A new condition for the genericity of ergodic measures on Riemannian manifolds

Abstract

This article investigates the genericity of ergodic probability measures for the geodesic flow on Riemannian manifolds. We demonstrate that if the metric splits as a product metric within a tubular neighborhood of a geodesically complete submanifold containing a closed geodesic, then the closure of the set of ergodic measures does not encompass all invariant probability measures. Our findings notably provide an answer to the question of genericity of ergodic measures concerning a specific example of 3-manifold introduced by Gromov.