WAVE 0-TRACE AND LENGTH SPECTRUM ON CONVEX CO-COMPACT HYPERBOLIC MANIFOLDS ´ ERIC ´ COLIN GUILLARMOU AND FRED NAUD Abstract. For convex co-compact hyperbolic quotients, we prove a formula relating the 0trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillop´ e and Zworski in dimension 2. As an application, we obtain asymptotics of the number of closed geodesics with an effective, exponentially small error term.

1. Introduction The purpose of this note is to explain the relation between the renormalized trace (called 0-trace) of the wave operator, the resonances, some conformal invariants of the boundary and the length spectrum for the Laplacian on a convex co-compact quotient of the hyperbolic space Hn+1 . We derive from Patterson-Perry [26], Bunke-Olbrich [6] works on Selberg zeta function a trace formula similar to the one given by Guillop´e and Zworski [12, 13] on surfaces. Actually, a Selberg formula relating resonances and length spectrum has already been obtained by Perry [30] in this setting but Perry did not use explicitly the 0-trace of the wave operator. The first motivation of studying this 0-trace is that our formula could be extended to more general settings like asymptotically hyperbolic manifolds. Another interest of this formula is also the conformal invariants of the boundary which appear in the expansion of the 0-trace of the wave operator and come from the divisors of Selberg’s zeta function at certain values. Then we compute the 0-trace of the wave operator in terms of the primitive geodesics to obtain asymptotic expansions for the counting function of the set of closed geodesics with precise error terms related to the spectrum of the Laplacian, improving asymptotics previously obtained by Perry [29]. To prove the trace formula we proceed as in Guillop´e-Zworski paper [12] although we don’t have some model operators and relative scattering operator in this case. Here, the role of the relative scattering phase is played by Selberg zeta function associated to the group, as emphasized by Perry [30]. Let us first recall standard notations and definitions. A discrete group Γ of (orientation preserving) isometries of the n + 1-dimensional real hyperbolic space Hn+1 is called convex cocompact if it admits a finite sided fundamental domain whose intersection with ∂Hn+1 does not touch the limit set of Γ. If we require that Γ has no elliptic elements, then X = Γ\Hn+1 is a hyperbolic manifold of infinite volume. The non-wandering set of the geodesic flow on the unit tangent bundle is then a compact set whose Hausdorff dimension is 2δ + 1, where δ is the dimension of the limit set. The dimension δ is also the topological entropy of the geodesic flow on its non-wandering set, see the work of Sullivan [33, 34]. Let P be the set of primitive closed geodesics γ on X, and if γ ∈ P, its length is denoted by l(γ). A wide class of convex co-compact groups are given by Schottky groups of isometries, and every convex co-compact surface is in fact obtained by a quotient of H2 by a fuchsian Schottky 2000 Mathematics Subject Classification. Primary 58J50, Secondary 35P25. 1

´ ERIC ´ COLIN GUILLARMOU AND FRED NAUD

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group. In higher dimensions, as pointed out by Maskit [20], the set of Schottky groups does not exhaust the set of convex co-compact groups. Indeed, a 3-manifold is Schottky if and only if its fundamental group is a free group, thus all convex co-compact manifolds obtained by a quotient of H3 by quasifuchsian groups cannot be Schottky. We also recall that on a convex co-compact hyperbolic quotient X = Γ\H n+1 equipped with 2 its hyperbolic metric g, the Laplacian ∆X has for continuous spectrum the half line [ n4 , +∞) and 2 a finite set σpp (∆X ) of eigenvalues included in (0, n4 ). The modified resolvent of the Laplacian R(s) := (∆X − s(n − s))−1 is meromorphic on {<(s) > n2 } with a finite number of poles related to σpp (∆X ), and extends to s ∈ C (from L2comp (X) to L2loc (X)) with poles of finite multiplicity called resonances, the multiplicity of a resonance s0 being defined by ms0 := rank(Ress0 R(s)).

(1.1)

¯ such We can also add that X can be compactified in a compact manifold with boundary X ¯ ¯ ¯ that, for x a boundary defining function of ∂ X in X (i.e. ∂ X = {x = 0} and dx|∂ X¯ 6= 0), ¯ In view of the non-uniqueness of the boundary the metric x2g on X extends smoothly to X. defining function, the boundary carries a natural conformal class of metric [h0 ] associated to g by taking the conformal class of h0 := x2 g|∂ X¯ . Graham, Jenne, Manson and Sparling [7] have defined some conformally covariant powers of the Laplacian on a conformal manifold and those were redefined by Graham and Zworski [9] using scattering theory. We will denote by P k this ¯ [h0]). k-th conformal power of the Laplacian on the conformal infinity (∂ X, On a convex co-compact hyperbolic manifold, although the trace of the wave operator is not well defined, there is a natural renormalization called 0-trace. Thus, following Joshi-Sa Barreto [19] or Guillop´e-Zworski [12, 13], we can define the 0-trace of the wave operator as a distribution on R∗ noted !! r n2 0-tr cos t ∆X − . 4 The definition of this renormalized trace will be detailed in next section. We recall that the set of closed geodesics P is in one-to-one correspondence with primitive conjugacy classes of hyperbolic isometries in Γ. Given an hyperbolic element h ∈ Γ associated to a closed geodesic γ, then there exists α ∈ Isom(Hn+1 ) such that for all (x, y) ∈ Hn+1 = Rn ×R+ , α−1 ◦ h ◦ α(x, y) = el(γ) (Oγ (x), y), where Oγ ∈ SOn (R). We will denote by α1(γ), . . . , αn(γ) the eigenvalues of Oγ , and we set n     Y Gγ (k) = det I − e−kl(γ) Oγk = 1 − e−kl(γ) αi(γ)k .

1

i=1

We can now state the wave 0-trace formula.

Theorem 1.1. Let X = Γ\Hn+1 a convex co-compact hyperbolic manifold, (Pk )k∈N the k-th ¯ then we have as distributions on R∗ conformal Laplacian on the boundary ∂ X, !! r cosh 2t 1X 1X n n2 0-tr cos t ∆X − = ms e−( 2 −s)|t| + dk e−k|t| − 2−n−1A(X) , |t| 4 2 2 (sinh )n+1 s∈R

where

dk := dim ker Pk ,

A(X) =



k∈N

0 χ(X) the Euler characteristic of X

2

if n + 1 is even . if n + 1 is odd

1If P is the Poincar´ e linear map asociated to the primitive periodic orbit γ of the geodesic flow on the unit γ tangent bundle, then ´ ` n | det I − Pγm |1/2 = e 2 ml(γ) Gγ (m).

WAVE 0-TRACE AND LENGTH SPECTRUM

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We also have r

n2 0-tr cos t ∆X − 4

!!

=

n ∞ XX cosh 2t l(γ)e− 2 ml(γ) δ(|t| − ml(γ)) + B(X) 2Gγ (m) (sinh |t| )n+1 γ m=1

2

where the sum runs over the primitive closed geodesics of X and  3(n+1) n+1 n!!2− 2 (−π)− 2 0-vol(X) if n + 1 is even B(X) = 0 if n + 1 is odd. Remark: the first formula look quite surprising in view of these additional terms dk coming from the boundary since in other settings, the trace formula involves only resonances. Of course, we expect that this one could be extended to compact perturbations of hyperbolic convex cocompact manifolds and possibly to asymptotically Einstein manifolds, at least when n+1 is even. Observe also that it corresponds to Guillop´e-Zworski trace formula in [12] in the sense that in dimension 2 we have dk = 0 for all k (in this case, dk must be interpreted as the dimension of the kernel of the residue of the scattering matrix at n2 + k). As a corollary of the trace formula, we can study the asymptotic behaviour of the primitive geodesics counting function. The basic counting function N (T ) for the closed geodesics is defined as usual for T ≥ 0 by N (T ) = #{γ ∈ P : l(γ) ≤ T }. In this paper, we show the following. Theorem 1.2. Let X = Γ\Hn+1 be a convex co-compact manifold as above such that δ > As T → +∞, we have  βn (δ)T  X e li(eαi T) + O N (T ) = li(eδT ) + , T

n 2.

βn (δ)<αi <δ

R x dt and βn (δ) = where li(x) = 2 log(t) by αi(n − αi) = λi ∈ σpp (∆X ).

n ( 1 +δ). n+1 2

The coefficients αi are in bijection with σpp (∆X )

δT

Remarks. The leading term N (T ) ∼ eδT has been obtained by Perry in [29] without any assumption on the dimension δ. The question to obtain similar asymptotics when δ ≤ n2 is to our knowledge still open when 2 n ≥ 2. In the particular case of surfaces (n = 1), the second author [24] has obtained unconditional exponentially small error terms using transfer operator techniques and thermodynamical formalism. We point out that this asymptotic expansion also holds for geometrically finite surfaces with cusps [25]. If we let δ → 1 then we essentially recover 3 the error term O(e 4 T /T ) known for finite area surfaces, see the work of Huber, Hejhal, Randol and Sarnak [17, 16, 31, 32]. For 3-manifolds, Theorem 1.2 can be applied for example to the case of stricly quasifuchsian groups Γ (i.e the limit set is a strict quasicircle) for which we know since Bowen [4] that δ > 1 = n2 . Acknowledgements: The first author would like to thank Robin Graham for helpful discussions and Sie Hung Tang for pointing out the appendix of Patterson-Perry [26] to compute the conformal Laplacians on hyperbolic compact manifolds. Also, we thank Laurent Guillop´e for his comments on this work. 2At this time, there is a preprint By L. Stoyanov [35] which implies an exponentially small error term for an

arbitrary convex co-compact manifold without any assumptions on δ. The proof is a highly non-trivial extension of [24], but the error estimate is not at all explicit.

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´ ERIC ´ COLIN GUILLARMOU AND FRED NAUD

2. 0-trace formula 2.1. 0-Renormalization. Let X = Γ\Hn+1 a convex co-compact hyperbolic manifold equipped ¯ be its compactification as a smooth manifold with with the hyperbolic metric g and let X boundary. On such a manifold and after having chosen a boundary defining function, we can ¯ by the formula define the 0-integral of a smooth function f on X Z 0 Z f := F P→0 f(m)dvolg (m), X

x(m)>

this depends of course on the function x and this definition can be extended for functions for which the finite part exists. However, it is shown for exemple by Graham [8] that the 0-volume of X, defined as the 0-integral of the function 1, is independent of the choice of x (here X is Einstein) if the dimension n + 1 of X is even. For an operator A on X which has a Schwartz kernel A(m, m0 ) which is smooth when restricted to the diagonal of X × X, one can also define its 0-trace by Z 0 0-tr A := A(w, w) X

when the 0-integral exists (see [12, 19, 26, 3] for details and examples). Note that those renormalizations are naturally related to the 0-calculus and 0-structure defined by Mazzeo-Melrose [21, 22, 23] (i.e. related to the ‘geometric operators’ on conformally compact manifolds). Following Joshi-Sa Barreto [19] (see also Guillop´e-Zworski [12, 13]), we can define its 0-trace as a distribution on R∗ !! r n2 . (2.1) u(t) := 0-tr cos t ∆X − 4 which means that for all ϕ ∈ C0∞ (R∗ ) r

n2 hu, ϕi := 0-trhcos • ∆X − 4

!

, varphii.

2.2. Resonances, scattering poles and conformal operators. If s is not a resonance and not in 21 Z, the generalized eigenfunctions of ∆X for the ‘eigenvalue’ s(n − s) have the following ¯ behaviour on X ¯ E(s) = xsF1 (s) + xn−sF2(s), Fi(s) ∈ C ∞ (X) and one can show (see [9] for example) that it is unique if one requires that F2 (s)|∂ X¯ = f0 for ¯ Thus one can define the scattering operator as the operator on a fixed function f0 ∈ C ∞ (∂ X). ∞ ¯ C (∂ X) S(s) : f0 → F1 (s)|∂ X¯ . It turns out that (see [27, 26]) for h0 := (x2g)|∂ X¯ the operator n −s+n e := 22s−n Γ(s − 2 ) (1 + ∆h0 ) −s+n 2 S(s)(1 + ∆h0 ) 2 S(s) n Γ( 2 − s)

has also a meromorphic extension to C with poles (called scattering poles) of finite multiplicity in C \ ( n2 + N), the multiplicity of a pole s0 being defined here by (2.2)

νs0 := −Tr(Ress0 (Se0 (s)Se−1 (s))).

The set of the scattering poles contained in {<(s) < n2 } will be denoted by S whereas the set of resonances will be denoted by R. In [10], we have given a formula relating the multiplicities (1.1) and (2.2) on an general asymptotically hyperbolic manifold. We recall this result in the present case and extend it slightly:

WAVE 0-TRACE AND LENGTH SPECTRUM

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Proposition 2.1. Let X = Γ\Hn+1 be a convex co-compact hyperbolic manifold with conformal ¯ [h0]) and let <(s0 ) < n , then we have the relation infinity (∂ X, 2 νs0 = ms0 − mn−s0 + 1l n2 −N (s0 ) dim ker P n2 −s0

(2.3)

with 1l n2 −N the characteristic function of n2 − N and Pk for k ∈ N is the k-th conformal Laplacian ¯ [h0]) defined in Graham-Zworski [9]. on the conformal manifold (∂ X, Proof : in [10] we dealt with all cases except when s0 ∈ {s; s(n − s) ∈ σpp (∆g )} ∩ ( n2 − N). In fact we can deal with these special points using the perturbation method of Borthwick-Perry [2]. Indeed, since the resolvent ans scattering operator are meromorphic with poles of finite multiplicity in {<(s) < n2 }, one can remark from [2] that if s0 ∈ n−k , it is possible to add a sufficiently 2 small non-negative compactly supported potential V on X, such that mn−s0 , ms0 and νs0 remains invariant and the eigenvalue s0 (n − s0 ) is pushed a little so that s0 (n − s0 ) ∈ / σpp (∆g + V ). The formula being now satisfied at s0 we have the result since P k when k is even depends only 2 on the k first derivatives (∂xj (x2 g)|x=0 )j=1...,k if g is the hyperbolic metric on X (see again [9]).  Now we recall a few words about the conformal powers of the Laplacian. For each conformal ¯ the conformal Laplacian Pk is a differential representative h in the conformal class [h0] on ∂ X, operator of order 2k with principal symbol σ0 (Pk ) = σ0 (∆kh ) and it satisfies a covariant rule when the conformal representative is changed. The first example n−2 ¯ and P2 R where R is the scalar curvature of h on ∂ X is the conformal Laplacian P1 = ∆h + 4(n−1) n is the so-called Paneitz operator. Actually if n = 1, 2 or if n+1 is odd and k > 2 these conformal operators Pk are not well-defined as conformally covariant operators on general asymptotically Einstein manifolds. However for conformally compact manifolds with constant curvature near infinity we learned from Robin Graham (see also [7]) that the Pk are well-defined as conformally covariant for all k and n > 2, even if n + 1 is odd. In the cases n = 1, 2, Pk must then be interpreted as the differential operator Pk := p2k obtained in the work of Graham-Zworski [9, ¯ [h0]) Prop. 3.5]. By convention, we will always call Pk the k-th conformal Laplacian on (∂ X, and dk =: dim ker Pk is a finite number which is conformally invariant on the conformal infinity (since Pk is Fredholm and satisfies a covariant rule). There is a special case where the operators Pk are computable, this is when the conformal infinity has a conformal representative with constant curvatures. Proposition 2.2. Let (M, [h0 ]) be a conformal manifold with H0(M ) connected component. If h0 is flat then Pk = ∆kh0 and dk = H0(M ) whereas if h0 has non-zero constant sectional curvatures then k   n  n Y −j +j−1 ∆h 0 + (2.4) Pk = 2 2 j=1

and if Sh0 := {s −

n 2;s

≥ n − 1, s(s − (n − 1)) ∈ σ(∆h0 )} then dk = ](Sh0 ∩ {j ∈ N0; j ≤ k − 1}).

Proof : the flat case is well known (see [9]). Since the local expression of Pk on a manifold with positive constant curvature is clearly the same than for the sphere S n , the result is obtained for example by calculating the residue of the scattering operator at n2 + k on the hyperbolic space, which is Pk on the sphere according to [9]. Using for example [11, Appendix], the scattering matrix on Hn+1 is q  n−1 2 n n n + ( ∆ Γ ) − + 1 + s S Γ( − s) 2 2 S(s) = 2n−2λ 2 n q  Γ(s − 2 ) Γ ∆ n + ( n−1 )2 + n + 1 − s S

2

2

´ ERIC ´ COLIN GUILLARMOU AND FRED NAUD

6

k −2k

2 and the residue is easily seen to be ck Pk with Pk in (2.4) and ck = (−1) k!(k−1)! . To deal with the case of negative constant curvature, one can use the expression of the scattering operator on the cylindrical manifold studied in [26, Appendix B] and we find the same result. The formula for dk is a straightforward consequence of the expression of Pk . 

We also learned from Robin Graham that a similar result is true if the boundary has an Einstein conformal representative and that the expression of Pk on the sphere was obtained previously by Thomas Branson [5] (see also Beckner [1] when k = n2 ). As an application, if we consider a convex co-compact quotient X = Γ\H3 , the boundary ¯ has a conformal representative h0 with constant curvature and one can use the M := ∂ X previous Proposition to explicit dk : if M = M1 t M2 with (M1 , h0) flat and (M2 , h0) having non-zero constant curvatures, then we have dk = ] ({s − 1; s ≥ 1, s(s − 1) ∈ σ(∆M2 )} ∩ {j ∈ N0 ; j ≤ k − 1}) + H0(M1 ) where H0(M1 ) is the number of connected components of M1 . In the generic case, dk will thus be equal to the number H0(M ) of connected components of M . Let us consider the counting function for resonances N(R) and for scattering poles N s (R) defined by X X N(R) := ms , Ns (R) := νs . |s|≤R, s∈R

|s|≤R, s∈S

We clearly have

Ns (R) ≥ N(R). Patterson-Perry [26] proved the upper bounds Ns (R) = O(Rn+1 ) which implies in view of (2.3) Lemma 2.3. Using the above notations, we have as R → +∞, N(R) = O(Rn+1 ),

R X

dk = O(Rn+1 ).

k=1

Note that those results are optimal in general in the sense that for some examples (take H n+1 with n + 1 even for the first bound and Hn+1 with n + 1 odd for second bound) this is an asymptotic. 2.3. The hyperbolic space model. We begin by a simple calculation for the wave kernel on the hyperbolic space Hn+1 which shows the relation between resonances and 0-trace. Recall that the set of resonances R0 for the Laplacian on Hn+1 is given by  {−k with multiplicity hn(k); k ∈ N0} if n + 1 is even R0 = ∅ if n + 1 is odd where hn(k) is the dimension of the space of spherical harmonics of degree k on S n+1 hn (k) := (2k + n)

(k + 1)(k + 2) . . . (k + n − 1) , n!

Let us write for simplicity R0 (s) := (∆Hn+1 − s(n − s))

−1

,

U0 (t) := cos t

r

hn (0) = 1

∆Hn+1

n2 − 4

!

and R0(s; w, w0), U0(t; w, w0) their respective kernel. Then we choose a boundary defining function x0 of Hn+1 and define the distribution on R∗ u0 (t) := 0-tr(U0 (t)) where the 0-trace is taken with respect to x0.

WAVE 0-TRACE AND LENGTH SPECTRUM

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Lemma 2.4. On Hn+1 , we have the following formula for t > 0  1 P∞
n hn (k)e−t( 2 +k) if n + 1 is even k=0 2 u0(t) = 0 if n + 1 is odd Proof : we use the formula of the wave kernel on Hn+1 (see Helgason [15]). In odd dimension the result is clear since the wave kernel vanishes on the diagonal (2.5)

U0 (t; w, w) = 0.

In even dimension it suffices to use the formula U0 (t; w, w) = (−π)−

(2.6)

n+1 2

n!!2−

3(n+1) 2

cosh 2t (sinh 2t )n+1

and check that ∞ X cosh 2t n n = 2 hn(k)e−t( 2 +k) t n+1 (sinh 2 ) k=0

(2.7) then u0(t) =



(−π)− 0

n+1 2

n!!2−

and using that 0-vol(Hn+1 ) =

n+3 2

P∞

k=0 hn (k)e

n+1 2

(−2π) n!!

−t( n 2 +k)




!

0-vol(Hn+1 )

if n + 1 is even if n + 1 is odd

when n + 1 is even (see [8]) this gives the result.



2.4. Proof of Theorem 1.1. We will now study the general case of a convex co-compact quotient of Hn+1 . Let X = Γ\Hn+1 be the quotient of the hyperbolic space by a convex cocompact group of isometries. We first define the distribution on R∗ !! r n2 uc(t) := 0-tr PX cos t ∆X − 4 with PX the projector on the continuous part of ∆X . Note that in Theorem 1.1, the cosh terms come trivially from the discrete spectrum. From [13, eq. 4.13]3, we have for ϕ ∈ C0∞ (R∗ ) Z Z (2.8) uc (t)ϕ(t)dt = (4π)−1 ϕ(z)θ(z)dz ˆ R

where

    n n + iz − RX − iz θ(z) := 2iz 0-tr RX 2 2 will be shown to be a tempered distribution on R. We recall that the Selberg zeta function Z(s) of X = Γ\Hn+1 is defined by the Euler product (which converges for <(s) > δ),   Y Y 1 − α1(γ)k1 . . . αn (γ)kn e−(s+|k|)l(γ) , Z(s) = γ∈P k1 ,...,kn ∈N

where |k| = k1 + . . . + kn . Let F be a fundamental domain of Γ in Hn+1 , then from Patterson-Perry formula [26, eq. 6.7] we have (2.9)

θ(z) = θ1 (z) + θ2 (z) Z 0 ( n2 + iz) Z 0 ( n2 − iz) + Z( n2 + iz) Z( n2 − iz)  i h n n dvol(w) + iz; w, w0 − R0 − iz; w, w0 2iz R0 2 2 w=w 0

θ1 (z) := θ2 (z) := F P→0

Z

{x>}∩F

3We believe there is a sign typo

´ ERIC ´ COLIN GUILLARMOU AND FRED NAUD

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Using the factorization of the Selberg zeta function in a Hadamard product by Patterson-Perry [26, Th. 1.9] (they also use Bunke-Olbrich results [6]), we observe that (   ∞ X 1 1 θ1 (z) = −i P (z) − χ(X) hn(k) − z − i( n2 + k) z + i( n2 + k) k=0 ) X  1 1 − + Q(z, s) + νs z − i( n2 − s) z + i( n2 − s) s∈S

with P (z), Q(z, s) some polynomials in z of degree less or equal to n and χ the Euler characteristic of X. This proves, using the arguments of Lemma 4.7 of [12], that θ1 is a tempered distribution on R, hence we can define its Fourier transform θˆ1 which is also a tempered distribution on R. Now we differentiate n + 1 times the last equation and obtain (   ∞ X 1 1 n+1 n − (−i∂z ) θ1 (z) = i (n + 1)! −χ(X) hn (k) (z − i( n2 + k))n+2 (z + i( n2 + k))n+2 k=0 ) X  1 1 νs + − (z − i( n2 − s))n+2 (z + i( n2 − s))n+2 s∈S

We combine this formula with the following −1 Ft→z (tn+1 eiζ|t|) = (2π)−1(n + 1)!in



1 1 − (z − ζ)n+2 (z + ζ)n+2



for =(ζ) ≥ 0 to conclude that (2.10)

t

n+1

θˆ1 (t) = −2πt

in view of the identity

n+1

(

χ(X)

∞ X

hn (k)e

−( n 2 +k)|t|

k=0

!

−

X

νs e

−( n 2 −s)|t|

s∈S

)


tn+1 θˆ1 (t) = Fz→t (−i∂z )n+1 θ1 (z) .

To study θ2 we use that (see [26]) n

θ2 (z) = π− 2

Γ( n2 ) Γ( n2 + iz)Γ( n2 − iz) 0-vol(X) Γ(n) Γ(iz)Γ(−iz)

which is a tempered distribution for z ∈ R since it is bounded by a polynomial (note that the 0-volume of X could depend of the defining function x when n + 1 is odd). We deduce that θ is a tempered distribution on R and by (2.8), its Fourier transform is a tempered distribution on R which, when restricted to R∗ , is 4πuc. To find the Fourier transform of θ2 , we remark by using again [12, eq. 4.13] that this is equivalent to calculate  Z if n + 1 is odd  0 3(n+1) t − U0 (t; w, w)dvol(w) = 4π 0-vol(X) F P→0 cosh 2  n!!2 n+1 (sinh t )2n+1 if n + 1 is even F ∩{x>} (−π)

2

2

where we used (2.5) and (2.6). But from Epstein formula for the 0-volume of X in [26] (−2π) 0-vol(X) = n!!

n+1 2

χ(X),

and (2.7) we deduce that  0 ˆ
P∞ (2.11) θ2 (t) = −( n 2 +k)|t| 2πχ(X) k=0 hn (k)e

if n + 1 is odd if n + 1 is even

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To achieve the proof of the formula relating 0-trace and resonances, it suffices to add (2.10) with (2.11), use (2.8) and we obtain X n 1X n νse−( 2 −s)|t| + u(t) = ms cosh(t( − s)) 2 2 s∈S

s∈R <(s)> n 2

if n + 1 is even and X cosh 2t n n 1X + νse−( 2 −s)|t| − 2−n−1χ(X) ms cosh(t( − s)) u(t) = |t| n+1 2 2 (sinh ) s∈S

s∈R <(s)> n 2

2

if n + 1 is odd. Now it suffices to use the formula (2.3) relating νs0 and ms0 , mn−s0 . To obtain the part with the length spectrum, we remark that for <(s) > δ, ! ∞ XX 1 e−sml(γ) Z(s) = exp − . m Gγ (m) γ m=1 As before s =

n 2

+ iz, so when the sum converges (this is the case if =(z) = 0 <

θ1 (z) =

X γ

thus θˆ1 (t) =

∞ XX γ

∞ X

n

n 2

− δ),

n

l(γ)Gγ (m)−1 (e−( 2 +iz)ml(γ) + e−( 2 −iz)ml(γ) )

m=1

n

2πl(γ)Gγ (m)−1 e− 2 ml(γ) (δ(t + ml(γ)) + δ(t − ml(γ)))

m=1

and we are done at least when δ < n2 since there are no L2 -eigenvalues in this case. The other cases are treated by the arguments of Perry [30, lem 2.3] using a contour deformation which makes the term X n − ms cosh(t( − s)) 2 s∈R <(s)> n 2

q appear in addition and this term cancels the term 0-tr((1 − PX ) cos(t ∆X −

n2 )). 4



3. The prime geodesics asymptotics In this section, we prove Theorem 1.2. The proof is based directly on the trace formula. The standard method using zeta functions and contour deformation could be applied, but a good estimate of the growth of |Z 0 (s)/Z(s)| in strips parallel to the imaginary axis is lacking and the error term obtained might be marginally large. However in the special case of Schottky δ manifolds, Guillop´e, Lin and Zworski [14] have shown that |Z(s)| = O(eC|=(s)| ). Using this estimate, it could be interesting to apply the method of regularization, contour deformation and then finite differences and compare the error term obtained with the remainder of Theorem 1.2. Let us define for x ≥ 1, the following counting functions. X 1 Π0 (x) = #{γ ∈ P : el(γ) ≤ x}, Π(x) = , Ψ(x) = k kl(γ) e

We have obviously N (T ) = Π0 (eT ) and Π(x) = of Perry [29], we have Π0 (x) = O

P+∞

≤x

1 1/k ). k=1 k Π0 (x



xδ log(x)



,

X

ekl(γ) ≤x

l(γ) Gγ (k)

Using the asymptotic formula

´ ERIC ´ COLIN GUILLARMOU AND FRED NAUD

10

which is enough to deduce Π(x) = Π0 (x) + O(xδ/2). The counting function Ψ(x) is related to Π(x) by remarking that Z x dΨ(u) = Π(x) + Φ(x) + O(1), 2 log(u) P where the remainder Φ(x) = kl(γ)≤log(x) k1 (Gγ (k)−1 − 1) can be estimated by  Z x X 1 −kl(γ) dΠ(u) |Φ(s)| ≤ CX . e =O k u 2 kl(γ)≤log(x)

δ

Since we have Π(x) = O(x ), a straightforward Stieltjes integration by parts shows that Φ(x) = O(xδ−1 ). In a nutshell, we have (3.1)

Π0 (x) =

Z

x 2

  δ dΨ(u) + O xmax( 2 ,δ−1) . log(u)

Our goal is now to get a precise asymptotic of Ψ(x). For this purpose we need to introduce the following family of test functions. Let l(X) denote the length of the shortest closed geodesic on X. In the following of the proof, x, y will be large parameters satisfying y = O(xα), where α < 1 will be chosen a posteriori to optimize the error term. Let ϕx,y be a C0∞ (R) positive even test function such that  h i l(X)   0 if u ∈ 0, 2 ϕx,y (u) = 1 if u ∈ [l(X), log(x)]   0 if u ∈ [log(x + y), +∞) Clearly such a function does exist and it can be chosen such that for all k ∈ N, there exists a constant Ck > 0 such that  y k (3.2) sup |ϕ(k) . x,y (u)| ≤ Ck x log(x)≤u≤log(x+y) n

Set hx,y (t) = e 2 |t|ϕx,y (t). Testing the trace formula on hx,y , we obtain the relation Z ∞ X X Z +∞ X l(γ) e−kthx,y (t)dt = est ϕx,y (t)dt + dk (3.3) ms ϕx,y (kl(γ)) G γ (k) 0 0 k∈N

s∈R

+ 2−n A(X) + 2B(X) Obviously, we have

and since t 7→

P

k∈N dk e

Z

+∞ 0

0≤kl(γ)≤log(x+y)




Z

+∞ 0

cosh(t/2) hx,y (t)dt. (sinh(t/2))n+1

cosh(t/2) hx,y (t)dt = O(log(x)), (sinh(t/2))n+1

is uniformly convergent and bounded on [ l(X) 2 , ∞), we can write X Z +∞ n
dk e−kthx,y (t)dt = O log(x)x 2 .

−kt

k∈N n 2 , we

0

know that the point spectrum of the Laplacian is a non-empty Since we assume that δ > n2 finite subset of (0, 4 ) whose bottom is the simple eigenvalue δ(n − δ). Let n < α0 ≤ α1 ≤ . . . < α p = δ 2 be the corresponding resonances with respect to the modified spectral parameter s(n − s). We have the partition n R = {α0, . . . , δ} ∪ {s ∈ R : <(s) ≤ }. 2

WAVE 0-TRACE AND LENGTH SPECTRUM

11

In the following, we will denote by R+ = {s ∈ R : <(s) ≤ n2 }. For all s ∈ R, we set Z +∞ ψx,y (s) = estϕx,y (t)dt. 0

On the spectral side of formula (3.3), we have X

X

ms ψx,y (s) =

s∈R

ms ψx,y (s) +

ψx,y (αk ).

k=0

s∈R+

For all 0 ≤ k ≤ p, we get directly

p X

 x αk y  y  x αk +O x αk = + O δ−1 . αk x αk x P It remains to estimate
carefully the spectral sum s∈R+ ms ψx,y (s). If s = 0, then we certainly n have ψx,y (0) = O x 2 . If s 6= 0 then integrating by parts k times yields (we use the fact that n <(s) ≤ 2 and the estimate (3.2)), k−1! Z n  (−1)k +∞ st (k) x2 x . (3.4) ψx,y (s) = e ϕx,y (t)dt = Ok sk |s|k y 0 ψx,y (αk ) =

Writing

X

(3.5)

ms ψx,y (s) =

X

ms ψx,y (s) +

|s|≤ x y

s∈R+

X

ms ψx,y (s),

|s|≥ x y

and using the estimate (3.4) for k = 1 and k = n + 2, we get  n+1 Z +∞ Z yx X n x dN(u) dN(u) n n 2 2 ms ψx,y (s) ≤ x +x + O(x 2 ). n+2 x u y u 1 + y s∈R

A Stieltjes integration by parts combined with Lemma 2.3 shows that  n  Z xy x dN(u) =O , u y 1 and a similar argument yields

Z

+∞ x y

y  dN(u) = O . un+2 x

Gathering all our previous estimates, we get for all x, y large (and satisfying the previously defined a priori properties),  p−1 n  X y x δ X x αk x3 2 l(γ) n
ϕx,y (kl(γ)) = + +O + + O (log x)x 2 . 1−δ n Gγ (k) δ αk x y k=0

0≤kl(γ)≤log(x+y)

Substracting the above formula from that of x + y instead of x and using the positivity of the left side, we observe that  n  p X X (x + y)αi − xαi l(γ) y x3 2 n
ϕx,y (kl(γ)) ≤ +O + n + O (log x)x 2 , 1−δ G (k) α x y γ i kl(γ) x≤e

i=0

≤x+y

 n   y  y x3 2 n
= O 1−δ + O + n + O (log x)x 2 . 1−δ x x y

In other words, the above standard argument shows that we can drop the terms over e kl(γ) without changing the asymptotics. We have thus obtained  p−1 n  y x3 2 n
x δ X x αk + +O + n + O (log x)x 2 . Ψ(x) = 1−δ δ αk x y k=0

12

´ ERIC ´ COLIN GUILLARMOU AND FRED NAUD

We recall that all the preceding estimates are valid for all y = O(xα) with α < 1. A straightforward computation shows that the choice 4 of 1

3

y = n n+1 x n+1

1−δ n 2 + n+1



n 1 minimizes the global error term which becomes O xβn (δ) where βn (δ) = n+1 2 + δ . Going back to formula (3.1), the final asymptotics follow by a direct Stieltjes integration by parts.  References [1] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Annals of Math. 138 (1993), 213-242. [2] D. Borthwick, P. Perry, Scattering poles for asymptotically hyperbolic manifolds, Trans. A.M.S. 354 (2002) 1215-1231. [3] D. Borthwick, C. Judge, P. Perry, Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces, preprint. ´ [4] R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Etudes Sci. Publ. Math. 50 1979, 11-25. [5] T. Branson, The functional determinant, Lecture notes series, No.4, Seoul National University, 1993. [6] U. Bunke, M. Olbrich Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group, Ann. Math 149 (1999), 627-689. [7] C.R. Graham, R. Jenne, L.J. Manson, G.A.J. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. (2) 46 (1992), 557-565. [8] C.R. Graham Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo, Ser.II, Suppl. 63 (2000), 31-42. [9] C.R. Graham, M. Zworski, Scattering matrix in conformal geometry, Invent. Math. 152 (2003), 89-118. [10] C. Guillarmou, Resonances and scattering poles on asymptotically hyperbolic manifolds, to appear in Math. Research Letter. [11] L. Guillop´ e, M. Zworski, Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Analysis 129 no.2 (1995), 364-389. [12] L. Guillop´ e, M. Zworski, Scattering asymptotics for Riemann surfaces, Ann. Math. 145 (1997), 597-660. [13] L. Guillop´ e, M. Zworski, The wave trace for Riemann surfaces, G.A.F.A. 9 (1999), 1156-1168. [14] L. Guillop´ e, K. Lin, M. Zworski, The Selberg zeta function for convex co-compact Schottky groups, Comm. Math. Phys. 245 Vol. 1 (2004), 149-176. [15] S. Helgason, Groups and geometric analysis, Academic Press (1984). [16] D. Hejhal, The Selberg trace formula for P SL(2, R), Lecture notes in Mathematics, Vol. 548, Springer-Verlag, Berlin, 1976. [17] H. Huber, Zur analytischen theorie hyperbolisher raumformen und bewegungsgruppen 2, Math. Ann., 142 (1961), 385-398. [18] M. Joshi, A. S´ a Barreto, Inverse scattering on asymptotically hyperbolic manifolds, Acta Math. 184 (2000), 41-86. [19] M. Joshi, A. S´ a Barreto, The wave group on asymptotically hyperbolic manifolds, Journ. Funct. Anal. 184 (2001), no.2, 291-312. [20] B. Maskit, A characterization of Schottky groups, J. Analyse Math. 19 (1967), 227-230. [21] R. Mazzeo, R. Melrose Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), 260-310. [22] R. Mazzeo, The Hodge cohomology of a conformally compact metric, J. Diff. Geom. 28 (1988), 309-339. [23] R. Mazzeo, Elliptic theory of differential edge operators. I, Commun. Partial Diff. Equations 16 (1991), 1615-1664. [24] F. Naud, Expanding maps on Cantor sets and analytic continuation of zeta functions, to appear in Ann. ´ Sci. Ecole Norm. Sup., 2005. [25] F. Naud, Precise asymptotics of the length spectrum for finite geometry Riemann surfaces, to appear in Int. Math. Res. Notices, 2005. [26] S. Patterson, P. Perry, The divisor of Selberg’s zeta function for Kleinian groups. Appendix A by Charles Epstein., Duke Math. J. 106 (2001), 321-391. [27] P. Perry The Laplace operator on a hyperbolic manifold II, Eisenstein series and the scattering matrix, J. Reine Angew. Math. 398 (1989), 67-91. [28] P. Perry The Selberg zeta function and a local trace formula for Kleinian groups, J. Reine Angew. Math. 410 (1990), 116-152. [29] P. Perry, Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume, GAFA, Geom. funct. anal., 11 (2001), 132-141. 4Remark that because n < δ, we have indeed the exponent 3 n + 1−δ < 1. 2 n+1 2 n+1

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13

[30] P. Perry, A Poisson formula and lower bounds for resonances on hyperbolic manifolds, Int. Math. Res. Notices 34 (2003), 1837-1851. [31] B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. Amer. Math. Soc., 233 (1977), 241-247. [32] P. Sarnak, Prime geodesic theorems, Phd Thesis. Stanford University 1980. ´ [33] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Etudes Sci. Publ. Math., 50 (1979), 171-202. [34] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups., Acta Math., 153, 3-4 (1984), 259-277. [35] L. Stoyanov, Ruelle zeta functions and spectra of transfer operators for some axiom A flows, Preprint (2005). Department of mathematics, Purdue University, 150 N. University Street, West-Lafayette, IN 47907, USA E-mail address: [email protected] Department of Mathematics, University of California Berkeley, Berkeley CA 94720-3840 USA E-mail address: [email protected]