American Institute of Mathematical Sciences

Advanced
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,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[5]

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

[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.  Google Scholar

[7]

, M. Hutchings,, email correspondence., ().   Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

show all references

References:
[1]

D. Cristofaro-Gardiner and M. Hutchings, From one Reeb orbit to two,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[5]

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

[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.  Google Scholar

[7]

, M. Hutchings,, email correspondence., ().   Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[1]

Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005
[2]

Mark Pollicott. Closed orbits and homology for $C^2$-flows. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 529-534. doi: 10.3934/dcds.1999.5.529
[3]

Simon Lloyd. On the Closing Lemma problem for the torus. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 951-962. doi: 10.3934/dcds.2009.25.951
[4]

Peter Albers, Urs Frauenfelder. Floer homology for negative line bundles and Reeb chords in prequantization spaces. Journal of Modern Dynamics, 2009, 3 (3) : 407-456. doi: 10.3934/jmd.2009.3.407
[5]

Lan Wen. A uniform $C^1$ connecting lemma. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 257-265. doi: 10.3934/dcds.2002.8.257
[6]

Al Momin. Contact homology of orbit complements and implied existence. Journal of Modern Dynamics, 2011, 5 (3) : 409-472. doi: 10.3934/jmd.2011.5.409
[7]

Shuhei Hayashi. A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2285-2313. doi: 10.3934/dcds.2020114
[8]

Marcelo R. R. Alves. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. Journal of Modern Dynamics, 2016, 10: 497-509. doi: 10.3934/jmd.2016.10.497
[9]

Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177
[10]

Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505
[11]

Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949
[12]

Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451
[13]

Ayadi Lazrag, Ludovic Rifford, Rafael O. Ruggiero. Franks' lemma for $\mathbf{C}^2$-Mañé perturbations of Riemannian metrics and applications to persistence. Journal of Modern Dynamics, 2016, 10: 379-411. doi: 10.3934/jmd.2016.10.379
[14]

Guji Tian, Qi Wang, Chao-Jiang Xu. $C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1023-1039. doi: 10.3934/dcds.2016.36.1023
[15]

Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331
[16]

Paolo Perfetti. Hamiltonian equations on $\mathbb{T}^\infty$ and almost-periodic solutions. Conference Publications, 2001, 2001 (Special) : 303-309. doi: 10.3934/proc.2001.2001.303
[17]

Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109
[18]

Jorge Rebaza. Bifurcations and periodic orbits in variable population interactions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2997-3012. doi: 10.3934/cpaa.2013.12.2997
[19]

Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057
[20]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences