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arXiv · 2110.02463

PFH spectral invariants and $C^\infty$ closing lemmas

Oliver Edtmair·Michael Hutchings
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Abstract

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 $\delta$ a periodic orbit must appear of period $O(\delta^{-1})$. We also prove a "Weyl law" describing the asymptotic behavior of PFH spectral invariants.

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Oliver Edtmair, Michael Hutchings. 2021-10-06. PFH spectral invariants and $C^\infty$ closing lemmas. https://arxiv.org/abs/2110.02463

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