Closed geodesics and the first Betti number
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912580759453696 |
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| author | Contreras, Gonzalo Mazzucchelli, Marco |
| author_facet | Contreras, Gonzalo Mazzucchelli, Marco |
| contents | We prove that, on any closed manifold of dimension at least two with non-trivial first Betti number, a $C^\infty$ generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length. We derive this existence result combining a theorem of Mañé together with the following new theorem of independent interest: the existence of minimal closed geodesics, in the sense of Aubry-Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable $C^\infty$-close Riemannian metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02995 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Closed geodesics and the first Betti number Contreras, Gonzalo Mazzucchelli, Marco Dynamical Systems Differential Geometry Symplectic Geometry 58E10, 53C22 We prove that, on any closed manifold of dimension at least two with non-trivial first Betti number, a $C^\infty$ generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length. We derive this existence result combining a theorem of Mañé together with the following new theorem of independent interest: the existence of minimal closed geodesics, in the sense of Aubry-Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable $C^\infty$-close Riemannian metric. |
| title | Closed geodesics and the first Betti number |
| topic | Dynamical Systems Differential Geometry Symplectic Geometry 58E10, 53C22 |
| url | https://arxiv.org/abs/2407.02995 |