1

Introduction

The purpose of this note is to prove a prime orbit theorem for an arbitrary Riemann surface of finite geometry and infinite volume. Let us recall the standard analytic and geometric notations which will be used in this paper. We denote by H2 the Poincar´e half plane H2 = {z = u + iv, v > 0} endowed with its 2 2 standard metric of constant negative curvature −1 given by ds2 = du v+dv . A geometrically 2 finite Riemann surface is a non-compact surface whose ends are of two types: cusps and funnels. A cusp is isometric to a strip {0 ≤ u ≤ h, v ≥ 1} ⊂ H2 whose edges {u = 0} and {u = h} are identified, and its boundary is a closed horocycle of length h. A funnel is isometric to an annulus {1 ≤ u2 + v 2 ≤ e2l , u ≥ 0} ⊂ H2 whose edges {u2 + v 2 = 1} and {u2 + v 2 = e2l } are identified, and its boundary is a closed geodesic of length l. A finite geometry surface M is then obtained by attaching f funnels to the convex core of the surface N called the nielsen region whose boudary is composed of f closed geodesics. The Nielsen region is itself builded by gluing c cusps through their horocyclic boundary to a compact surface N0 with totally geodesic boundary having f + c connected components. Such a surface can be obtained as a quotient M = Γ\H2 of the hyperbolic space by a Fuchsian group Γ which is torsion free, non cyclic and with a finite presentation. If there is at least one funnel, the Fuchsian group Γ is of the second kind and the Hausdorff dimension δ of the limit set Λ ⊂ ∂H2 has several interpretations. The dimension δ is also the critical exponent of the group and the measure theoretic entropy of the geodesic flow on its recurrent set, endowed with the Patterson-Sullivan measure (see [15, 23, 22]). The usual positive Laplacian acting on functions is denoted by ∆M . The set P of primitive periodic orbits γ of the geodesic flow least period l(γ) is in one to one correspondence with the conjugacy classes of hyperbolic elements in Γ. The counting function N (T ) for the prime geodesics is defined as usual for T ≥ 0 by N (T ) = #{γ ∈ P : l(γ) ≤ T }. A well known result is the following.

1

Theorem 1.1 Let M = Γ\H2 be a geometrically finite Riemann surface with finite volume, then as T → +∞, we have p

3 X li eαk T + O e 4 T , N (T ) = li eT + k=1

R x dt 1 where li(x) = 2 log t and the exponents 2 ≤ αp ≤ αp−1 ≤ . . . < α0 = 1 are in bijection with the eigenvalues of the Laplacian included in [0, 1/4] by the formula αk (1 − αk ) = λk . The asymptotic analysis of the periodic geodesics of Riemann surfaces is build on the trace formula proved by Selberg [21] and the first precise expansions for compact surfaces were obtained later by Huber, Hejhal and Randol [9, 8, 18]. In the non-compact and finite volume case, the first asymptotics were proved by Sarnak [20], see also the book by Iwaniec [10]. In the infinite volume case, much less is already known. In the convex co-compact case (f ≥ 1, c = 0) the first asymptotics were proved by Guillop´e [4] and Lalley [11] where they obtained the leading term (1)

lim N (T )

T →+∞

δT = 1. eδT

Here the dimension of the limit set δ satisfies 0 < δ < 1 (see Beardon [1, 2]). If there are cusps, then by a result of Beardon [1], we have δ > 21 and the spectral methods show that the asymptotic (1) is still valid (see Guillop´e [5]). For a more recent work in the general setup of CAT (-1) spaces where the leading term (1) still holds, we refer to the work of Roblin [19]. In the higher dimensional case, a similar asymptotic has been established by Perry [17]. In this paper, we build on the work of Guillop´e-Zworski, i.e. the wave trace formula [3], and the bounds on the number of resonances [7], to obtain the following. Theorem 1.2 Let M = Γ\H2 be a geometrically finite Riemann surface such that δ > f ≥ 1. Then as T → +∞, we have

N (T ) = li e

δT

+

p X k=1

1 2

and

li eαk T + O e(δ/2+1/4)T ,

where the exponents 21 < αp ≤ αp−1 ≤ . . . < α0 = δ are related to the point spectrum of ∆M included in (0, 1/4) by the formula αk (1 − αk ) = λk . 3

If we let δ → 1 we remark that the error term tends to O(e 4 T ) which is what we already know in the finite volume case. It is natural to ask if, as it is for the prime number theorem assuming the classical Riemann hypothesis, one could improve this error term to O(e(1/2+ε)T ). Even for finite volume Fuchsian groups, this problem is still open. A better bound is known for the modular surface PSL2 (Z)\H2 where Luo and Sarnak [12] have proved that O(e(7/10+ε)T ) holds. The case δ ≤ 21 is more difficult and requires some additional informations on the resonances. Let us recall briefly a definition of the resonances in the setup of finite geometry surfaces. In the 2

infinite volume case, the Laplace-Beltrami operator ∆M has [1/4, +∞) as continuous spectrum and the point spectrum of ∆M is at most finite and included in (0, 1/4). Let R(λ) be the the resolvent (acting on L2 (M )) −1 1 2 , R(λ) = ∆M − − λ 4 which is a meromorphic bounded operator valued function on the lower half plane {Im(λ) < 0}. When acting on C0∞ (M ) into C ∞ (M ), the resolvent R(λ) admits a meromorphic extension [13] to the whole complex plane. The resonances are the poles of the meromorphic extension to {Im(λ) ≥ 0}, denoted by R+ M . For other definitions of the resonances involving Eisenstein series and scattering operators, we refer to [7]. If δ ≤ 21 , then using the correspondance between zeros of the Selberg zeta function and the resonances proved by Patterson and Perry [16], s0 = i( 21 − δ) is a simple resonance and we have R+ M ⊂ {Im(s) ≥

1 2

− δ}.

In the paper [14], the author has in fact proved the following. Theorem 1.3 If δ ≤ 21 , then there exists ε > 0 such that R+ M \ {s0 } ⊂ {Im(s) ≥ Equivalently, we have β :=

sup s∈R+ M \{s0 }

1 2

1 2

− δ + ε}.

− Im(s) < δ.

This result, based on the transfer operator techniques of Dolgopyat (see the references given in [14]), is enough to show the next asymptotic expansion. Theorem 1.4 Let M = Γ\H2 be a geometrically finite Riemann surface such that δ ≤ f ≥ 1. Then as T → +∞, we have + N (T ) = li eδT + O e(δ+β )T /2 ,

1 2

and

where β + = max(0, β).

In the above statement, M is actually convex co-compact since δ ≤ 21 . We point out that if δ β ≤ 0 then the error term becomes O(e 2 T ), which is the best we can expect. In that direction, the numerical experiments in [6] could be of interest to find manifolds having this property.

2

Distribution of closed geodesics and resonances trace formula

We use the notations introduced in the preceding section. We assume that M = Γ\H2 and f ≥ 1. According to the statement of Theorem 1.2, we will denote the point spectrum (when it is non empty) by 0 < δ(1 − δ) = λ0 < λ1 ≤ . . . ≤ λp < 1/4. 3

The possible poles of the resolvent in the lower half-plane {Im(s) < 0} are denoted by r 1 sk = −i − λk , k = 0, . . ., p 4 If αk (1 − αk ) = λk , k = 0, . . ., p, then sk = −i(αk − 21 ). We set in the following RM = R + M ∪ {s0 , s1 , . . . , sp }. As usual, the points in RM are repeated according to their multiplicity. q The regularized trace (see [7, 3]) of the wave operator 0-tr cos t ∆M − 41 is a distribution on the real line and can be explicitly computed [3]. More precisely, for j = 1, . . ., c, let hj be the lengths of the horocyclic of the c cuspidal ends of M , let C denote the Euler P boundaries N 1 constant C = limN →+∞ n=1 n − log N .

We recall that the principal value of an odd function f ∈ L1loc (R \ {0}) with a singularity at x = 0 of type O(1/x) is a distribution acting on C0∞ (R) by the formula Z hPV[f ], ϕi = lim f (x)ϕ(x)dx. ε→0 R\[−ε,ε]

Then in the distributional sense, we have on the real line r 1 c Vol(N ) 1 0 (t) + [sgn(t) log | sinh(t/2)(t)|]0 PV 0-tr cos t ∆M − = 4 4π sinh(t/2) 4 ! c XX X l(γ) δ0 (|t| − kl(γ)). + c(C − log 2) − log hi δ0 (t) + 21 2 sinh(kl(γ)/2) γ∈P k≥1

i=1

The regularized trace is itself related to the point spectrum and the resonances by the so-called Poisson formula [3]: for t 6= 0, we have (again in the distributional sense), r X 1 0-tr cos t ∆M − = 12 eis|t| . 4 s∈RM

The combination of the above two formulas gives a resonances trace formula relating resonances and point spectrum to the length spectrum. Before we give a proof of Theorem 1.2 and 1.4, we need to introduce the following counting functions, defined for X ≥ 1. ψ(X) =

X

kl(γ)≤log X

l(γ) ; π0(X) = tanh(kl(γ)/2)

X

kl(γ)≤log X

π(X) = #{γ ∈ P : el(γ) ≤ X}.

1 ; k

Clearly we have N (T ) = π(eT ). In addition, using the leading asymptotic [11, 5, 19] π(X) ∼

Xδ , δ log(X) 4

we can write (we actually only need the upper bound), π0 (X) =

+∞ X 1 k=1

k

π(X 1/k) = π(X) + O(X δ/2).

Using Stieltjes integrals, we write Z X Z X Z X dψ(t) dψ(t) t − 1 dψ(t) π0(X) = + O(1) = −2 + O(1). t + 1 log t log t 2 2 2 (t + 1) log t Moreover, we have also by convergence of the series Z X X dψ(t) = O 2 (t + 1) log t

kl(γ)≤log X

and thus

π(X) =

Z

X

2

Using the summation by parts Z X 2

e−kl(γ) = O(1),

dψ(t) + O X δ/2 . log t

dψ(t) ψ(X) = + log t log X

Z

X

2

ψ(t)dt + O(1), t log2 t

Theorem 1.2 and 1.4 are a direct consequence of the next asymptotics. Proposition 2.1 Using the above notations we have the following. • If δ >

1 2

then ψ(X) =

X Xδ + δ

1 2 <αi <δ

X αi + O X δ/2+1/4 , αi

where αi (1 − αi ) are the eigenvalues of ∆M in the interval (0, 41 ). • If δ ≤

1 2

then ψ(X) =

Xδ + O X (β+δ)/2 . δ

Proof. Let l0 denote the length of the shortest closed geodesic on M . Let h be an even C0∞ compactly supported function satisfying the following conditions. For all x ≥ 0, we have h(x) ≥ 0 and x ∈ [0, l0/2] 0 if 1 if x ∈ [l0, log X] h(x) = 0 if x ≥ log(X + Y ), 5

where X, Y > 0 satisfy Y = O(X α), with α < 1. Such a function clearly exists and it is not difficult to construct it so that additionally the derivatives h(k) enjoy the upper bound |h(k) (x)| ≤ Ck

sup x∈[log X,log(X+Y )]

X Y

k

.

We actually need these estimates only for the first three derivatives. Let us consider the compactly supported function g(x) = 2 cosh(x/2)h(x). Using this test function g(x), the trace formula yields the identity Z Z X Z +∞ g(t) c +∞ Vol(N ) +∞ cosh(t/2) ist dt (2) g(t)dt + e g(t)dt = − 2 4π 2 0 tanh(t/2) sinh (t/2) 0 0 s∈RM

X

+

kl(γ)≤log(X+Y )

l(γ) h(kl(γ)), tanh(kl(γ)/2)

where the right sum is running over all prime geodesics γ ∈ P and all positive integers k such that kl(γ) ≤ log(X + Y ). First Case: δ > 12 . On the left side, for all sj = −i(αj − 21 ), 0 ≤ j ≤ p, we can write Z

+∞

isj t

e

g(t)dt =

Z

log(X)

αj t

e

+e

(αj −1)t

l0

0

+2

Z

log(X+Y )

=

αj

+O

dt +

Z

l0

eisj t g(t)dt

0

1

e(αj − 2 )t cosh(t/2)h(t)dt

log X

X αj

Y (X + Y )αj X

X αj = +O αj

Y X 1−δ

.

R +∞ ist P For all s ∈ R+ e g(t)dt. Let us estimate the sum s∈R+ w(s). If s = 0 M , we set w(s) = 0 M √ then clearly |w(s)| = O( X). If s 6= 0, then integrating by parts yields w(s) = −

Z

log X l0

eist sinh(t/2)dt − is

Z

log(X+Y ) log X

eist 0 g (t)dt − is

Z

l0 0

eist 0 g (t)dt. is

Hence a rough estimate shows (we recall that Im(s) ≥ 0) √ ! √ ! 1 X X Y X |w(s)| = O +O log 1 + +O |s| |s| X Y |s| =O

√ ! X . |s| 6

Integrating by parts twice more, we get in the same way Z

+∞

isj t

e

g(t)dt =

0

Z

log X

0

!

eist (3) g (t)dt + is3

Z

log(X+Y )

log(X)

√ 2! X X +O =O =O |s|3 Y √ We have therefore shown that |w(s)| = O |s|X min 1, |s|12 √

X |s|3

R+ M = {ξn , n ≥ 0},

eist (3) g (t)dt is3

√ 2 ! X X . |s|3 Y X 2 . Let us now set Y

where 0 ≤ |ξ0| ≤ |ξ1| ≤ |ξ2| ≤ . . . ≤ |ξn | ≤ . . .. We recall that it is proved in [7] that there exists C > 0 such that for all R > C, 2 C −1 R2 ≤ #{s ∈ R+ M : |s| ≤ R} ≤ CR .

e > 0 such that for all ξn ∈ R+ with n > n0 = mult(0), This implies easily that there exists C M √ √ e n. e−1 n ≤ |ξn | ≤ C C

We can write X

s∈R+ M

|w(s)| ≤ mult(0)|w(0)| +

≤ mult(0)|w(0)| + C1

X

0<|ξn |≤X/Y

X

0<|ξn |≤X/Y

|w(ξn)| +

√ 2 X X + C2 |ξn | Y

X

|w(ξn )|

X

√ X , |ξn |3

|ξn |>X/Y

|ξn |>X/Y

where C1 , C2 > 0 are absolute constants. Hence we get X

s∈R+ M

√

|w(s)| ≤ mult(0)|w(0)| + C1 X

e2 (X/Y )2 C

X

n=n0 +1

2 e √ C X √ + C2 X n Y

X

e−2 (X/Y )2 n≥C

e3 C , n3/2

and a standard estimate of the above series with suitable integrals yields ! X X 3/2 |w(s)| = O . Y + s∈RM

In addition, we have obviously Z Z +∞ √ c +∞ cosh(t/2) g(t) g(t)dt = O(log X) and X log X . dt = O 2 0 tanh(t/2) sinh2 (t/2) 0 7

Gathering all the previous estimates into formula (2), we get (3)

p

X

kl(γ)≤log(X+Y )

l(γ) X δ X X αi h(kl(γ)) = + +O tanh(kl(γ)/2) δ αi i=1

Y X 3/2 + X 1−δ Y

!

.

Using the positivity of the left sums in (3), and substracting (3) from that of X + Y in place of X, we get X

X≤ekl(γ) ≤X+Y

l(γ) hX (kl(γ)) ≤ tanh(kl(γ)/2)

p X (X + Y )αi − X αi +O ≤ αi

X

X≤ekl(γ) ≤X+2Y

Y X 3/2 + X 1−δ Y

i=0

!

l(γ) hX+Y (kl(γ)) tanh(kl(γ)/2)

=O

Y X 3/2 + X 1−δ Y

!

.

As a conclusion, we can drop the terms in the left sum over ekl(γ) ≥ X without changing the error term. We have obtained ! p X δ X X αi Y X 3/2 . ψ(X) = + +O + δ αi X 1−δ Y i=1

Choosing Y = X 5/4−δ/2 minimizes the error term which becomes O(X 1/4+δ/2), and the proof is done. Second case: δ ≤ 12 . + In that case, RM = R+ M and RM \ {s0 } ⊂ {Im(s) ≥ have Z +∞

w(s0 ) =

=

Z

log X

δt

(e + e

(δ−1)t

)dt +

l0

Z

− β}, with β > δ and s0 = i( 12 − δ). We

1

e(δ− 2 )tg(t)dt

0

l0

1 (δ− 2 )t

e

0

=

1 2

Xδ +O δ

g(t)dt +

Y X 1−δ

Z

log(X+Y )

1

e(δ− 2 )t g(t)dt

log X

.

1 For all s ∈ R+ M \ {s0 }, we have Im(s) ≥ 2 − β and following the same ideas as in the previous case, we get by integration by parts Z log(X) Z log(X+Y ) 1 1 1 1 (β− 2 )t 0 |g (t)|dt + |w(s)| ≤ e e(β− 2 )t|g 0(t)|dt |s| 0 |s| log X β X =O . |s|

Integrating by parts twice more gives similarly Z log(X+Y ) 1 1 e(β− 2 )t |g (3)(t)|dt |w(s)| ≤ 3 |s| 0 8

=O

Xβ |s|3

X Y

2 !

.

The same trick as in the previous case allows us to write X

s∈R+ M \{s0 }

|w(s)| ≤ C1

X

|s|≤X/Y

=O

Xβ + C2 |s|

X β+1 Y

X Y

2

X

|s|>X/Y

Xβ |s|3

.

Since δ ≤ 21 , then by Beardon [1] we must have c = 0 (M is in fact convex co-compact) and the trace formula (2) yields X Y Xδ X β+1 l(γ) h(kl(γ)) = +O + + O(log(X)). tanh(kl(γ)/2) δ X 1−δ Y kl(γ)≤log(X+Y )

Using the same arguments as above gives Xδ +O ψ(X) = δ

Y X β+1 + X 1−δ Y

,

and choosing Y = X 1+(β−δ)/2 ends the proof.

Acknowledgments The author thanks the referee for his valuable comments on this paper.

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[8] Dennis A. Hejhal. The Selberg trace formula for PSL(2, R). Vol. I. Springer-Verlag, Berlin, 1976. Lecture Notes in Mathematics, Vol. 548. [9] Heinz Huber. Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II. Math. Ann., 142:385–398, 1960/1961. [10] Henryk Iwaniec. Spectral methods of automorphic forms, volume 53 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2002. [11] Steven P. Lalley. Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits. Acta Math., 163(1-2):1–55, 1989. [12] W. Luo and P. Sarnak. Quantum ergodicity of eigenfunctions on SL2 (Z)\H2. IHES Publ. Math., 81:207–237, 1995. [13] Rafe R. Mazzeo and Richard B. Melrose. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal., 75(2):260–310, 1987. [14] Fr´ed´eric Naud. Expanding maps on Cantor sets and analytic continuation of zeta functions. To ´ appear in Ann. Sci. Ecole Norm. Sup., 2004. [15] S. J. Patterson. The limit set of a Fuchsian group. Acta Math., 136(3-4):241–273, 1976. [16] S. J. Patterson and Peter A. Perry. The divisor of Selberg’s zeta function for Kleinian groups. Duke Math. J., 106(2):321–390, 2001. Appendix A by Charles Epstein. [17] P. A. Perry. Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume. Geom. Funct. Anal., 11(1):132–141, 2001. [18] Burton Randol. On the asymptotic distribution of closed geodesics on compact Riemann surfaces. Trans. Amer. Math. Soc., 233:241–247, 1977. [19] Thomas Roblin. Ergodicit´e et ´equidistribution en courbure n´egative. M´emoires de la SMF, 95:1–96, 2003. [20] Peter Sarnak. Prime geodesic theorems. PhD thesis. Stanford University, 1980. [21] A. Selberg. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.), 20:47–87, 1956. [22] Dennis Sullivan. The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes ´ Etudes Sci. Publ. Math., (50):171–202, 1979. [23] Dennis Sullivan. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math., 153(3-4):259–277, 1984.

Fr´ ed´ eric Naud Department of Mathematics University of California Berkeley Berkeley CA 94720-3840 email: [email protected]

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