PFH spectral invariants and $C^\infty$ closing lemmas
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866914740125564928 |
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| author | Edtmair, Oliver Hutchings, Michael |
| author_facet | Edtmair, Oliver Hutchings, Michael |
| contents | We develop the theory of spectral invariants in periodic Floer homology (PFH) of area-preserving surface diffeomorphisms. We use this theory to prove $C^\infty$ closing lemmas for certain Hamiltonian isotopy classes of area-preserving surface diffeomorphisms. In particular, we show that for a $C^\infty$-generic area-preserving diffeomorphism of the torus, the set of periodic points is dense. Our closing lemmas are quantitative, asserting roughly speaking that for a given Hamiltonian isotopy, within time $δ$ a periodic orbit must appear of period $O(δ^{-1})$. We also prove a "Weyl law" describing the asymptotic behavior of PFH spectral invariants. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_02463 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | PFH spectral invariants and $C^\infty$ closing lemmas Edtmair, Oliver Hutchings, Michael Symplectic Geometry Dynamical Systems We develop the theory of spectral invariants in periodic Floer homology (PFH) of area-preserving surface diffeomorphisms. We use this theory to prove $C^\infty$ closing lemmas for certain Hamiltonian isotopy classes of area-preserving surface diffeomorphisms. In particular, we show that for a $C^\infty$-generic area-preserving diffeomorphism of the torus, the set of periodic points is dense. Our closing lemmas are quantitative, asserting roughly speaking that for a given Hamiltonian isotopy, within time $δ$ a periodic orbit must appear of period $O(δ^{-1})$. We also prove a "Weyl law" describing the asymptotic behavior of PFH spectral invariants. |
| title | PFH spectral invariants and $C^\infty$ closing lemmas |
| topic | Symplectic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2110.02463 |