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arXiv · 1512.06336

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

Masayuki Asaoka·Kei Irie
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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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Masayuki Asaoka, Kei Irie. 2015-12-20. A $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. https://arxiv.org/abs/1512.06336

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