Journal of Modern Dynamics (JMD)

Dense existence of periodic Reeb orbits and ECH spectral invariants

Pages: 357 - 363, Volume 9, 2015      doi:10.3934/jmd.2015.9.357

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Kei Irie - Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan (email)

Abstract: In this paper, we prove: (1) for any closed contact three-manifold with a $C^\infty$-generic contact form, the union of periodic Reeb orbits is dense; (2) for any closed surface with a $C^\infty$-generic Riemannian metric, the union of closed geodesics is dense. The key observation is a $C^\infty$-closing lemma for three-dimensional Reeb flows, which follows from the fact that the embedded contact homology (ECH) spectral invariants recover the volume.

Keywords:  Periodic Reeb orbits, embedded contact homology, $C^\infty$-closing lemma.
Mathematics Subject Classification:  Primary: 37J45; Secondary: 53D42, 53D25.

Received: July 2015;      Revised: August 2015;      Available Online: December 2015.