Ergodic properties of some negatively curved manifolds with infinite measure

TL;DR

This paper investigates the ergodic and mixing properties of geodesic flows on certain negatively curved manifolds with infinite measure, extending known results to divergent Schottky groups with infinite Bowen-Margulis measure.

Contribution

It provides a detailed analysis of ergodic properties and symbolic coding for divergent Schottky groups with infinite Bowen-Margulis measure, expanding understanding beyond finite measure cases.

Findings

Proves ergodicity and mixing for geodesic flow with infinite measure

Characterizes asymptotic behavior of closed geodesics and orbital counting

Uses symbolic coding to analyze ergodic properties

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 $T^1M$, the asymptotic equivalent as $R\longrightarrow+\infty$ of the number of closed geodesics on $M$ of length less than $R$ and of the orbital counting function $\sharp\{\gamma\in\Gamma\ |\ d(o,\gamma.o)\le R\}$. These properties are well known when the Bowen-Margulis measure on $T^1M$ 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.