# American Institute of Mathematical Sciences

February  2012, 32(2): 679-701. doi: 10.3934/dcds.2012.32.679

## Closed trajectories on symmetric convex Hamiltonian energy surfaces

 1 Key Laboratory of Pure and Applied Mathematics, School of Mathematical Science, Peking University, Beijing, 100871, China

Received  August 2010 Revised  May 2011 Published  September 2011

In this article, let $\Sigma\subset\mathbf{R}^{2n}$ be a compact convex Hamiltonian energy surface which is symmetric with respect to the origin, where $n\ge 2$. We prove that there exist at least two geometrically distinct symmetric closed trajectories of the Reeb vector field on $\Sigma$.
Citation: Wei Wang. Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 679-701. doi: 10.3934/dcds.2012.32.679
##### References:
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##### References:
 [1] A. Borel, "Seminar on Transformation Groups," Annals of Mathematics Studies, No. 46, Princeton Univ. Press, Princeton, 1960.  Google Scholar [2] I. Ekeland, Une théorie de Morse pour les systèmes hamiltoniens convexes, Ann. IHP. Anal. non Linéaire, 1 (1984), 19-78.  Google Scholar [3] I. Ekeland, "Convexity Methods in Hamiltonian Mechanics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 19, Springer-Verlag, Berlin, 1990.  Google Scholar [4] I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories, Comm. Math. Phys., 113 (1987), 419-469. doi: 10.1007/BF01221255.  Google Scholar [5] I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées d'un systéme hamiltoniens convexes, Ann. IHP. Anal. non Linéaire, 4 (1987), 307-335.  Google Scholar [6] E. Fadell and P. Rabinowitz, Generalized comological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174. doi: 10.1007/BF01390270.  Google Scholar [7] H. Hofer, K. Wysocki, and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math., 148 (1998), 197-289. doi: 10.2307/120994.  Google Scholar [8] Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.  Google Scholar [9] Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Advances in Math., 154 (2000), 76-131. doi: 10.1006/aima.2000.1914.  Google Scholar [10] Y. Long, "Index Theory for Symplectic Paths with Applications," Progress in Math., 207, Birkhäuser Verlag, Basel, 2002.  Google Scholar [11] C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbfR^{2n}$, Math. Ann., 323 (2002), 201-215. doi: 10.1007/s002089100257.  Google Scholar [12] Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $\mathbfR^{2n}$, Ann. of Math., 155 (2002), 317-368. doi: 10.2307/3062120.  Google Scholar [13] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.  Google Scholar [14] A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France, 116 (1988), 171-197.  Google Scholar [15] C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math. Soc., 311 (1989), 621-655. doi: 10.1090/S0002-9947-1989-0978370-5.  Google Scholar [16] W. Wang, X. Hu and Y. Long, Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462. doi: 10.1215/S0012-7094-07-13931-0.  Google Scholar [17] W. Wang, Stability of closed characteristics on compact convex hypersurfaces in $R^6$, J. Eur. Math. Soc., 11 (2009), 575-596. doi: 10.4171/JEMS/161.  Google Scholar [18] W. Wang, Symmetric closed characteristics on symmetric compact convex hypersurfaces in $R^{2n}$, J. Diff. Equa., 246 (2009), 4322-4331. doi: 10.1016/j.jde.2008.10.003.  Google Scholar [19] A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518. doi: 10.2307/1971185.  Google Scholar
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