2007, 1(4): 597-613. doi: 10.3934/jmd.2007.1.597

A note on Reeb dynamics on the tight 3-sphere

1. 

Département de Mathématique, Université Libre de Bruxelles CP 218, Boulevard du Triomphe, 1050 Bruxelles, Belgium

2. 

Mathematisches Institut der LMU München, Theresienstr. 39, 80333 München, Germany

3. 

Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden

Received  March 2007 Revised  July 2007 Published  July 2007

We show that a nondegenerate tight contact form on the 3-sphere has exactly two simple closed Reeb orbits if and only if the differential in linearized contact homology vanishes. Moreover, in this case the Floquet multipliers and Conley-Zehnder indices of the two Reeb orbits agree with those of a suitable irrational ellipsoid in 4-space.
Citation: Frédéric Bourgeois, Kai Cieliebak, Tobias Ekholm. A note on Reeb dynamics on the tight 3-sphere. Journal of Modern Dynamics, 2007, 1 (4) : 597-613. doi: 10.3934/jmd.2007.1.597
[1]

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

[2]

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

[3]

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

[4]

M. Ollé, J.R. Pacha, J. Villanueva. Dynamics close to a non semi-simple 1:-1 resonant periodic orbit. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 799-816. doi: 10.3934/dcdsb.2005.5.799

[5]

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

[6]

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

[7]

Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080

[8]

Peter Giesl, James McMichen. Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation. Journal of Computational Dynamics, 2016, 3 (2) : 191-210. doi: 10.3934/jcd.2016010

[9]

Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315

[10]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883

[11]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485

[12]

Peter Giesl. Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 355-373. doi: 10.3934/dcds.2007.18.355

[13]

Alain Léger, Elaine Pratt. On the equilibria and qualitative dynamics of a forced nonlinear oscillator with contact and friction. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 501-527. doi: 10.3934/dcdss.2016009

[14]

Mark Lewis, Daniel Offin, Pietro-Luciano Buono, Mitchell Kovacic. Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137

[15]

Radu Saghin. Note on homology of expanding foliations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 349-360. doi: 10.3934/dcdss.2009.2.349

[16]

Islam A. Moneim, David Greenhalgh. Use Of A Periodic Vaccination Strategy To Control The Spread Of Epidemics With Seasonally Varying Contact Rate. Mathematical Biosciences & Engineering, 2005, 2 (3) : 591-611. doi: 10.3934/mbe.2005.2.591

[17]

Philippe Pécol, Pierre Argoul, Stefano Dal Pont, Silvano Erlicher. The non-smooth view for contact dynamics by Michel Frémond extended to the modeling of crowd movements. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 547-565. doi: 10.3934/dcdss.2013.6.547

[18]

Octav Cornea and Francois Lalonde. Cluster homology: An overview of the construction and results. Electronic Research Announcements, 2006, 12: 1-12.

[19]

Henri Berestycki, Jean-Michel Roquejoffre, Luca Rossi. The periodic patch model for population dynamics with fractional diffusion. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 1-13. doi: 10.3934/dcdss.2011.4.1

[20]

Viktor L. Ginzburg, Başak Z. Gürel. On the generic existence of periodic orbits in Hamiltonian dynamics. Journal of Modern Dynamics, 2009, 3 (4) : 595-610. doi: 10.3934/jmd.2009.3.595

2016 Impact Factor: 0.706

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (10)

[Back to Top]