In this paper, we study a class of Ruelle dynamical zeta functions related to uniformly expanding maps on Cantor sets. We show that under a non-local integrability condition, the zeta function enjoys a non-vanishing analytic continuation in a strip on the left of the line of absolute convergence. Applying these results to Fuchsian Schottky groups and Julia sets yields precise asymptotics of the number of closed geodesics for convex co-compact surfaces and the distribution of periodic points for a family of Cantor-like Julia sets. 2005 Elsevier SAS RÉSUMÉ. -Dans cet article, on s'intéresse à une classe de fonctions zêta de Ruelle associées aux applications markoviennes uniformément dilatantes générant des ensembles de Cantor. On montre, sous une hypothèse de non intégrabilité locale, que ces fonctions zêta admettent un prolongement analytique sans zéros dans une bande à gauche de l'axe de convergence absolue. Appliqué aux ensembles limites de groupes de Schottky fuchsiens, ce résultat implique une asymptotique précise de la fonction de comptage des géodésiques périodiques sur les surfaces convexes co-compactes. On donne également un exemple d'application à des résultats de comptage pour une famille d'ensembles de Julia quadratiques de type Cantor. 2005 Elsevier SAS
Let X = Γ\H 2 be a convex co-compact hyperbolic surface. We show that the density of resonances of the Laplacian ∆ X in strips {σ ≤ Re(s) ≤ δ} with |Im(s)| ≤ T is less than O(T 1+δ−ε(σ) ) with ε(σ) > 0 as long as σ > δ 2 . This improves the previous fractal Weyl upper bound of Zworski [28] and is in agreement with the conjecture of [13] on the essential spectral gap.
For convex co-compact hyperbolic quotients Γ\H n+1 , we obtain a formula relating the 0-trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. Then, by writing this 0-trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of Γ is greater than n/2.
Abstract. Let Γ be a convex co-compact Fuchsian group. We formulate a conjecture on the critical line i.e. what is the largest half-plane with finitely many resonances for the Laplace operator on the infinite area hyperbolic surface X = Γ\H2 . An upper bound depending on the dimension δ of the limit set is proved which is in favor of the conjecture for small values of δ and in the case when δ > 1 2 and Γ is a subgroup of an arithmetic group. New omega lower bounds for the error term in the hyperbolic lattice point counting problem are derived.
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