Approximation by invariant Dirac measures on non-positively curved manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908537056133120 |
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| author | Mella, Paul |
| author_facet | Mella, Paul |
| contents | 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 Coudène 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 $ε>0$, its $ε$-stable set and $ε$-unstable set both intersect the set $Ω_{NF}$ 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 $Ω_{NF}$. 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 $Ω_{NF} \subset \mathcal{R}_w$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04916 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximation by invariant Dirac measures on non-positively curved manifolds Mella, Paul Dynamical Systems Differential Geometry 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 Coudène 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 $ε>0$, its $ε$-stable set and $ε$-unstable set both intersect the set $Ω_{NF}$ 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 $Ω_{NF}$. 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 $Ω_{NF} \subset \mathcal{R}_w$ 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. |
| title | Approximation by invariant Dirac measures on non-positively curved manifolds |
| topic | Dynamical Systems Differential Geometry |
| url | https://arxiv.org/abs/2509.04916 |