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.
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Asaoka, M., Irie, K. A \({C^\infty}\) closing lemma for Hamiltonian diffeomorphisms of closed surfaces. Geom. Funct. Anal. 26, 1245–1254 (2016). https://doi.org/10.1007/s00039-016-0386-3
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DOI: https://doi.org/10.1007/s00039-016-0386-3