A new condition for the genericity of ergodic measures on non-positively curved Riemannian manifolds

TL;DR

This paper establishes a new condition showing that ergodic measures are not generically dense among invariant measures for geodesic flows on certain non-positively curved manifolds, answering an open question in the field.

Contribution

It introduces a geometric condition involving product manifolds with $S^1$ factors that prevents ergodic measures from being generic, advancing understanding of measure dynamics on non-positively curved manifolds.

Findings

Ergodic measures are not dense in the space of invariant measures under certain conditions.

The presence of an open isometric embedding of a product with an $S^1$ factor obstructs ergodic genericity.

Provides a solution to an open problem related to a specific 3-manifold example by Heintze.

Abstract

This article investigates the genericity of ergodic probability measures for the geodesic flow on non-positively curved Riemannian manifolds. We demonstrate that the existence of an open isometric embedding of a product manifold with a factor isometric to $S^1$ implies that the closure of the set of ergodic measures does not encompass all invariant measures, thus the genericity of ergodic probability measures fails. Our findings notably provide an answer to an open question concerning a specific example of 3-manifold attributed to Heintze.