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

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.