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We prove a $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closedsurfaces. This is a consequence of a $C^\infty$ closing lemma for Reeb flows onclosed contact three-manifolds, which was recently proved as an application ofspectral invariants in embedded contact homology. A key new ingredient of thispaper is an analysis of an area-preserving map near its fixed point, which isbased on some classical results in Hamiltonian dynamics: existence of KAMinvariant circles for elliptic fixed points, and convergence of the Birkhoffnormal form for hyperbolic fixed points.

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