What question did this study set out to answer?

The aim is to explore under what conditions ergodic probability measures are generic for geodesic flows on Riemannian manifolds.

July 1, 2026

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

Key Points

The aim is to explore under what conditions ergodic probability measures are generic for geodesic flows on Riemannian manifolds.
Investigated the metric's structure as a product metric within a tubular neighborhood of a submanifold.
Analyzed the implications for invariant probability measures and ergodic measures.
Considered specific examples, particularly in relation to Gromov's 3-manifold.
Demonstrated that closure of ergodic measures does not include all invariant probability measures.
Established conditions under which ergodicity fails to be generic in the case of certain Riemannian manifolds.

Abstract

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.

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