Generic density of periodic orbits of area-preserving maps on punctured surfaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909401094291456 |
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| author | Zhou, Shaoyang |
| author_facet | Zhou, Shaoyang |
| contents | We study the dynamics of area-preserving maps in a non-compact setting. We show that the $C^{\infty}$-closing lemma holds for area-preserving diffeomorphisms on a closed surface with finitely many points removed. As a corollary, a $C^{\infty}$-generic area-preserving diffeomorphism on such a surface has a dense set of periodic points. For area-preserving maps on a finitely punctured 2-sphere, we establish a more quantitative result regarding the equidistribution of periodic orbits. The proof of this result involves a PFH Weyl law for rational area-preserving homeomorphisms, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15429 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic density of periodic orbits of area-preserving maps on punctured surfaces Zhou, Shaoyang Dynamical Systems Symplectic Geometry We study the dynamics of area-preserving maps in a non-compact setting. We show that the $C^{\infty}$-closing lemma holds for area-preserving diffeomorphisms on a closed surface with finitely many points removed. As a corollary, a $C^{\infty}$-generic area-preserving diffeomorphism on such a surface has a dense set of periodic points. For area-preserving maps on a finitely punctured 2-sphere, we establish a more quantitative result regarding the equidistribution of periodic orbits. The proof of this result involves a PFH Weyl law for rational area-preserving homeomorphisms, which may be of independent interest. |
| title | Generic density of periodic orbits of area-preserving maps on punctured surfaces |
| topic | Dynamical Systems Symplectic Geometry |
| url | https://arxiv.org/abs/2411.15429 |