Generic measures for hyperbolic flows on non compact spaces

TL;DR

This paper investigates invariant measures for hyperbolic flows on non-compact spaces, establishing density results and providing tools applicable to non-positively curved manifolds, advancing understanding of ergodic properties in these settings.

Contribution

It introduces general methods for analyzing hyperbolic systems on non-compact spaces and proves the density of ergodic measures supported on the non-wandering set.

Findings

Dense G_delta subset of ergodic measures exists

Invariant measures fully supported on the non-wandering set

Tools developed for hyperbolic systems on non-compact spaces

Abstract

We consider the geodesic flow on a complete connected negatively curved manifold. We show that the set of invariant borel probability measures contains a dense $G_\delta$-subset consisting of ergodic measures fully supported on the non-wandering set. We also trat the case of non-positively curved manifolds and provide general tools to deal with hyperbolic systems defined on non-compact spaces.