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A C ∞ CLOSING LEMMA FOR HAMILTONIAN DIFFEOMORPHISMS SURFACES

. We prove a C ∞ closing lemma for Hamiltonian diﬀeomorphisms of closed surfaces. This is a consequence of a C ∞ 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.