Symmetrical symplectic capacity with applications

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June 2012, 32(6): 2253-2270. doi: 10.3934/dcds.2012.32.2253

Symmetrical symplectic capacity with applications

Chungen Liu ^1, and Qi Wang ^1,

1.	School of Mathematics and LPMC, Nankai University, Tianjin 300071, China

Received January 2011 Revised October 2011 Published February 2012

Abstract
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In this paper, we first introduce the concept of symmetrical symplectic capacity for symmetrical symplectic manifolds, and by using this symmetrical symplectic capacity theory we prove that there exists at least one symmetric closed characteristic (brake orbit and $S$-invariant brake orbit are two examples) on prescribed symmetric energy surface which has a compact neighborhood with finite symmetrical symplectic capacity.

Keywords: Symmetrical symplectic manifolds, brake orbits., symmetrical symplectic capacity.

Mathematics Subject Classification: 53D40, 37J4.

Citation: Chungen Liu, Qi Wang. Symmetrical symplectic capacity with applications. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2253-2270. doi: 10.3934/dcds.2012.32.2253

References:

[1]	A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Mathematics (2), 152 (2000), 881-901. doi: 10.2307/2661357.
[2]	I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378. doi: 10.1007/BF01215653.
[3]	I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics. II, Math. Z., 203 (1990), 553-567. doi: 10.1007/BF02570756.
[4]	M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.
[5]	D. Hermann, Holomorphic curves and Hamiltonian systems in an open set with restricted contact-type boundary, Duke Mathematical Journal, 103 (2000), 335-374. doi: 10.1215/S0012-7094-00-10327-4.
[6]	M.-R. Herman, Differentiabilité optimale et contre-exemples à la fermeture en topologie $C^\infty$ des orbites recurrentes de flots Hamiltoniens, Comptes Rendus de l'Académie des Sciences Serie-I. Mathématique, 313 (1991), 49-51.
[7]	M.-R. Herman, Exemples de flots Hamiltoniens dont aucune perturbation en topologie $C^\infty$ n'a d'orbites periodiques sur un ouvert de surfaces d'énergies, Comptes Rendus de l'Académie des Sciences Serie-I. Mathématique, 312 (1991), 989-994.
[8]	H. Hofer, On the topolgical properties of symplectic maps, Proc. Roy. Soc. Edinburgh, 115 (1990), 25-38. doi: 10.1017/S0308210500024549.
[9]	H. Hofer and C. Viterbo, The Weinstein conjecture in the presence of holomorpgic spheres, Comm. Pure Appl. Math, 45 (1992), 583-622. doi: 10.1002/cpa.3160450504.
[10]	H. Hofer and E. Zehnder, A new capacity for symplectic manifolds, in "Analysis, et Cetera" (eds. P. Rabinowitz and E. Zehnder), Academic Press, Boston, MA, (1990), 405-427.
[11]	H. Hofer and E. Zehnder, "Symelectic Invariants and Hamiltonian Dynamics," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994.
[12]	M.-Y. Jiang, Hofer-Zehnder symplectic capatcity for two dimensional manifolds, Proc. Royal Soc. Edinb., 123 (1993), 945-950. doi: 10.1017/S0308210500029590.
[13]	C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbbR^{2n}$, Math. Ann., 323 (2002), 201-215. doi: 10.1007/s002089100257.
[14]	Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $\mathbbR^{2n}$, Ann. Math. (2), 155 (2002), 317-368. doi: 10.2307/3062120.
[15]	G. Lu, Finiteness of the Hofer-Zehnder capacity of neighborhoods of syplectic submanifolds, IMRN, 2006, Art. ID 76520, 33 pp.
[16]	C. Li and C. Liu, Brake subharmonic solutions of first order Hamiltonian systems, Science in China (Mathematics), 53 (2010), 2719-2732. doi: 10.1007/s11425-010-4105-5.
[17]	C. Liu, Q. Wang and X. Lin, An index theory for symplectic paths associated with Lagrangian subspaces with applications, Nonlinearity, 24 (2011), 43-70. doi: 10.1088/0951-7715/24/1/002.
[18]	C. Liu and D. Zhang, Iteration theory of $L$-index and multiplicity of brake orbits,, preprint., ().
[19]	Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Adv. Math., 203 (2006), 568-635. doi: 10.1016/j.aim.2005.05.005.
[20]	L. Macarini, Hofer-Zehnder capacity and Hamiltonian circle actions, Communications in Contemporary Mathematics, 6 (2004), 913-945. doi: 10.1142/S0219199704001550.
[21]	D. McDuff and D. Salamon, "Introduction to Symplectic Topology," Second edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.
[22]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.
[23]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems on a prescribed energy surface, J. Diff. Equ., 33 (1979), 336-352. doi: 10.1016/0022-0396(79)90069-X.
[24]	P. H. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems, Nonlinear Anal., 11 (1987), 599-611. doi: 10.1016/0362-546X(87)90075-7.
[25]	M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface, Bol. Soc. Bras. Mat. (N.S.), 20 (1990), 49-58.
[26]	A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann., 283 (1989), 241-255. doi: 10.1007/BF01446433.
[27]	C. Viterbo, A proof of the Weinstein conjecture in $\mathbbR^{2n}$, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 4 (1987), 337-356.
[28]	C. Viterbo, Capacités symplectiques et applications (d'aprés Ekeland- Hofer, Gromov), Astérisque No., 177-178 (1989), 345-362.
[29]	C. Viterbo, Functors and computations in Floer homology with applications, I and II, Geom. Funct. Anal., 9 (1999), 985-1033. doi: 10.1007/s000390050106.
[30]	A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. Math. (2), 108 (1978), 507-518. doi: 10.2307/1971185.
[31]	A. Weinstein, On the hypotheses of Rabinowitz's periodic orbit theorems, J. Diff. Equ., 33 (1979), 353-358. doi: 10.1016/0022-0396(79)90070-6.
[32]	D. Zhang, Relative Morse index and multiple brake orbits of asymptotically linear Hamiltonian systems in the presence of symmetries, J. Differential Equations, 245 (2008), 925-938. doi: 10.1016/j.jde.2008.04.020.

show all references

References:

[1]	A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Mathematics (2), 152 (2000), 881-901. doi: 10.2307/2661357.
[2]	I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378. doi: 10.1007/BF01215653.
[3]	I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics. II, Math. Z., 203 (1990), 553-567. doi: 10.1007/BF02570756.
[4]	M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.
[5]	D. Hermann, Holomorphic curves and Hamiltonian systems in an open set with restricted contact-type boundary, Duke Mathematical Journal, 103 (2000), 335-374. doi: 10.1215/S0012-7094-00-10327-4.
[6]	M.-R. Herman, Differentiabilité optimale et contre-exemples à la fermeture en topologie $C^\infty$ des orbites recurrentes de flots Hamiltoniens, Comptes Rendus de l'Académie des Sciences Serie-I. Mathématique, 313 (1991), 49-51.
[7]	M.-R. Herman, Exemples de flots Hamiltoniens dont aucune perturbation en topologie $C^\infty$ n'a d'orbites periodiques sur un ouvert de surfaces d'énergies, Comptes Rendus de l'Académie des Sciences Serie-I. Mathématique, 312 (1991), 989-994.
[8]	H. Hofer, On the topolgical properties of symplectic maps, Proc. Roy. Soc. Edinburgh, 115 (1990), 25-38. doi: 10.1017/S0308210500024549.
[9]	H. Hofer and C. Viterbo, The Weinstein conjecture in the presence of holomorpgic spheres, Comm. Pure Appl. Math, 45 (1992), 583-622. doi: 10.1002/cpa.3160450504.
[10]	H. Hofer and E. Zehnder, A new capacity for symplectic manifolds, in "Analysis, et Cetera" (eds. P. Rabinowitz and E. Zehnder), Academic Press, Boston, MA, (1990), 405-427.
[11]	H. Hofer and E. Zehnder, "Symelectic Invariants and Hamiltonian Dynamics," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994.
[12]	M.-Y. Jiang, Hofer-Zehnder symplectic capatcity for two dimensional manifolds, Proc. Royal Soc. Edinb., 123 (1993), 945-950. doi: 10.1017/S0308210500029590.
[13]	C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbbR^{2n}$, Math. Ann., 323 (2002), 201-215. doi: 10.1007/s002089100257.
[14]	Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $\mathbbR^{2n}$, Ann. Math. (2), 155 (2002), 317-368. doi: 10.2307/3062120.
[15]	G. Lu, Finiteness of the Hofer-Zehnder capacity of neighborhoods of syplectic submanifolds, IMRN, 2006, Art. ID 76520, 33 pp.
[16]	C. Li and C. Liu, Brake subharmonic solutions of first order Hamiltonian systems, Science in China (Mathematics), 53 (2010), 2719-2732. doi: 10.1007/s11425-010-4105-5.
[17]	C. Liu, Q. Wang and X. Lin, An index theory for symplectic paths associated with Lagrangian subspaces with applications, Nonlinearity, 24 (2011), 43-70. doi: 10.1088/0951-7715/24/1/002.
[18]	C. Liu and D. Zhang, Iteration theory of $L$-index and multiplicity of brake orbits,, preprint., ().
[19]	Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Adv. Math., 203 (2006), 568-635. doi: 10.1016/j.aim.2005.05.005.
[20]	L. Macarini, Hofer-Zehnder capacity and Hamiltonian circle actions, Communications in Contemporary Mathematics, 6 (2004), 913-945. doi: 10.1142/S0219199704001550.
[21]	D. McDuff and D. Salamon, "Introduction to Symplectic Topology," Second edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.
[22]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.
[23]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems on a prescribed energy surface, J. Diff. Equ., 33 (1979), 336-352. doi: 10.1016/0022-0396(79)90069-X.
[24]	P. H. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems, Nonlinear Anal., 11 (1987), 599-611. doi: 10.1016/0362-546X(87)90075-7.
[25]	M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface, Bol. Soc. Bras. Mat. (N.S.), 20 (1990), 49-58.
[26]	A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann., 283 (1989), 241-255. doi: 10.1007/BF01446433.
[27]	C. Viterbo, A proof of the Weinstein conjecture in $\mathbbR^{2n}$, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 4 (1987), 337-356.
[28]	C. Viterbo, Capacités symplectiques et applications (d'aprés Ekeland- Hofer, Gromov), Astérisque No., 177-178 (1989), 345-362.
[29]	C. Viterbo, Functors and computations in Floer homology with applications, I and II, Geom. Funct. Anal., 9 (1999), 985-1033. doi: 10.1007/s000390050106.
[30]	A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. Math. (2), 108 (1978), 507-518. doi: 10.2307/1971185.
[31]	A. Weinstein, On the hypotheses of Rabinowitz's periodic orbit theorems, J. Diff. Equ., 33 (1979), 353-358. doi: 10.1016/0022-0396(79)90070-6.
[32]	D. Zhang, Relative Morse index and multiple brake orbits of asymptotically linear Hamiltonian systems in the presence of symmetries, J. Differential Equations, 245 (2008), 925-938. doi: 10.1016/j.jde.2008.04.020.

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