator with principal symbol 3

a ∈ C ∞ 0 (X, T * X), then as t → +∞ AE X ( n 2 + it; •, ξ), E X ( n 2 + it; •, ξ) L 2 (X) = S * X a dµ ξ + O(t -min(1,n-2δ Γ ) )
where µ ξ is a g t -invariant measure supported on the fractal subset L Γ ξ ⊂ S * X. Notice that the fractal behaviour of the semi-classical limit µ ξ can only be observed at the microlocal level. This is to our knowledge the first example in the mathematical literature of a high energy limit of eigenfunctions which is concentrated microlocally on a genuine fractal trapped set.

In the theoretical physics literature, we mention some work of Ishio-Keating [START_REF] Ishio | Semiclassical wave functions in chaotic scattering systems[END_REF] on somehow related questions for billards.

By averaging over the boundary with respect to the volume measure on ∂X induced by the defining function x, we obtain as

t → +∞ (2) ∂X X a(m) E X ( n 2 + it; m, ξ) 2 dv(m)dv ∂X (ξ) = vol(S n ) X a(m)dv(m) + O(t 2δ Γ -n ) and ∂X AE X ( n 2 + it; •, ξ), E X ( n 2 + it; •, ξ) L 2 (X) dv ∂X (ξ) = S * X a dµ + O(t -min(n-2δ Γ ,1) )
where µ denotes the Liouville measure. This is the perfect analog of the previously known results for the modular surface (actually with a remainder in our case).

It is actually possible to prove a much stronger asymptotic expansion for the averaged case. We show the following. Theorem 3. Let X = Γ\H n+1 be a convex co-compact hyperbolic manifold and assume that the limit set of Γ has Hausdorff dimension δ Γ < n/2. Let a ∈ C ∞ 0 (X) and C be the set of all closed geodesics. Then for all N ∈ N, one has the expansion as t → +∞

(3) ∂X X a(m)|E X ( n 2 + it; m, ξ)| 2 dv(m) dv ∂X (ξ) = vol(S n ) X a(m)dv(m) +L(t) γ∈C e -( n 2 +it)ℓ(γ) N k=0 H k (ℓ(γ)) | det(Id -e -ℓ(γ) R -1 γ )| t -k γ P k a dµ+ +L(-t) γ∈C e -( n 2 -it)ℓ(γ) N k=0 H k (ℓ(γ)) | det(Id -e -ℓ(γ) R -1 γ )| (-t) -k γ P k a dµ + O(t -N -n-1 ).
where H k (ℓ(γ)) is an explicit bounded function of ℓ(γ), P k are differential operators of order 2k with coefficient uniformly bounded in terms of γ ∈ C, dµ is the Riemannian measure induced on γ, R γ ∈ SO(n) is the holonomy along γ and

L(t) = t -(n-1)/2 2 n 2 +3-2it |Γ(it)| 2 Γ(it + 1 2 )Γ( n 2 -it) = O(t -n ) P 0 = 1, H 0 (ℓ(γ)) = 2 n 2 π n-1 2 e i n-1 4 π . 3 a ∈ C ∞ 0 (X, T * X) means that a ∈ C ∞ (T * X
) is compactly supported on the base but not necessarily in the fibers.

Notice that the asymptotic expansion is valid at all orders and, except for the first term, involves oscillating sums over closed geodesics. This formula is similar to what physicists call "the Gutzwiller formula" in the literature related to Quantum Chaos. In the mathematics literature it is of course quite similar to the Selberg trace formula (proved in [START_REF] Patterson | The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein[END_REF][START_REF] Bunke | Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group[END_REF][START_REF] Guillopé | The wave trace for Riemann surfaces[END_REF] in this setting, see also [START_REF] Guillarmou | Wave 0-trace and length spectrum on convex co-compact hyperbolic manifolds[END_REF]) and to trace formulae proved in [START_REF] De Verdière | Spectre du Laplacien et longueurs des géodésiques périodiques I[END_REF][START_REF] Chazarain | Formule de Poisson sur les variété riemanniennes[END_REF][START_REF] Duistermaat | The spectrum of positive elliptic operators and periodic geodesics[END_REF][START_REF] Brummelhuis | A trace formula for Schrödinger operators[END_REF][START_REF] Meinraken | Semi-classical principal symbols and Gutzwiller's trace formula[END_REF].

Let us describe the organization of the paper. The section §2 is devoted to the necessary analytic and geometric background. The proofs start in §3 and take advantage of the convergence of Poincaré series garanteed by the hypothesis δ Γ < n/2. We first prove the "configuration space" equidistribution result using a direct approach which is based on an "off-diagonal" cancelation phenomena due to some non-stationnary phase effect. To get a sharp remainder estimates in terms of δ, we need to analyze differently the elements in the group with large and small translation lengths. This technique is robust enough to extend to the pseudo-differential (phase space) equidistribution case. We treat the problem using semiclassical analysis, namely some standard results on the action of h-Pseudos on h-Lagrangian distributions. Theorem 2 is based on a completely different technique and make use of Stone's formula and the high frequency asymptotic for the resolvent on the universal covering H n+1 . The result follows from a repeated application of stationnary phase expansions (with parameter) and requires some precise estimate in terms of the group elements.

We believe that extensions of these results to non-constant negative curvatures setting can be obtained for manifolds with asymptotically hyperbolic (or asymptotically Euclidean) ends under a condition on the topological pressure of the flow on the trapped set like in [START_REF] Nonnenmacher | Quantum decay rates in chaotic scattering[END_REF]. This will be pursued elsewhere.

Generalities

2.1. Definition of Eisenstein series. Let X = Γ\H n+1 be a convex co-compact hyperbolic manifold. The group Γ is a discrete group of orientation preserving isometries of H n+1 with only hyperbolic transformations (i.e. fixing 2 points on the sphere S n = ∂H n+1 ). The limit set is Λ Γ := Γ.o ∩ S n where o ∈ H n+1 is the center in the unit ball model B o (1) ⊂ R n+1 of H n+1 . The discontinuity set is the complement Ω Γ := S n \ Λ Γ and Γ acts properly discontinously on [START_REF] Beardon | The Geometry of discrete groups[END_REF]. The manifold X compactifies smoothly to a manifold X with boundary ∂X by setting X = Γ\(H n+1 ∪ Ω Γ ). If g denotes the hyperbolic metric on X, there exists a smooth boundary defining function x such that x 2 g extends to a smooth metric on X and that |dx| x 2 g = 1 on ∂X. The Laplacian ∆ on (X, g) has absolutely continuous spectrum on [n 2 /4, ∞) and a possibly non-empty finite set of eigenvalues in (0, n 2 /4). By [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF][START_REF] Guillopé | Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity[END_REF], the resolvent of the Laplacian R X (s) := (∆s(ns)) -1 defined in the half plane Re(s) > n/2 admits a meromorphic continuation to the whole complex plane C, with poles of finite rank (i.e. the polar part at a pole is a finite rank operator), as a family of bounded operators

H n+1 ∪ Ω Γ ⊂ B o
R X (s) : C ∞ 0 (X) → C ∞ (X).
Using [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF][START_REF] Guillopé | Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity[END_REF], the resolvent integral kernel R X (s; m, m ′ ) near the boundary ∂X has an asympotic expansion given as follows: for any m ∈ X fixed

m ′ → R X (s; m, m ′ )x(m ′ ) -s ∈ C ∞ (X)
and similarly for m ′ ∈ X fixed and m ′ → ∂X. The Eisenstein series are functions in C ∞ (X × ∂X) defined by 4

(4)

E X (s; m, ξ) = 2π n 2 Γ(s -n 2 + 1) 2 -s Γ(s) [x(m ′ ) -s R X (s; m, m ′ )]| m ′ =ξ , ξ ∈ ∂X.
The terminology 'series' will become clear later since they can be obtained by series over the group. Notice that E X (s; •, ξ) depends a priori on the choice of the boundary defining function x however any other choice x ′ = ψ(ξ)x + O(x) with ψ ∈ C ∞ (∂X) would simply amount to multiply E X (s; m, ξ) by ψ(ξ) -s . Moreover, they are generalized eigenfunctions of ∆ X

(∆ X -s(n -s))E X (s; •, ξ) = 0, ∀ξ ∈ ∂X.
The functions E X (s, •, ξ) replace the plane wave functions z → e isz.ξ in R n (with ξ ∈ S n-1 ) which satisfy (∆ R ns 2 )e isz.ξ = 0, they provide a spectral representation of the continous spectrum for ∆ X .

Let us define the constant

(5) C(s) = π -n 2 2 -s Γ(s) Γ(s -n 2 )
.

The spectral measure 2tdΠ X (t) of the Laplacian ∆ X is related to Eisenstein series by

2tdΠ X (t; m, m ′ ) = is π (R X ( n 2 + it; m, m ′ ) -R X ( n 2 -it; m, m ′ )) = |C( n 2 + it)| 2 2π ∂X E X ( n 2 + it; m, ξ)E X ( n 2 -it; m ′ , ξ)dv ∂X (ξ) 2tdΠ X (t; m, m) = |C( n 2 + it)| 2 2π ∂X |E X ( n 2 + it; m, ξ)| 2 dv ∂X (ξ) (6)
with dv ∂X the Riemannian measure on ∂X induced by the metric h 0 = (x 2 g)| T ∂X . The first identity is Stone's formula, the second identity is an application of Green's formula proved in [10, Prop .1] or [START_REF] Perry | The Laplace operator on a hyperbolic manifold II, Eisenstein series and the scattering matrix[END_REF]. We notice that the density |E X ( n 2 + it; m, ξ)| 2 dv ∂X (ξ) is independent of the choice of boundary defining function x used for the definition (4), as long as h 0 is defined by x 2 g| T ∂X .

2.2.

Eisenstein functions on H n+1 . To fix the ideas, and since it will be used later, let us recall the case of H n+1 . The resolvent kernel is given (away from the diagonal m = m ′ ) by [START_REF] Guillopé | Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity[END_REF]Sec. 2]:

R H n+1 (s; m, m ′ ) = π -n 2 2 -2s-1 Γ(s) Γ(s -n 2 + 1) cosh -2s d(m, m ′ ) 2 × F s, s - n -1 2 , 2s -n + 1; cosh -2 d(m, m ′ ) 2 (7)
4 The normalization factor is not the usual one, but it is the natural one for our equidistribution purpose;

this corresponds somehow to the normalization given by the Weyl law in the co-compact setting.

where d(m, m ′ ) is the hyperbolic distance between m and m ′ , and for z

∈ C \ [1, +∞), F (a, b, c; z) = Γ(c) Γ(b)Γ(c -b) 1 0 t b-1 (1 -t) c-b-1 (1 -tz) -a dt,
is the usual hypergeometric function. In the ball model B o (1) := {m ∈ R n+1 ; |m| < 1} (with center denoted by o), the function [START_REF] Evans | Lectures on semi-classical analysis[END_REF] x(m)

:= 2e -d(m,o) = 2 1 -|m| 1 + |m|
is a boundary defining function for the closed ball B o (1). Moreover, using

(9) sinh 2 d(m, m ′ ) 2 = |m -m ′ | 2 (1 -|m| 2 )(1 -|m ′ | 2 )
,

we obtain for E 0 (s; m, ξ) := E H n+1 (s; m, ξ) defined with the boundary defining function x

(10) E 0 (s; m, ξ) = 1 -|m| 2 4|m -ξ| 2 s = e sφ ξ (m) with φ ξ (m) := log 1 -|m| 2 4|m -ξ| 2 .
Let s h := n/2 + i/h with h > 0 small, then as a function of m, E 0 (s h ; m, ξ) is a semi-classical Lagrangian distribution associated to the Lagrangian submanifold

L ξ := {(m, dφ ξ (m)) ∈ T * m H n+1 ; m ∈ H n+1 } ⊂ T * H n+1 .
If H ξ (m) is the horosphere tangent to ξ and containing m,

then φ ξ (m) = ±d(H ξ (m), H ξ (o)) is the hyperbolic distance between H ξ (m) and H ξ (o). Moreover, since φ Rξ (m) = φ ξ (R -1 m) for any R ∈ SO(n+1)
, the norm |dφ ξ (m)| g with respect to the hyperbolic metric g is independent of ξ. By going through the half-space model R + x × R n y of H n+1 , and choosing ξ = ∞, we have φ ∞ (x, y) = log(y) and thus |dφ ∞ (m)| g = 1 for all m ∈ H n+1 , which implies that [START_REF] Guillopé | Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity[END_REF] |dφ ξ (m)| g = 1 for all ξ ∈ S n , m ∈ H n+1 .

The level sets of φ ξ are the horospheres tangent to ξ, and the covector dφ ξ (m) is dual via the metric to the gradient ∇φ ξ (m) which is the unit vector tangent to the geodesic going from m to ξ. The manifold L ξ is the stable manifold associated to ξ.

Let us describe the action of isometries on Eisenstein funtions.

Lemma 4. Let γ ∈ Isom + (H n+1 ) be an isometry of H n+1 , then for all m ∈ H n+1 and ξ ∈ S n , E 0 (s; γm, γξ) = E 0 (s; m, ξ)|Dγ(ξ)| -s
where |Dγ(ξ)| is the Euclidean norm of Dγ at the point ξ ∈ S n .

Proof. We use the fact that the function ( 9) is invariant with respect to γ, i.e. for all m, m

′ |γm -γm ′ | (1 -|γm ′ | 2 )(1 -|γm ′ | 2 ) = |m -m ′ | 2 (1 -|m| 2 )(1 -|m ′ | 2 )
and thus multiplying this inequality by (1

-|m ′ | 2 ) and letting m ′ → ξ ∈ S n , we obtain E 0 (s; γm, γξ)/E 0 (s; m, ξ) = lim m ′ →ξ ( 1-|γm ′ | 2 1-|m ′ | 2 ) -s . This limit is given by |Dγ(ξ)| -s .
Notice that for any Möbius transformation γ, |Dγ(ξ)v| = |Dγ(ξ)| for any v of Euclidean norm 1. Let us finally give the formula for the integral of the hyperbolic Poisson kernel: for all m ∈ H n+1 , (12)

S n E 0 (n; m, ξ)dξ = 1/C(n) = vol(S n ).
where C(n) is defined in [START_REF] De Verdière | Spectre du Laplacien et longueurs des géodésiques périodiques I[END_REF] and dξ is the canonical measure on S n .

Eisenstein series on Γ\H

n+1 . Lemma 5. Let Γ be a convex co-compact group of isometries of H n+1 with δ Γ < n/2 and π Γ := H n+1 → Γ\H n+1 be the quotient map. Let ξ ∈ Ω Γ and m ∈ H n+1 , then for Re(s) > δ Γ the series E(s; m, ξ) := γ∈Γ E H n+1 (s; γm, ξ)
converges absolutely and is a Γ-automorphic function on H n+1 satisfying 8) and ( 10), together with the estimate |γm -ξ| > ǫ > 0 for some ǫ uniform in γ ∈ Γ, we deduce the existence of a constant C s uniform in γ so that (γm,o) . This shows that the series converges in Re(s) > δ Γ by [START_REF] Beardon | The Geometry of discrete groups[END_REF].

E X (s; π Γ (m), π Γ (ξ)) = E(s; m, ξ) if E X (s; •, •) is defined by (4) with a boundary defining function x so that, in a neighbourhood of ξ, π * Γ (x) = (1 -|m|)/(1 + |m|). Proof. By combining (
|E 0 (s; γ, ξ)| ≤ C s e -sd
The resolvent kernel for Re(s

) > n/2 on the quotient is given, for m, m ′ ∈ H n+1 so that π Γ (m) = π Γ (m ′ ), by the sum R X (s; π Γ (m), π Γ (m ′ )) = γ∈Γ R H n+1 (γm, m ′ ),
and this series converges absolutely in Re(s) > δ Γ by ( 7) and (1), moreover the convergence is uniform on compact sets in the variable m, m ′ (in any C k norms). The proof is standard and is left to the reader. By using ( 8) and ( 9), it is easily seen that after we multiply this last equation by x(π γ (m ′ )) -s , we take the limit m ′ → ξ and it is given by 2s

-n C(s) γ∈Γ lim m ′ →ξ (R H n+1 (s; γm, m ′ )(2 1-|m ′ | 1+|m ′ | ) -s ) = E(s; m, ξ).
This concludes the proof.

2.4. The Liouville measure. If (X, g) is an oriented n + 1 dimensional Riemannian manifold, recall that the Liouville form (or measure) on the unit sphere bundle SX is given by the 2n + 1 form µ := θ ∧ dθ n where θ is the contact form on SX

θ u (v) := g(u, π * v) if π : SX → X is the projection. Isometries γ of X act on SX by γ.(m, v) := (γm, dγ(m)v)
and clearly γ * θ = θ. In particular, this implies that γ * µ = µ. Since the group Isom + (H n+1 ) acts transitively on SH n+1 , the Liouville form is determined by its value at one point, for instance (m, v) = (o, e 1 ) where e 1 is a fixed unit vector of S o H n+1 .

Let us define the map ψ :

H n+1 × S n → SH n+1 by (13) ψ : (m, ξ) → (m, ∇φ ξ (m))
where ∇ denotes the gradient with respect to the hyperbolic metric g. This map is a diffeomorphism and it identifies the point (m, v) ∈ SH n+1 to the point (m, ξ) ∈ H n+1 × S n so that ξ is the end point of the geodesic starting at m with tangent vector v.

γ * ψ * µ = ψ * γ * µ = ψ * µ.
By [START_REF] Ishio | Semiclassical wave functions in chaotic scattering systems[END_REF], this implies that both ν and ψ * µ are preserved by the transitive action of Isom + (H n+1 ), and thus there exists a constant C n such that ψ * µ, and an easy computation at m = 0, ξ = e 1 gives (15)

ψ * µ = ν.
The same is obviously true with the Liouville measure on S * H n+1 by duality, provided ψ is defined with dφ ξ (m) instead of ∇φ ξ (m). On a quotient X = Γ\H n+1 , we also have

S * X = Γ\S * H n+1 ≃ Γ\(H n+1 × S n
) and the Liouville measure on the quotient lifts to the Liouville measure µ on H n+1 .

Proof of equidistribution of Eisenstein series

3.1. Equidistribution on X. In this section, we show equidistribution in space of the Eisenstein series.

Theorem 6. Let X = Γ\H n+1 be a convex co-compact quotient with δ Γ < n/2, and let x be a boundary defining function of X. Let a ∈ C ∞ 0 (X) and let E X (s; •, ξ) be the Eisenstein series defined by (4) using the defining function x and a point ξ ∈ ∂X at infinity. Then we have as

t → ∞, X a(m) E X ( n 2 + it; m, ξ) 2 dv(m) = X a(m)E X (n; m, ξ)dv(m) + O(t 2δ-n ).
Proof. Let us consider the ball model

B o (1) = {m ∈ R n+1 ; |m| < 1} of H n+1
, with boundary the sphere S n . Consider a fundamental domain F of Γ and let ξ ∈ Ω Γ ∩ F be a point in the domain of discontinuity. The boundary defining function x of X lifts to an automorphic function x on H n+1 and we define η(ξ) := 2 lim m→ξ (1 -|m|)/ x(m). By Lemma 5, the Eisenstein series on X = Γ\H n+1 at parameter s such that Re(s) > δ Γ is given by ( 16)

E X (s; π Γ (m), π Γ (ξ)) = γ∈Γ E 0 (s; γm, ξ)η(ξ) s = γ∈Γ E 0 (1; γm, ξ)η(ξ) s E 0 (1; m, ξ) := 1 -|m| 2 4|m -ξ| 2 ,
for m ∈ H n+1 . For simplicity, we shall identify π Γ (m) with m in the fundamental domain. We are interested in the pointwise norm (for t ∈ R)

E X ( n 2 + it; m, ξ) 2 = γ∈Γ E 0 (1; γm, ξ)η(ξ) n + η(ξ) n γ =γ ′ E 0 (1; γm, ξ)E 0 (1; γ ′ m, ξ) n 2 E 0 (1; γm, ξ) E 0 (1; γ ′ m, ξ) it E X ( n 2 + it; m, ξ) 2 =E X (n; m, ξ) + η(ξ) n γ =γ ′ E 0 (1; γm, ξ)E 0 (1; γ ′ m, ξ) n 2 E 0 (1; γm, ξ) E 0 (1; γ ′ m, ξ) it Let a ∈ C ∞ 0 (X) and consider the off-diagonal sum γ =γ ′ I γ,γ ′ (t) where (17) I γ,γ ′ (t) := X a(m) E 0 (1; γm, ξ)E 0 (1; γ ′ m, ξ) n 2 E 0 (1; γm, ξ) E 0 (1; γ ′ m, ξ) it dv(m)
To end the proof, we need to show the

Lemma 7. As t → ∞, the off-diagonal sum satisfies γ =γ ′ I γ,γ ′ (t) = O(t 2δ Γ -n )
Proof. The proof reduces to some non-stationary phase in t on a finite number of terms in each factor of Γ × Γ, essentially those γ which satisfy d(o, γo) ≤ log t. Let us first define the Buseman function

(18) B ξ (m, m ′ ) := log(1 -|m| 2 ) -log(|m -ξ| 2 ) -log(1 -|m ′ | 2 ) + log(|m ′ -ξ| 2 ) which satisfies the cocycle condition B ξ (m, m ′′ ) = B ξ (m, m ′ ) + B ξ (m ′ , m ′′ ) and for any γ ∈ Isom(H n+1 ) B γξ (γm, γm ′ ) = B ξ (m, m ′ ). One can rewrite (17) as X a(m) E 0 (1; γm, ξ)E 0 (1; γ ′ m, ξ) n 2 e itB ξ (γm,γ ′ m) dv(m) The phase B ξ (γm, γ ′ m) = B γ -1 ξ (m, γ -1 o) + B γ ′-1 ξ (γ ′ -1 o, m) has for gradient (with respect to Euclidean metric) in m ∇ m B ξ (γm, γ ′ m) = 2 m -γ -1 ξ |m -γ -1 ξ| 2 - m -γ ′ -1 ξ |m -γ ′ -1 ξ| 2
with Euclidean norm given by ( 19)

∇ m B ξ (γm, γ ′ m) = 2|γ -1 ξ -γ ′ -1 ξ| |m -γ ′ -1 ξ|.|m -γ -1 ξ| . Remark that ∇ m B ξ (γm, γ ′ m) = Q m (γ -1 ξ, γ -1 ξ ′ )/(|m-γ ′ -1 ξ| 2 .|m-γ -1 ξ| 2 ) for some Q m (u, v) polynomial in (u, v) and with Q m (u, u) = 0, moreover it is smooth in m in a compact set K ⊂ H n+1 , we then easily deduce that for each α ∈ N n+1 there is C α > 0 such that for all m ∈ K and α ∈ N n+1 (20) ∂ α m ∇ m B ξ (γm, γ ′ m) ≤ C α |γ -1 ξ -γ ′ -1 ξ|.
Let o be the center of the unit ball in R n+1 . For γ a Möbius transformation preserving the unit ball, we define a γ = γ -1 (∞) ∈ R n+1 the center of the isometric sphere S(a γ , r γ ) of γ,

where r γ = 1/ sinh( 1 2 d(o, γo)) is the radius of S(a γ , r γ ) (here d(•, •) is the hyperbolic distance). By §3.5 in Beardon [START_REF] Beardon | The Geometry of discrete groups[END_REF], we have for any ξ, ξ ′ ∈ B o (1) [START_REF] Patterson | On a lattice-point problem for hyperbolic space and related questions in spectral theory[END_REF] |γξ -

γξ ′ | = 1 sinh 2 ( 1 2 d(o, γo)) |ξ -ξ ′ | |ξ -a γ | |ξ ′ -a γ | .
Since |a γ | is bounded uniformly, we can set ξ ′ = γ -1 • γ ′ ξ and we deduce directly that |γξγ ′ ξ| 2 ≥ Ce -d(o,γo) and by symmetry in γ, γ ′ this gives

(22) |γξ -γ ′ ξ| ≥ C max(e -d(o,γo) , e -d(o,γ ′ o) )
uniformly in γ = γ ′ and ξ ∈ Ω Γ . By combining [START_REF] Nonnenmacher | Quantum decay rates in chaotic scattering[END_REF], ( 20)and ( 22), we deduce that for m in a compact set K of H n+1 one has

(23) ∂ α m ∇ m B ξ (γm, γ ′ m) |∇ m B ξ (γm, γ ′ m)| 2 ≤ C α min(e d(o,γo) , e d(o,γ ′ o) ).
For each γ, γ ′ ∈ Γ, we can therefore integrate by part to get

I γ,γ ′ (t) = 1 it X a(m) E 0 (1; γm, ξ)E 0 (1; γ ′ m, ξ) n 2 ∇ m (e itB ξ (γm,γ ′ m) ). ∇ m B ξ (γm, γ ′ m) |∇ m B ξ (γm, γ ′ m)| 2 dv(m) = 1 it X e itB ξ (γm,γ ′ m) ∇ * a(m) E 0 (1; γm, ξ)E 0 (1; γ ′ m, ξ) n 2 ∇ m B ξ (γm, γ ′ m) |∇ m B ξ (γm, γ ′ m)| 2 dv(m)
where ∇ * is the adjoint of ∇ with respect to the Riemannian measure dv(m) For ξ ∈ Ω Γ , there exists C > 0 depending only on K such that for all m ∈ K and γ ∈ Γ, one has

|γm -ξ| > C, (1 -|γm| 2 ) ≤ Ce -d(o,γo) , |Dγ(m)| ≤ Ce -d(o,γo) .
The last identity comes for instance from [START_REF] Patterson | On a lattice-point problem for hyperbolic space and related questions in spectral theory[END_REF]. Combining these estimates with [START_REF] Perry | The Laplace operator on a hyperbolic manifold II, Eisenstein series and the scattering matrix[END_REF], we obtain that there is C > 0 such that for all γ = γ ′ and m ∈ K

∇ * a(m) E 0 (1; γm, ξ)E 0 (1; γ ′ m, ξ) n 2 ∇ m B ξ (γm, γ ′ m) |∇ m B ξ (γm, γ ′ m)| 2 ≤ Ce -n 2 (d(o,γo)+d(o,γ ′ o))
min(e d(o,γo) , e d(o,γ ′ o) ). We thus have the estimate, which is uniform in

γ = γ ′ |I γ,γ ′ (t)| ≤ Ct -1 e -n 2 (d(o,γo)+d(o,γ ′ o)) min(e d(o,γo) , e d(o,γ ′ o)
). We can iterate this argument and get for j ∈ N

(24) |I γ,γ ′ (t)| ≤ Ct -j e -n 2 (d(o,γo)+d(o,γ ′ o)) min(e jd(o,γo) , e jd(o,γ ′ o) ). Let us decompose Γ into Γ = Γ ≤ ⊔ Γ > : {γ ∈ Γ; d(o, γ) ≤ log(t)} ⊔ {γ ∈ Γ; d(o, γ) > log(t)}
and then the double sum (γ,γ ′ )∈Γ×Γ\diag I γ,γ ′ (t) into 4 parts involving Γ α × Γ β \ diag for α, β ∈ {≤, >}. For the elements in Γ ≤ , we can now take advantage of the bound (24) with j taken larger than n/2δ Γ . Using the counting estimate (proved by Patterson [START_REF] Patterson | On a lattice-point problem for hyperbolic space and related questions in spectral theory[END_REF]) [START_REF] Schnirelman | Ergodic properties of eigenfunctions[END_REF] ♯{γ

∈ Γ; d(o, γo) ≤ T } = O(e δ Γ T )
it is easily seen that for each α, β ∈ {≤, >}, we have

(γ,γ ′ )∈Γα×Γ β \diag I γ,γ ′ (t) = O(t 2δ Γ -n ).
Let us for instance write the case α = β =≤, then by ( 24) and ( 25) we have for any j > n/2-δ Γ integer

d(o,γo)≤log t d(o,γ ′ o)≤log t,γ ′ =γ I γ,γ ′ (t) =O(t 2j ) d(o,γo)≤log t e (j-n 2 )d(o,γo) d(o,γ ′ o)≤log t e (j-n 2 )d(o,γ ′ o) =O(t n-2δ Γ )
and the other cases are similar.

This achieves the proof of the Theorem.

Notice that E X (n; m, ξ) is a harmonic function, more precisely this is the harmonic function on X with distributional boundary value the Dirac mass δ ξ at ξ.

As a corollary, we obtain Corollary 8. Let (X, g) = Γ\H n+1 be a convex co-compact quotient with δ Γ < n/2, let x be a boundary defining function of X and let dv ∂X be the Riemannian measure on the conformal boundary induced by the smooth metric x 2 g| ∂X . Let a ∈ C ∞ 0 (X) and let E X (s; •, ξ) be the Eisenstein series defined by (4) using the defining function x and a point ξ ∈ ∂X at infinity. Then, we have as

t → ∞ ∂X X a(m) E X ( n 2 + it; m, ξ) 2 dv(m)dv ∂X (ξ) = vol(S n ) X a(m)dv(m) + O(t 2δ Γ -n ).
Proof. The statement in Theorem ( 6) is uniform in ξ ∈ ∂X, this is clear from the proof of that Theorem. If F is a fundamental domain of X and η is defined as in the proof of Theorem [START_REF] De Verdière | Ergodicité et fonctions propres du Laplacien[END_REF] in F ∩ S n , then the measure dv ∂X pulls back to F ∩ S n by π Γ to the measure η(ξ) -n dξ where dξ is the canonical measure on S n . Since E X (n; π γ (m), π γ (ξ)) = η(ξ) n Γ E 0 (n; γm, ξ), we have

F∩S n E X (n; π γ (m), π γ (ξ))η(ξ) -n dξ = γ∈Γ F∩S n E 0 (n; m, γ -1 ξ)|Dγ(γ -1 ξ)| -n dξ = γ∈Γ γ(F∩S n ) E 0 (n; m, ξ)dξ ∂X E X (n; m, ξ)dv ∂X (ξ) = Ω Γ E 0 (n; m, ξ)dξ = S n E 0 (n; m, ξ)dξ = vol(S n )
where in the second line we used Lemma 4, in the third we used that the Jacobian of γ at ξ ∈ S n is |Dγ(ξ)| n , the fourth line we used that the set of discontinuity Ω Γ = ∪ γ γ(F ∩ S n ) has full measure in S n (for the standard measure on S n ). This ends the proof.

3.2.

Equidistribution in phase space. In this section, we extend the previous equidistribution result to "microlocal lifts" of the measures

E X ( n 2 + it; m, ξ) 2 dv(m).
To this end, we recall a few basic facts on pseudo-differential operators, including semi-classical ones. The proof of the micro-local extension will be based on semi-classical quantization and action on Lagrangian states.

3.3.

Pseudo-differential and semi-classical pseudo-differential operators. Let X be a complete manifold of dimension d. For s ∈ R, we define the space Ψ s c (X) of compactly supported pseudo-differential operators of order s on X to be the set of bounded operators A : C ∞ (X) → C ∞ 0 (X), with Schwartz kernel compactly supported in X × X, and which decomposes as a sum A = A sm + A sg where A sm has a smooth Schwartz kernel and A sg can be written in any local chart U with coordinate z under the following form: for all f ∈ C ∞ 0 (X) supported in U ,

A sg f (z) = 1 (2π) d e i(z-z ′ )ξ a(z, ξ)f (z ′ )dz ′ dξ with a(z, ξ) is a classical symbol of order s in U . A classical symbol a is an element in C ∞ (U × R d
) such that for all α, γ, there exists C > 0 so that

|∂ γ z ∂ α ξ a(z, ξ)| ≤ C(1 + |ξ|) s-|α| , a(z, ξ) ∼ |ξ|→∞ ∞ j=0 a j (z, ξ)
with a j some homogeneous functions of degree sj in ξ. The principal symbol of A is defined to be a 0 , and it is invariantly defined as a homogeneous section of degree s on T * X.

Let h ∈ (0, 1) be a small parameter. For s, k ∈ R, we define the space Ψ s,k c (X) of compactly supported semi-classical pseudo-differential operators of order s to be the set of h-dependent bounded operators A h : C ∞ (X) → C ∞ 0 (X), with Schwartz kernel compactly supported in X × X, and which decomposes as a sum A h = A h,sm + A h,sg where A h,sm has a smooth Schwartz kernel satisfying ||A h,sm || L 2 →L 2 = O(h ∞ ) and A h,sg can be written in any local chart U with coordinate z under the following form: for all f ∈ C ∞ 0 (X) supported in U ,

A h,sg f (z) = 1 (2πh) d e i (z-z ′ ).ξ h a(h; z, ξ)f (z ′ )dz ′ dξ with a(h; z, ξ) a semi-classical symbol of order s in U . A semi-classical symbol a(h; •, •) is an h-dependent element in C ∞ (U × R d
) such that for all α, γ and all N , there exists C > 0 so that

|∂ γ z ∂ α ξ a(h; z, ξ)| ≤ Ch -k (1 + |ξ|) s-|α| , a(h; z, ξ) = N -1 j=0 h -k+j a j (z, ξ) + h N r N (h; z, ξ)
where a j , r N are smooth functions satisfying

|∂ γ z ∂ α ξ a j (z, ξ)| ≤ C(1 + |ξ|) s-|α| , |∂ γ z ∂ α ξ r N (h; z, ξ)| ≤ Ch -k (1 + |ξ|) s-N -|α| .
The space Ψ 0,0 c (X) is an algebra under composition of operators, and the principal symbol of a composition is the product of their principal symbol. The semi-classical principal symbol of A h is defined to be h -k a 0 , and it is invariantly defined as a section of T * X. The microsupport

WF h (A h ) of A h is the complement of the set of points (z, ξ) ∈ T * X such that |∂ α a h | = O(h ∞ ) near (z, ξ), for all α. Lemma 9. Let B ∈ Ψ 0 c (X) and A h ∈ Ψ 0,0 c (X) with microsupport satisfying dist(WF h (A h ), T * 0 X) > ǫ,
for some ǫ > 0 uniform in h, and where T * 0 X is the zero section in T * X. Let a 0 , b 0 be the semiclassical principal symbol and classical principal symbol of A h and B. Then BA h ∈ Ψ 0,0 c (X) and the principal symbol of BA h is b 0 a 0 .

Proof. We decompose B = B sm + B sg , and using an adequate change of variables B sg is given locally by

Bf (z) = 1 (2πh) d e i (z-z ′ ).ξ h b(z, ξ/h)f (z ′ )dξdz ′ .
Let χ ∈ C ∞ 0 (R d ) so that its support does not intersect WF h (A h ). Then

B 1 sg : f → B 1 sg f (z) = 1 (2πh) d e i (z-z ′ ).ξ h (1 -χ(ξ))b(z, ξ/h)f (z ′ )dξdz ′
is an operator in Ψ 0,0 c (X) with semi-classical symbol (1χ(ξ))b 0 (z, ξ). On the other hand, by using non-stationary phase and the fact that supp(χ) ∩ WF h (A h ) = ∅, we deduce that (B sg -B 1 sg )A h has a smooth kernel and is an O(h ∞ ) as an operator on L 2 (X). The composition B sm A h has the same properties by applying again non-stationary phase (using that B sm has kernel supported away from the diagonal and a(h; z, ξ) = 0 near ξ = 0). We conclude that BA h ∈ Ψ 0,0 c (X) and its principal symbol is (1χ)b 0 a 0 = b 0 a 0 .

Action of semi-classical operators on semi-classical Lagrangian distributions.

Let us consider a Lagrangian submanifold L ⊂ T * X which is the graph of dϕ for some smooth function ϕ on X L = {(m, dϕ(m)) ∈ T * X; m ∈ X}.

A (compactly supported) semi-classical Lagrangian distribution with Lagrangian L is a function on X which can be written, for all N ∈ N, under the form

u(m) = e i ϕ(m) h ( N -1 j=1 a j (m)h j + h N r N,h (m))
for some a j , r N ∈ C ∞ 0 (X) and |∂ α m r N,h | = O(1) for all α. We call a 0 the principal symbol of u. Semi-classical pseudo-differential operators preserve the space of lagrangian distributions with Lagragian L, as is proved for example in [START_REF] Nonnenmacher | Quantum decay rates in chaotic scattering[END_REF]Lemma 4.1], which we recall here Lemma 10. Let T ∈ Ψ 0,0 c (X) be a compactly supported semi-classical operator with principal symbol α 0 and u be a compactly supported semi-classical Lagrangian distribution with Lagrangian L and principal symbol a 0 . Then T u is a compactly supported Lagrangian distribution with Lagrangian L and with principal symbol b 0 (m) := α 0 (m, dϕ(m))a 0 (m). More precisely

T u(m) = e i ϕ(m) h (b 0 (m) + hr 1,h (m)) with r 1,h ∈ C ∞ 0 (X) satisfying ||r 1,h || L 2 ≤ C(||u|| L 2 + ||a 0 || C n+3 )
for some C > 0 depending only on semi norms of the local symbol of T and on semi-norms of ϕ.

We will use the semi-classical notations in this section: let

s h = n 2 + i h , then for ξ ∈ S n fixed (26) E 0 (s h ; m, ξ) = E 0 ( n 2 ; m, ξ)e i φ ξ (m) h with φ ξ (m) := log 1 -|m| 2 4|m -ξ| 2 .
As a function of m, E 0 (s h ; m, ξ) is a semi-classical Lagrangian distribution associated to the Lagrangian submanifold

L ξ := {(m, dφ ξ (m)) ∈ T * m H n+1 ; m ∈ H n+1 } ⊂ T * H n+1 .
Let Γ be a convex co-compact group of isometries of H n+1 . For a given ξ ∈ Ω Γ , let us define the (closure of) discrete union of Lagrangians ( 27)

L Γ ξ := γ∈Γ L γξ .
If π Γ : H n+1 → Γ\H n+1 = X is the projection to the quotient, then

π Γ (L Γ ξ ) = γ∈Γ π Γ (L γξ )
is the closure of a countable superposition of Lagrangian submanifolds of T * X. Over a point m ∈ X, π Γ (L Γ ) ∩ S * m X corresponds to the (closure of) set of directions v(m) ∈ S * m X such that the geodesic starting at (m, v(m)) tends to π Γ (ξ) ∈ Γ\Ω Γ = ∂X. We point out that since Γ is discrete, for all ξ ∈ Ω Γ , we have

Γ.ξ =   γ∈Γ γξ   ⊔ Λ Γ ,
see Beardon [START_REF] Beardon | The Geometry of discrete groups[END_REF], Theorem 5.3.9. In other words, the set of accumulation points of the orbit Γ.ξ is the full limit set Λ Γ . Since L Γ ξ is locally diffeomorphic to H n+1 × Γ.ξ, some basic facts of dimension theory show that the Hausdorff dimension of L Γ ξ is n + 1 + δ Γ . We show the following Theorem 11. Let X = Γ\H n+1 be a convex co-compact hyperbolic manifold, and assume δ Γ < n 2 . Let A ∈ Ψ 0 c (X) be a compactly supported classical pseudo-differential operator of order 0 with principal symbol a ∈ C ∞ 0 (X, T * X). Then for ξ ∈ ∂X = Γ\Ω Γ , we have as h → 0 and for if

s h := n/2 + i/h AE(s h ; •, ξ), E(s h ; •, ξ) L 2 (X) = S * X a dµ ξ + O(h min(1,n-2δ) )
where dµ ξ is a measure supported on π Γ (L Γ ξ ) as defined in [START_REF] Zelditch | Uniform distribution of eigenfunctions on compact hyperbolic surfaces[END_REF]. More precisely, denoting also a and ξ for their lifts to respectively S * H n+1 and Ω Γ through π Γ , and if F is a fundamental domain of Γ, S * X adµ ξ can be written as a converging series (28)

S * X a(m, v)dµ ξ (m, v) := F γ∈Γ a(m, dφ γξ (m))E 0 (n; m, γξ)|Dγ(ξ)| n dv(m).
Proof. First we study AE X (s h ; •, ξ). Let χ, χ ′ ∈ C ∞ 0 (X) be functions such that χAχ = A and χ ′ χ = χ, and let

ψ h ∈ Ψ 0,0 c (X) be a semi-classical pseudo-differential operator with WF h (ψ h ) ⊂ {(m, v) ∈ T * M ; ||v| -1| ≤ 1/2} and with semi-classical principal symbol ψ equal to 1 on {(m, v) ∈ T * M ; ||v| -1| ≤ 1/4}. We first claim that χ ′ (1 -ψ h )χE X (s h ; •, ξ) = O(h) in L 2 . Indeed, let P h := h 2 (∆ X -n 2 /4) -1, then using that P h E X (s h ; •, ξ) = 0, we deduce that P h χ ′ (1 -ψ h )χE X (s h ; •, ξ) = [P h , χ ′ (1 -ψ h )χ]E X (s h ; •, ξ) which has L 2 norm O(h) since the commutator is in Ψ 0,-1 c (X) (the semi-classical symbol of ψ h being compactly supported) and ||χE X (s h , •, ξ)|| L 2 = O(1)
. The claim follows by the invertibility of the semi-classical principal symbol of P h (1-ψ h ) on the microsupport of χE X (s h ; •, ξ).

Therefore

(29) AE X (s h ; •, ξ) = Aψ h χE X (s h ; •, ξ) + O L 2 (h).
By Lemma 9, A h := Aψ h χ ′ ∈ Ψ 0,0 c (X) is a semi-classical operator with semi-classical principal symbol a(m, v)ψ(m, v).

Now we lift all objects to the covering H n+1 and we can assume without loss of generality that A has kernel compactly supported in F ×F where F is a fundamental domain of Γ. Recall that E X (s h ; m, ξ) can be expressed as a converging series

E X (s h ; m, ξ) = γ∈Γ E 0 ( n 2 ; m, γξ)|Dγ -1 (γξ)| -s h e iφ γξ (m)/h = γ∈Γ E 0 ( n 2 ; m, γξ)|Dγ(ξ)| s h e iφ γξ (m)/h
For the second line, we have used that γ is a Möbius transform, so that

|Dγ -1 (γξ)| -1 = |Dγ(ξ)|, series being convergent when δ Γ < n/2 since |Dγ(ξ)| ≤ Ce -d(o,γo) uniformly for ξ ∈ F ∩ S n , by (21)
. In particular, one has ||χE

X (s h ; •, ξ)|| L 2 = O(1), uniformly in ξ ∈ ∂X.
The action of A h on each term of the series, ie. A h (χe iφ γξ /h E 0 ( n 2 ; •, γξ)), is described by Lemma 10, A h (χe iφ γξ /h E 0 ( n 2 ; •, γξ)) is a Lagrangian distribution on H n+1 associated to the Lagrangian L γξ , and compactly supported in F:

A h (χe iφ γξ /h E 0 ( n 2 ; •, γξ)) = e iφ γξ /h χ(m)E 0 ( n 2 ; m, γξ)a(m, dφ γξ (m))ψ(m, dφ γξ (m))
+ hr h (m, γξ)

with ||r h (•, γξ)|| L 2 = O(1) bounded by a constant times finitely many semi-norms of a, E 0 ( n 2 , •, ξ) and of the phase φ γξ . But it is easily seen that for all α ∈ N n+1 there exists C α > 0 such that for all γ ∈ Γ and ξ ∈ F ∩ S n , (31) sup

m∈supp(χ) |∂ α m φ γξ (m)| ≤ C α , sup m∈supp(χ) |∂ α m E 0 ( n 2 ; m, γξ)| ≤ C α therefore ||r h (•, γξ)|| L 2
is uniformly bounded with respect to h, γ and ξ. Notice, by [START_REF] Guillopé | Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity[END_REF] and since ψ = 1 on {(m, v) ∈ T * X; ||v| -1| < 1/4}, we have ψ(m, dφ γξ (m)) = 1. We thus deduce that there exists C > 0 uniform in h, γ, ξ such that

e iφ γξ /h χE 0 ( n 2 ; •, γξ)a(•, dφ γξ (•)) + hr h (•, γξ) L 2 ≤ C. Consequently, using again δ Γ < n/2 with |Dγ(ξ)| ≤ Ce -d(o,γo) , γ∈Γ e iφ γξ /h |Dγ(ξ)| s h | χE 0 ( n 2 ; •, γξ)a(•, dφ γξ (•)) + hr h (•, γξ) L 2 < ∞
and we deduce that

A h χE X (s h ; m, ξ) = γ∈Γ A h (χe iφ γξ /h E 0 ( n 2 ; •, γξ))(m)|Dγ(ξ)| s h = γ∈Γ e i φ γξ(m) h |Dγ(ξ)| s h E 0 ( n 2 ; m, γξ)a(m, dφ γξ (m)) + hr h (m, γξ) . Integrating this against E X (s h , •, ξ) = E X (s h , •, ξ), we have (32) A h χE X (s h ; m, ξ), E X (s h , •, ξ) = F γ∈Γ |Dγ(ξ)| n E 0 (n; m, γξ)a(m, dφ γξ )dv(m) + O(h) + F γ =γ ′ e i B ξ (γm,γ ′ m) h |Dγ(ξ)| s h |Dγ ′ (ξ)| s h E 0 (1; m, γξ)E 0 (1; m, γ ′ ξ) n 2 a(m, dφ γξ (m))dv(m)
where B ξ (m, m ′ ) is the Buseman function of ( 18) and the O(h) term comes from the remainder

h γ∈Γ e iφ γξ /h r h (•, γξ)|Dγ(ξ)| s h , E X (s h , •, ξ)
which is uniformly bounded in ξ ∈ ∂X by the discussion above. Now it remains to apply Lemma 7 to deal with the off-diagonal term in the second line of (32), this gives that this term is a O(h n-2δ Γ ). Notice that the amplitude a(m, dφ γξ (m)) here depends on γ, but this does not affect the argument of Lemma 7, since the derivatives of the amplitude are uniformly bounded in γ, ξ by (31). The proof of the Theorem is then finished by combining this with (29).

As a corollary, we also get Corollary 12. Let X = Γ\H n+1 be a convex co-compact hyperbolic manifold with δ Γ < n 2 . Let E X (s; •, ξ) and dv ∂X be the Eisenstein series and the Riemannian measure on ∂X defined with a boundary defining function x as before. If A ∈ Ψ 0 c (X) is a compactly supported classical pseudo-differential operator of order 0 with principal symbol a ∈ C ∞ 0 (X, T * X), then setting

s h = n/2 + i/h, we have as h → 0 ∂X AE X (s h ; •, ξ), E X (s h ; •, ξ) L 2 (X) dv ∂X (ξ) = S * X a(m, v)dµ(m, v) + O(h min(n-2δ,1) )
where S * X is the unit cotangent bundle and dµ the Liouville measure.

Proof. We integrate [START_REF] Zelditch | Mean lindelöf hypothesis and Equidistribution of Cusps forms and Eisenstein series[END_REF] in the ξ ∈ ∂X variable with respect to dv ∂X like in the proof of Corollary 8 by lifting to the covering: we have, modulo O(h min (1,n-2δ) ).

∂X AE X (s h ; •, ξ), E X (s h ; •, ξ) L 2 (X) dv ∂X (ξ) = F γ∈Γ ξ∈F∩S n a(m, dφ γξ (m))E 0 (n; m, γξ)|Dγ(ξ)| n dv(m)dξ = F ξ∈Ω Γ a(m, dφ ξ (m))E 0 (n; m, ξ)dv(m)dξ
and the last term is exactly S * X a(m, v)dµ(m, v) by using ( 14) and (15).

Gutzwiller type formula

In this section, we shall give a more precise description of the equidistribution in space when the Eisenstein series are averaged over the boundary. Through the Stone identity [START_REF] De Verdière | Ergodicité et fonctions propres du Laplacien[END_REF], this reduces to a sort of Gutzwiller trace formula. We prove the following Theorem 13. Let X = Γ\H n+1 be a convex co-compact hyperbolic manifold and assume that the limit set of Γ has Hausdorff dimension δ Γ < n/2. Let a ∈ C ∞ 0 (X) and C be the set of all closed geodesics, then for all N ∈ N, one has the expansion as t → ∞,

∂X X a(m)|E( n 2 + it; m, ξ)| 2 dm dξ ∂ X = vol(S n ) X a dv +L(t) γ∈C e -( n 2 +it)ℓ(γ) N k=0 H k (ℓ(γ)) | det(Id -e -ℓ(γ) R -1 γ )| t -k γ P k a dµ+ +L(-t) γ∈C e -( n 2 -it)ℓ(γ) N k=0 H k (ℓ(γ)) | det(Id -e -ℓ(γ) R -1 γ )| (-t) -k γ P k a dµ + O(t -N -n-1 ). (33)
where H k (ℓ(γ)) is an explicit bounded function of ℓ(γ), P k are differential operators of order 2k with coefficients uniformly bounded in terms of γ ∈ C, dµ is the Riemannian measure induced on γ, R γ ∈ SO(n) is the holonomy along γ and

L(t) = t -(n-1)/2 2 n 2 +3-2it |Γ(it)| 2 Γ(it + 1 2 )Γ( n 2 -it) = O(t -n ) P 0 = 1, H 0 (ℓ(γ)) = 2 n 2 π n-1 2 e i n-1 4 π
Proof. We write µ t (a) for the left hand side of (3). We will assume, without losing generality, that the support of a is small enough so that there exists a fundamental domain F ⊂ H n+1 with ∂F ∩ supp(a) = ∅. We have by ( 6)

µ t (a) = X ∂X a(m)E X ( n 2 + it; m, y)E X ( n 2 -it; m, y)dv(m)dv ∂X (y) = 4πt |C( n 2 + it)| 2 X a(m)dΠ X (t; m, m)dv(m) µ t (a) = 4πt |C( n 2 + it)| 2 γ∈Γ F a(m)dΠ 0 (t; γm, m)dv(m).
where dΠ X is the spectral measure of ∆ on X and dΠ 0 the spectral measure of ∆ on H n+1 . For the γ = Id term, the spectral measure on H n+1 restricted on the diagonal is a constant function given by

α(t) := 4πt dΠ 0 (t; m, m) = π -n 2 Γ(n/2) Γ(n) Γ( n 2 + it)Γ( n 2 -it) Γ(it)Γ(-it) = |C( n 2 + it)| 2 vol(S n ).
The terms with γ = Id are actually lower order as t → ∞: we claim that (34

) γ∈Γ\Id F a(m)tdΠ 0 (t; γm, m)dv(m) = O(1)
and we will actually obtain a much better description.

From the explicit formula of the resolvent for the Laplacian on hyperbolic space in [START_REF] Duistermaat | The spectrum of positive elliptic operators and periodic geodesics[END_REF], we have that 4πtdΠ 0 (t; γm, m) = -4tIm(F t (σ(d(m, γm))))

where σ(r) = 1/ cosh(r), d(•, •) denotes the hyperbolic distance and

F t (σ) := M (t)σ n 2 +it 1 0 (u(1 -u)) it-1 2 (σ(1 -2u) + 1) -n 2 -it du M (t) := π -n+1 2 2 1-n 2 -it Γ( n 2 + it) Γ( 1 2 + it) = O(t n- 1
2 ) Lemma 14. Let ǫ > 0, and σ(r) = cosh(r) -1 , then there exist some smooth functions c k ∈ C ∞ ([0, 1ǫ)) such that for all N , we have as t → ∞

F t (σ(r)) = M (t)2 -it e -itr σ(r) n 2 N k=0 t -1 2 -k c k (σ(r)) + t -1 2 -N -1 R N (σ(r), t) .
with the remainder

R N satisfying |∂ ℓ σ R N (σ, t)| ≤ C ℓ , for all ℓ ∈ N 0 , σ ∈ [0, 1 -ǫ), t > t 0 > 0, and c 0 (σ) = π 1 2 e -iπ/4 (1 -σ 2 ) -n/4 .
Proof. Assuming that σ < 1-ǫ, then in the integral defining F t (u), one can use the stationary phase method (e.g. [START_REF] Evans | Lectures on semi-classical analysis[END_REF]Th 3.10] ) to get an asymptotic expansion in t → ∞: the phase has a unique nondegenerate critical point in (0, 1) given by u σ = 1 2σ (1 + σ -√ 1σ 2 ), the Hessian is a smooth function of σ ∈ [0, 1ǫ) which is bounded from below by a positive constant depending only on ǫ > 0. Using a cutoff function near the point u σ and using integration by parts for the non-stationary part, one deduces as t → ∞

σ n 2 +it 1 0 (u(1 -u)) it-1 2 (σ(1 -2u) + 1) -n 2 -it du ∼ σ n 2 ∞ k=0 t -1 2 -k c k (σ) 1 - √ 1 -σ 2 2σ
it with c k (σ) some smooth function of σ ∈ [0, 1ǫ] and a straightforward computation gives

(35) c 0 (σ) = π 1 2 e -iπ/4 (1 -σ 2 ) -n/4 . Since 1- √ 1-σ(r) 2 2σ(r) = 1
2 e -r , we have an expansion, for all N ∈ N, as t → ∞ (36)

F t (σ(r)) = M (t)2 -it e -itr σ(r) n 2 N k=0 t -1 2 -k c k (σ(r)) + O σ (t -1 2 -N -1 ) .
with a remainder uniform in σ < 1ǫ, as well as its derivatives with respect to σ.

Notice that here d(m, γm) > C 0 for some C 0 > 0 uniform in m (by the convex cocompactness of Γ), then σ(d(m, γm)) < 1ǫ for some uniform ǫ and thus F t (σ(m, γm)) is smooth in m ∈ K for any fixed compact set K ⊂ H n+1 .

The displacement function r(m, γm), or more precisely sinh( 1 2 r(m, γm)), is easily computed in dimension n + 1 = 2, it is given by sinh( 12 r(m, γm)) = cosh(d(m, axis(γ))) sinh( 1 2 ℓ(γ)). In higher dimension it is more complicated. Consider the half plane model {x > 0, y ∈ R n } of H n+1 . We can assume that ∞ is not in the limit set of Γ, so that a fundamental domain can be taken in a ball x 2 + |y| 2 ≤ R for some R. For all γ ∈ Γ, there is an isometry h γ such that (37)

h γ • γ • h -1 γ (x, y) = e ℓ(γ) (x, R γ y)
where ℓ(γ) > 0 is the translation length of γ and R γ ∈ SO(n) the holonomy associated to γ.

Since σ(d(m, γm)) = σ(d(h γ m, h γ • γ • h -1 γ h γ m)), one has (38) σ(d(h -1 γ m, γ • h -1 γ m)) = e -ℓ(γ) 2x 2 x 2 (1 + e -2ℓ(γ) ) + |(e -ℓ(γ) Id -R γ )y| 2
where m = (x, y). A consequence of this and Lemma 14 is Corollary 15. With h γ defined by (37), we have as

t → ∞ F a(m)F t (σ(d(m, γm)))dv(m) = M (t) 2 it N k=0 t -k-1 2 hγ(F ) e -itrγ (m) b k (σ γ (m))a(h -1 γ (m)) dm x n+1 + O( max m∈supp(a) e -n 2 d(m,αm) t -N +n-2 ) ( 39
)
where r γ (m) =:

d(h -1 γ m, γ • h -1 γ m), σ γ (m) := σ(d(h -1 γ m, γ • h -1 γ m)) and b k (σ) := σ n 2 c k (σ).
We now want to use stationary phase in the space variable m ∈ F in the integral (39). Let us define the following function Φ γ (m) := log(σ(d(h

-1 γ m, γ • h -1 γ m)), so that (40) dr γ (m) = dΦ γ (m) tanh(r γ (m))
.

We first calculate the critical points of the phase r γ (m), which in turn are the critical points of the function Φ γ (m) in the m = (x, y) coordinates of H n+1

dΦ γ (x, y) = 2|(e -ℓ(γ) Id -R γ )y| 2 x 2 (1 + e -2ℓ(γ) ) + |(e -ℓ(γ) Id -R γ )y| 2 dx x -x 2(e -ℓ(γ) Id -R γ ) T (e -ℓ(γ) Id -R γ )y x 2 (1 + e -2ℓ(γ) ) + |(e -ℓ(γ) Id -R γ )y| 2 . dy x . ( 41
)
Therefore the critical points are located on the line L = {y = 0} which is the image of the axis of γ by h γ . For each slice x = cst, we will integrate in the y variable in (39) by using the stationary phase, we then first need to check that det(Hess(Φ γ (x, 0))) = 0 for all x, where the Hessian is with respect to the y ∈ R n variable. The Hessian (in y variable) at y = 0 is given by Hess

(Φ γ (x, 0)) = -2 (e -ℓ(γ) Id -R γ ) T (e -ℓ(γ) Id -R γ ) x 2 (1 + e -2ℓ(γ) )
and thus, using that r γ (x, 0) = ℓ(γ),

(-r γ (x, 0)) = 2 x 2 (1 -e -2ℓ(γ) ) (e -ℓ(γ) Id -R γ ) T (e -ℓ(γ) Id -R γ ). with | det(e -ℓ(γ) Id -R γ )| 2 = | det(Id -e -ℓ(γ) R -1 γ )| 2 > 0. (42) Hess
Therefore the phase is non-stationary if the axis of γ does not intersect the support of a(m), contained in the fundamental domain F, and the integral is then a O(t -∞ ) in this case (constants are depending on γ). We shall split the sum over γ ∈ Γ into conjugacy classes [γ] in the group, as is usual in Selberg type analysis. For each primitive element γ, the conjugacy class of γ corresponds to a primitive geodesic on X, which is given by

π Γ βγβ -1 ∈[γ]
β(axis γ ) ∩ F where π Γ : H n+1 → Γ\H n+1 is the natural projection, [γ] denotes the conjugacy class of γ and axis γ is the axis of γ. Moreover, there are only finitely many geodesics β(axis γ ) that intersect F, we then partition

[γ] into A 1 (γ) ∪ A 2 (γ) so that α = βγβ -1 ∈ A 2 (γ) iff d(β(axis γ ), F) > 1.
If γ is not primitive, one can do the same thing with γ k 0 for some primitive γ 0 . We first show the Lemma 16. Let γ ∈ Γ be a primitive element, then that for all N > 0 there is C N depending only on N such that for all α ∈ A 2 (γ)

(43) F a(m)F t (σ(d(m, αm)))dv(m) ≤ C N t -N +(n-1)/2 max m∈supp(a) e -n 2 d(m,αm) .
Proof. For α ∈ A 2 (γ), we then use non-stationary phase to evaluate (39), i.e. integration by parts:

(44)

hα(F) e -itrα(m) b k (σ α (m))a(h -1 α (m)) dxdy x n+1 = - 1 it hα(F) ∂ x (e -itrα(m) ) ∂ x Φ α (m) tanh(r α (m))b k (σ α (m))a(h -1 α (m)) dxdy x n+1 = 1 it hα(F) e -itrα(m) (x∂ x -n) tanh(r α (m))b k (σ α (m))a(h -1 α m) x∂ x Φ α (m) dxdy x n+1 = 1 (it) N hα(F) e -itrα(m) (x∂ x -n) tanh(r α (m)) x∂ x Φ α N b k (σ α (m))a(h -1 α m) dxdy x n+1
where N ∈ N will be chosen later. To estimate the terms in the last integral, we need an expression for h α : if p 1 α , p 2 α ∈ R n are the two fixed points of α, define p α := p 2 αp 1 α , then h α will be the composition of the translation t α : m → mp 1 α with the reflection in the sphere {m; |m -

p α | = |p α |} followed by the dilation m → |p α | -1 m (45) g α : m → p α |p α | + |p α | (m -p α ) |m -p α | 2 , p α = p 2 α -p 1 α
and this defines h α := g α • t α . This map is an orientation reversing isometry of H n+1 which maps p 1 α to 0 and p 2 α to ∞, it conjugates α to a model form (37

). If m = (x, y) ∈ R + × R n are coordinates on H n+1 , then (46) x(h α (m)) = |p α |x |m -p 2 α | 2 , y(h α (m)) = p α |p α | + |p α | (y -p 2 α ) |m -p 2 α | 2 .
In particular, one has max m∈supp(a) |h α (m)| ≤ R for some R depending only on supp(a). We notice that there exists C > 0 uniform in γ, α such that for all γ, α

(47) min{|y|; m = (x, y) ∈ h α (supp(a)), α ∈ A 2 (γ)} ≥ C.
Indeed, by definition of A 2 (γ) and the fact that h α is an isometry, the hyperbolic distance between h α (supp(a)) and the line y = 0 is bounded below by 1, but

C 0 > x(h α (m)) > C 1 |p α |
for some C 0 , C 1 > 0 depending only on the support of supp(a). Then for ǫ > 0 small and α such that |p α | ≥ ǫ > 0, there is a constant C depending on ǫ such that (47) holds for those α. For the remaining α for which |p α | ≤ ǫ, it is easily seen from the expression (45) that |h α (m)| = 1 + O(ǫ) for m ∈ supp(a) and therefore if ǫ is small enough |y(h α (m))| > 1/2. We now want to bound the terms in the right hand side of (44). Using that |m| is uniformly (with respect to α) bounded on h α (supp(a)) and using (41), (47), we deduce that there exists C N > 0 depending only on N and C > 0 such that for all [γ], all α ∈ [γ], and all j ≤ N min m∈hα(supp(a))

|(x∂ x )Φ α (m)| ≥ C, sup m∈hα(supp(a)) |Y 1 . . . Y j Φ α (m)| ≤ C N , sup m∈hα(supp(a)) |Y 1 . . . Y j σ α (m) n 2 | ≤ C N sup supp(a) σ n 2 (d(m, αm)) sup m∈hα(supp(a)) |Y 1 . . . Y j σ α (m)| ≤ C N sup supp(a) σ(d(m, αm)) if Y i ∈ {x∂ x , x∂ y } (recall that σ α = e Φα ) Since g -1 α (m) = |p α |g α (|p α |m) and dg α (m) = |p α | Id |m -p α | 2 -2 (m -p α ) m -p α , • |m -p α | 4
then we have We need to consider now the finitely many α ∈ A 1 (γ) which contributes to the stationary phase (notice that there are roughly O(ℓ(γ)) elements in A 1 (γ)). Notice that, by the arguments above, the compact set h α (supp(a)) remains in a bounded region {C ≥ |y|, C > |x| > 1/C} of H n+1 , uniformly in γ, α ∈ A 1 (γ). This follows since the distance in the boundary between the fixed points of α is uniformly bounded below if the axis of α intersects supp(a). For the same reason, we obtain from the expression (45) that for α ∈ A 1 (γ) (49) max m∈hα(supp(a))

dg -1 α (g α (m)) = |p α | -1 (|m -p α | 2 Id -2(m -p α ) m -p α ,
|∂ µ x ∂ ν y h -1 α (m)| ≤ C
for some C depending on µ, ν but independent of γ, α. The phase in the integral are bounded uniformly. The uniform boundedness of semi-norms of r α in a fixed compact set of H n+1 (containing h α (supp(a))) follows directly from the expressions (40) and (41), while determinant of Hessian boundedness is clear from (42) since ℓ(γ) is bounded below by a positive constant (the radius of injectivity). By the explicit formula (41), and using (40), it is direct that |dr α (x, y)| ≥ C|y| for some C uniform in α and (x, y) in a fixed compact set of H n+1 (using again that ℓ(γ) is bounded below by a uniform positive constant). Similarly all terms in the left hand side of (50) is bounded by sup m∈supp(a) e -n 2 (d(m,αm)) . Notice that σ α (m) = cosh -1 (ℓ(γ)) is constant on m ∈ {y = 0}, and thus we have proved Lemma 17. With I k,α defined in (50), we have that for all N ∈ N, there exists C N > 0 such that for all [γ], all α ∈ A 1 (γ), and all t > 1 for some differential operator Q 2j with coefficients bounded uniformly in α, [γ], and dµ is the measure induced by the riemannian measure on axis(γ).

In particular from (50) and the usual expression of A 0 (x, y, ∂ y ) (see Th 3.14 in [START_REF] Evans | Lectures on semi-classical analysis[END_REF]) and (42), we deduce where dµ is the measure induced by the riemannian measure on axis(α). From (35), one also has c 0 (cosh(ℓ(γ)) -1 )) = √ πe -iπ/4 (tanh ℓ(γ)) -n/2 . converges uniformly for m in compact subsets of H n+1 . This ends the proof.

Summing

m -p 2 α|m -p 2 α

 22 • ) and combining with (46) and using dh α(m) = dg α (mp 1 α ) (48) h * α (xY )(m) = Y -2 (mp 2 α ) |mp 2 α | | 2 , Y , Y ∈ {∂ x , ∂ y }.Combining with (39), this shows that there is C N depending only on N such that for all α ∈ A 2 (γ)supp(a) a(m)F t (σ(d(m, αm)))dv(m) ≤ C N t -N +(n-1)/2 supp(a) σ(d(m, αm)) n 2 max j≤N |(h * α (x∂ x )) N a(m)|dv(m).and by (48) we have sup m∈supp(a) |(h * α (x∂ x )) N a(m)| ≤ C N for some C N > 0 independent of α, γ. We thus have proved the Lemma.

IA

  k,α (t) := hα(F) e -itrγ (x,y) b k (σ α (x, y))a(h -1 α (x, y)) dxdy x n+1 , is stationary at y = 0, and thus integrating on each slice x = cst, we have by stationary phase ([13, Th. 7.7.5])(50)I k,α (t)e -itℓ(γ) 2j (x, y, ∂ y )[a(h -1 α (x, y))b k (σ α (x, y)β (b k • σ α .a • h -1 α )|| L ∞ .for some smooth differential operator A 2j (x, y, ∂ y ) of order 2j. By [13, Th. 7.7.5], the constant C is shown to be bounded uniformly with respect to α, γ if ||r α || C 3N+3n (hα(supp(a))) and sup (x,y)∈hα(supp(a)) |y| |dr α (x, y)| are bounded uniformly with respect to α ∈ A 1 (γ) and with respect to [γ]. The differential operators A 2j have coefficients bounded uniformly in α, γ if | det(Hess(r α ))| -1 and ||r α || C 3N+3n (hα(supp(a)))

  The constant C in (50) is therefore uniform in α ∈ A 1 (γ) and the conjugacy class [γ]. Now we claim that the amplitudes ||∂ β (b k • σ α .a • h -1 α )|| L ∞ ≤ C max m∈supp(a) e -n 2 d(m,αm) with some C uniform in [γ], α ∈ A 1 (γ). By (49), the terms involving a • h α are clearly bounded uniformly. Moreover, since b k (σ) = σ n 2 c k (σ), it suffices to bound the derivatives of Φ α = log(σ α ). By using that h α supp(a) is in a uniform compact set of H n+1 with respect to α ∈ A 1 (γ) and [γ], we have |∂ β dΦ| ≤ C and thus sup m∈hα(supp(a)) |∂ β b k (σ α (m))| ≤ sup m∈supp(a) e -n 2 (d(m,αm)) .

I

  k,α (t) -c k ((cosh ℓ(γ)) -1 )) (cosh ℓ(γ))

I 2 (2π) n 2 e i nπ 4 (

 24 k,α (t) =e -i(t+ n 2 )ℓ(γ) t -n tanh ℓ(γ)) n 2 c k (cosh(ℓ(γ)) -1 ) | det(Ide -ℓ(γ) R -1 γ )| axis(α)∩F a(m)dµ + O( sup m∈supp(a)e -n 2 d(m,αm) t -n/2-1 ).

  Lemma 17 and 16 over all α ∈ [γ], we obtain the integrals over the closed geodesic associated to the conjugacy class [γ], and then summing over the set of conjugacy classes [γ], we obtain the result claimed in the Theorem. The sums over the group converges since each estimate involving α ∈ [γ] has a bound in terms of sup supp(a) e -n 2 (d(m,αm)) , and the series [γ] α∈[γ] e -n 2 (d(m,αm))

  The group of isometries Isom + (H n+1 ) acts transitively on H n+1 × S n by γ.(m, ξ) := (γm, γξ) and using Lemma 4, we obtain for all γ ∈ Isom + (H n+1 ) and dξ are the canonical volume forms of H n+1 and S n . But we also have ψ(γm, γξ) = γ.ψ(m, ξ) and therefore we deduce that for all γ ∈ Isom + (H n+1 )

	(14)	γ

* ν = ν, where ν := E(n; m, ξ)dv(m) ∧ dξ where dv(m)

Acknowledgement. We both thank D. Jakobson for pointing out previous works on equidistribution of Eisenstein series and S. Nonnenmacher for helpful discussions on that topic, as well as M. Zworski for his interest and urging us to write this down. Both authors are supported by ANR grant ANR-09-JCJC-0099-01.