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

2025/08/10 by Mella, Paul · 1 citation
#Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.07378

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 S¹ 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.

Citations

• On density of ergodic measures and generic points
• Generic measures for geodesic flows on nonpositively curved manifolds
• Ergodic universality of some topological dynamical systems
• Counterexamples in nonpositive curvature
• On the topological entropy of saturated sets

Cited by

• Approximation by invariant Dirac measures on non-positively curved manifolds

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