Closed trajectories on symmetric convex Hamiltonian energy surfaces

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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

Abstract
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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 $\Sigma$.

Keywords: Hamiltonian systems., closed characteristics, Compact convex hypersurfaces.

Mathematics Subject Classification: Primary: 58E05, 37J45; Secondary: 37C7.

Citation: Wei Wang. Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 679-701. doi: 10.3934/dcds.2012.32.679

References:

[1]	A. Borel, "Seminar on Transformation Groups,", Annals of Mathematics Studies, (1960).
[2]	I. Ekeland, Une théorie de Morse pour les systèmes hamiltoniens convexes,, Ann. IHP. Anal. non Linéaire, 1 (1984), 19.
[3]	I. Ekeland, "Convexity Methods in Hamiltonian Mechanics,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 19 (1990).
[4]	I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories,, Comm. Math. Phys., 113 (1987), 419. doi: 10.1007/BF01221255.
[5]	I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées d'un systéme hamiltoniens convexes,, Ann. IHP. Anal. non Linéaire, 4 (1987), 307.
[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. doi: 10.1007/BF01390270.
[7]	H. Hofer, K. Wysocki, and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces,, Ann. of Math., 148 (1998), 197. doi: 10.2307/120994.
[8]	Y. Long, Bott formula of the Maslov-type index theory,, Pacific J. Math., 187 (1999), 113.
[9]	Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics,, Advances in Math., 154 (2000), 76. doi: 10.1006/aima.2000.1914.
[10]	Y. Long, "Index Theory for Symplectic Paths with Applications,", Progress in Math., 207 (2002).
[11]	C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbfR^{2n}$,, Math. Ann., 323 (2002), 201. doi: 10.1007/s002089100257.
[12]	Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $\mathbfR^{2n}$,, Ann. of Math., 155 (2002), 317. doi: 10.2307/3062120.
[13]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 31 (1978), 157. doi: 10.1002/cpa.3160310203.
[14]	A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems,, Bull. Soc. Math. France, 116 (1988), 171.
[15]	C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems,, Trans. Amer. Math. Soc., 311 (1989), 621. doi: 10.1090/S0002-9947-1989-0978370-5.
[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. doi: 10.1215/S0012-7094-07-13931-0.
[17]	W. Wang, Stability of closed characteristics on compact convex hypersurfaces in $R^6$,, J. Eur. Math. Soc., 11 (2009), 575. doi: 10.4171/JEMS/161.
[18]	W. Wang, Symmetric closed characteristics on symmetric compact convex hypersurfaces in $R^{2n}$,, J. Diff. Equa., 246 (2009), 4322. doi: 10.1016/j.jde.2008.10.003.
[19]	A. Weinstein, Periodic orbits for convex Hamiltonian systems,, Ann. of Math., 108 (1978), 507. doi: 10.2307/1971185.

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References:

[1]	A. Borel, "Seminar on Transformation Groups,", Annals of Mathematics Studies, (1960).
[2]	I. Ekeland, Une théorie de Morse pour les systèmes hamiltoniens convexes,, Ann. IHP. Anal. non Linéaire, 1 (1984), 19.
[3]	I. Ekeland, "Convexity Methods in Hamiltonian Mechanics,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 19 (1990).
[4]	I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories,, Comm. Math. Phys., 113 (1987), 419. doi: 10.1007/BF01221255.
[5]	I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées d'un systéme hamiltoniens convexes,, Ann. IHP. Anal. non Linéaire, 4 (1987), 307.
[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. doi: 10.1007/BF01390270.
[7]	H. Hofer, K. Wysocki, and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces,, Ann. of Math., 148 (1998), 197. doi: 10.2307/120994.
[8]	Y. Long, Bott formula of the Maslov-type index theory,, Pacific J. Math., 187 (1999), 113.
[9]	Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics,, Advances in Math., 154 (2000), 76. doi: 10.1006/aima.2000.1914.
[10]	Y. Long, "Index Theory for Symplectic Paths with Applications,", Progress in Math., 207 (2002).
[11]	C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbfR^{2n}$,, Math. Ann., 323 (2002), 201. doi: 10.1007/s002089100257.
[12]	Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $\mathbfR^{2n}$,, Ann. of Math., 155 (2002), 317. doi: 10.2307/3062120.
[13]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 31 (1978), 157. doi: 10.1002/cpa.3160310203.
[14]	A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems,, Bull. Soc. Math. France, 116 (1988), 171.
[15]	C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems,, Trans. Amer. Math. Soc., 311 (1989), 621. doi: 10.1090/S0002-9947-1989-0978370-5.
[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. doi: 10.1215/S0012-7094-07-13931-0.
[17]	W. Wang, Stability of closed characteristics on compact convex hypersurfaces in $R^6$,, J. Eur. Math. Soc., 11 (2009), 575. doi: 10.4171/JEMS/161.
[18]	W. Wang, Symmetric closed characteristics on symmetric compact convex hypersurfaces in $R^{2n}$,, J. Diff. Equa., 246 (2009), 4322. doi: 10.1016/j.jde.2008.10.003.
[19]	A. Weinstein, Periodic orbits for convex Hamiltonian systems,, Ann. of Math., 108 (1978), 507. doi: 10.2307/1971185.

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