On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866910917353013248 |
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| author | Orita, Ryuma |
| author_facet | Orita, Ryuma |
| contents | We show that the presence of a non-contractible one-periodic orbit of a Hamiltonian diffeomorphism of a connected closed symplectic manifold $(M,ω)$ implies the existence of infinitely many non-contractible simple periodic orbits, provided that the symplectic form $ω$ is aspherical and the fundamental group $π_1(M)$ is either a virtually abelian group or an $\mathrm{R}$-group. We also show that a similar statement holds for Hamiltonian diffeomorphisms of closed monotone or negative monotone symplectic manifolds under the same conditions on their fundamental groups. These results generalize some works by Ginzburg and Gürel. The proof uses the filtered Floer--Novikov homology for non-contractible periodic orbits. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1703_01731 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds Orita, Ryuma Symplectic Geometry Dynamical Systems 53D40, 37J10, 37J45, 20F19 We show that the presence of a non-contractible one-periodic orbit of a Hamiltonian diffeomorphism of a connected closed symplectic manifold $(M,ω)$ implies the existence of infinitely many non-contractible simple periodic orbits, provided that the symplectic form $ω$ is aspherical and the fundamental group $π_1(M)$ is either a virtually abelian group or an $\mathrm{R}$-group. We also show that a similar statement holds for Hamiltonian diffeomorphisms of closed monotone or negative monotone symplectic manifolds under the same conditions on their fundamental groups. These results generalize some works by Ginzburg and Gürel. The proof uses the filtered Floer--Novikov homology for non-contractible periodic orbits. |
| title | On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds |
| topic | Symplectic Geometry Dynamical Systems 53D40, 37J10, 37J45, 20F19 |
| url | https://arxiv.org/abs/1703.01731 |