A $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces

TL;DR

This paper establishes a $C^ Infty$ closing lemma for Hamiltonian diffeomorphisms on closed surfaces, extending techniques from contact topology and Hamiltonian dynamics to demonstrate generic periodic points.

Contribution

It introduces a $C^ Infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces, leveraging spectral invariants and classical Hamiltonian analysis near fixed points.

Findings

Proves a $C^ Infty$ closing lemma for Hamiltonian diffeomorphisms.

Connects closing lemmas for Reeb flows and Hamiltonian diffeomorphisms.

Utilizes classical Hamiltonian dynamics results like KAM theory and Birkhoff normal form.

Abstract

We prove a $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an area-preserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.