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

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.

v5: various minor corrections, clarifications, and expanded explanations in response to referee comments

References in corpus (1)

• Relative asymptotic behavior of pseudoholomorphic half-cylinders

Cited by in corpus (3)

• Quantitative Heegaard Floer cohomology and the Calabi invariant
• Periodic points of rational area-preserving homeomorphisms
• Lagrangian intersections and a conjecture of Arnol'd