Franks' dichotomy for toric manifolds, Hofer-Zehnder conjecture, and gauged linear sigma model
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913191676608512 |
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| author | Bai, Shaoyun Xu, Guangbo |
| author_facet | Bai, Shaoyun Xu, Guangbo |
| contents | We prove that for any compact toric symplectic manifold, if a Hamiltonian diffeomorphism admits more fixed points, counted homologically, than the total Betti number, then it has infinitely many simple periodic points. This provides a vast generalization of Franks' famous two or infinity dichotomy for periodic orbits of area-preserving diffeomorphisms on the two-sphere, and establishes a conjecture attributed to Hofer-Zehnder in the case of toric manifolds. The key novelty is the application of gauged linear sigma model and its bulk deformations to the study of Hamiltonian dynamics of symplectic quotients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07991 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Franks' dichotomy for toric manifolds, Hofer-Zehnder conjecture, and gauged linear sigma model Bai, Shaoyun Xu, Guangbo Symplectic Geometry Dynamical Systems We prove that for any compact toric symplectic manifold, if a Hamiltonian diffeomorphism admits more fixed points, counted homologically, than the total Betti number, then it has infinitely many simple periodic points. This provides a vast generalization of Franks' famous two or infinity dichotomy for periodic orbits of area-preserving diffeomorphisms on the two-sphere, and establishes a conjecture attributed to Hofer-Zehnder in the case of toric manifolds. The key novelty is the application of gauged linear sigma model and its bulk deformations to the study of Hamiltonian dynamics of symplectic quotients. |
| title | Franks' dichotomy for toric manifolds, Hofer-Zehnder conjecture, and gauged linear sigma model |
| topic | Symplectic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2309.07991 |