Dense existence of periodic Reeb orbits and ECH spectral invariants

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

Dense existence of periodic Reeb orbits and ECH spectral invariants

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

Received July 2015 Revised August 2015 Published December 2015

Abstract
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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, 53D2.

Citation: Kei Irie. Dense existence of periodic Reeb orbits and ECH spectral invariants. Journal of Modern Dynamics, 2015, 9: 357-363. doi: 10.3934/jmd.2015.9.357

References:

[1]	D. Cristofaro-Gardiner and M. Hutchings, From one Reeb orbit to two,, , ().
[2]	D. Cristofaro-Gardiner, M. Hutchings and V. G. B. Ramos, The asymptotics of ECH capacities,, Invent. Math., 199 (2015), 187. doi: 10.1007/s00222-014-0510-7.
[3]	V. L. Ginzburg and B. Z. Gürel, On the generic existence of periodic orbits in Hamiltonian dynamics,, J. Mod. Dyn., 3 (2009), 595. doi: 10.3934/jmd.2009.3.595.
[4]	M.-R. Herman, Exemples de flots hamiltoniens dont aucune perturbation en topologie $C^\infty$ n'a d'orbites périodiques sur un ouvert de surfaces d'énergies,, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), 989.
[5]	M. Hutchings, Quantitative embedded contact homology,, J. Differential Geom., 88 (2011), 231.
[6]	M. Hutchings, Lecture notes on embedded contact homology,, in Contact and Symplectic Topology, (2014), 389. doi: 10.1007/978-3-319-02036-5_9.
[7]	, M. Hutchings,, email correspondence., ().
[8]	C. C. Pugh and C. Robinson, The $C^1$ closing lemma, including Hamiltonians,, Ergodic Theory Dynam. Systems, 3 (1983), 261. doi: 10.1017/S0143385700001978.
[9]	L. Rifford, Closing geodesics in $C^1$ topology,, J. Differential Geom., 91 (2012), 361.
[10]	M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds,, Pacific J. Math., 193 (2000), 419. doi: 10.2140/pjm.2000.193.419.

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References:

[1]	D. Cristofaro-Gardiner and M. Hutchings, From one Reeb orbit to two,, , ().
[2]	D. Cristofaro-Gardiner, M. Hutchings and V. G. B. Ramos, The asymptotics of ECH capacities,, Invent. Math., 199 (2015), 187. doi: 10.1007/s00222-014-0510-7.
[3]	V. L. Ginzburg and B. Z. Gürel, On the generic existence of periodic orbits in Hamiltonian dynamics,, J. Mod. Dyn., 3 (2009), 595. doi: 10.3934/jmd.2009.3.595.
[4]	M.-R. Herman, Exemples de flots hamiltoniens dont aucune perturbation en topologie $C^\infty$ n'a d'orbites périodiques sur un ouvert de surfaces d'énergies,, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), 989.
[5]	M. Hutchings, Quantitative embedded contact homology,, J. Differential Geom., 88 (2011), 231.
[6]	M. Hutchings, Lecture notes on embedded contact homology,, in Contact and Symplectic Topology, (2014), 389. doi: 10.1007/978-3-319-02036-5_9.
[7]	, M. Hutchings,, email correspondence., ().
[8]	C. C. Pugh and C. Robinson, The $C^1$ closing lemma, including Hamiltonians,, Ergodic Theory Dynam. Systems, 3 (1983), 261. doi: 10.1017/S0143385700001978.
[9]	L. Rifford, Closing geodesics in $C^1$ topology,, J. Differential Geom., 91 (2012), 361.
[10]	M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds,, Pacific J. Math., 193 (2000), 419. doi: 10.2140/pjm.2000.193.419.

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