PFH spectral invariants and C^∞ closing lemmas

2021/10/06 by Oliver Edtmair, Edtmair, Oliver, Michael Hutchings +1 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2110.02463

openalex publication_date 2021/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^∞ closing lemmas for certain Hamiltonian isotopy classes of area-preserving surface diffeomorphisms. In particular, we show that for a C^∞-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.

Citations

• Cobordism maps in embedded contact homology

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• Lagrangian intersections and a conjecture of Arnol'd

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