Approximation by invariant Dirac measures on non-positively curved manifolds

Abstract

We study the topology of the space of probability measures invariant under the geodesic flow, defined on the unit-tangent bundle of a compact Riemannian manifold with non-positive curvature. Building on a previous work by Coud\`ene and Schapira we introduce the set of \textit{weakly regular} vectors, denoted by $\mathcal{R}_w$: a vector in the unit tangent bundle of a Riemannian manifold is weakly regular if for all $\epsilon>0$, its $\epsilon$-stable set and $\epsilon$-unstable set both intersect the set $\Omega_{NF}$ of non-wandering vectors whose orbit does not bound a flat strip. We show that every ergodic probability measure supported on $\mathcal{R}_w$ can be approximated by Dirac measures supported on periodic orbits in $\Omega_{NF}$. As a consequence, ergodicity is a generic property in the space of invariant measures supported on $\mathcal{R}_w$. We illustrate our findings using a famous example of rank-one manifold attributed to Heintze and Gromov, demonstrating that in this setting the inclusion $\Omega_{NF} \subset \mathcal{R}_w$ is proper and $\mathcal{R}_w$ is the maximal subset of the unit-tangent bundle satisfying the density property stated above. Finally, as a consequence of our main result, we describe the topology of the closure of the set of ergodic probability measures and provide a complete decomposition of the space of finite invariant measures on the unit-tangent bundle of the Heintze-Gromov manifold.