Expanding maps on Cantor sets and analytic continuation of zeta functions

Abstract

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.