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A \({C^\infty}\) closing lemma for Hamiltonian diffeomorphisms of closed surfaces

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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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Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Masayuki Asaoka

  2. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Kei Irie

  3. Simons Center for Geometry and Physics, State University of New York, Stony Brook, NY, 11794-3636, USA

    Kei Irie

Authors
  1. Masayuki Asaoka
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  2. Kei Irie
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Correspondence to Masayuki Asaoka.

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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

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