Ergodicity of the geodesic flow for groups with a contracting element

TL;DR

This paper studies the dynamical behavior of geodesic flows in metric spaces acted upon by groups with contracting elements, establishing conditions under which the flow exhibits ergodic or dissipative behavior.

Contribution

It extends hyperbolic geometry results to spaces with contracting isometries, proving a dichotomy for the geodesic flow's ergodic properties.

Findings

Geodesic flow is either conservative and ergodic or dissipative.

Existence of a contracting isometry implies negative curvature-like properties.

Extended the Hopf-Tsuji-Sullivan dichotomy to this setting.

Abstract

In this article we investigate the dynamical properties of the geodesic flow for a proper metric space endowed with a proper action by isometries of a group with a contracting element. We show that the existence of a contracting isometry is a sufficient evidence of negative curvature to carry in this context various results borrowed from hyperbolic geometry. In particular, we extend the so-called Hopf-Tsuji-Sullivan dichotomy proving that the geodesic flow is either conservative and ergodic or dissipative.