Ergodic properties of some negatively curved manifolds with infinite measure

Abstract

Let $M=X/\Gamma$ be a geometrically finite negatively curved manifold with fundamental group $\Gamma$ acting on $X$ by isometries. The purpose of this paper is to study the mixing property of the geodesic flow on $T^{1}M$, the asymptotic equivalent as $R⟶+\infty$ of the number of closed geodesics on $M$ of length less than $R$ and of the orbital counting function $♯\{\gamma \in \Gamma |d(o,\gamma .o)\le R\}$. These properties are well known when the Bowen-Margulis measure on $T^{1}M$ is finite. We consider here divergent Schottky groups whose Bowen-Margulis measure is infinite and ergodic, and we precise these ergodic properties using a suitable symbolic coding.