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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

Kei Irie 1,

1. 

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

Received  July 2015 Revised  August 2015 Published  December 2015

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), 187-214. 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), 595-610. 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), 989-994.

[5]

M. Hutchings, Quantitative embedded contact homology, J. Differential Geom., 88 (2011), 231-266.

[6]

M. Hutchings, Lecture notes on embedded contact homology, in Contact and Symplectic Topology, Bolyai Soc. Math. Stud., 26, János Bolyai Math. Soc., Budapest, 2014, 389-484. 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), 261-313. doi: 10.1017/S0143385700001978.

[9]

L. Rifford, Closing geodesics in $C^1$ topology, J. Differential Geom., 91 (2012), 361-381.

[10]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., 193 (2000), 419-461. doi: 10.2140/pjm.2000.193.419.

show all references

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), 187-214. 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), 595-610. 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), 989-994.

[5]

M. Hutchings, Quantitative embedded contact homology, J. Differential Geom., 88 (2011), 231-266.

[6]

M. Hutchings, Lecture notes on embedded contact homology, in Contact and Symplectic Topology, Bolyai Soc. Math. Stud., 26, János Bolyai Math. Soc., Budapest, 2014, 389-484. 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), 261-313. doi: 10.1017/S0143385700001978.

[9]

L. Rifford, Closing geodesics in $C^1$ topology, J. Differential Geom., 91 (2012), 361-381.

[10]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., 193 (2000), 419-461. doi: 10.2140/pjm.2000.193.419.

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