American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[2]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378. doi: 10.1007/BF01215653.  Google Scholar

[3]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics. II, Math. Z., 203 (1990), 553-567. doi: 10.1007/BF02570756.  Google Scholar

[4]

M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

H. Hofer, On the topolgical properties of symplectic maps, Proc. Roy. Soc. Edinburgh, 115 (1990), 25-38. doi: 10.1017/S0308210500024549.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Hofer and E. Zehnder, "Symelectic Invariants and Hamiltonian Dynamics," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994.  Google Scholar

[12]

M.-Y. Jiang, Hofer-Zehnder symplectic capatcity for two dimensional manifolds, Proc. Royal Soc. Edinb., 123 (1993), 945-950. doi: 10.1017/S0308210500029590.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

G. Lu, Finiteness of the Hofer-Zehnder capacity of neighborhoods of syplectic submanifolds, IMRN, 2006, Art. ID 76520, 33 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

C. Liu and D. Zhang, Iteration theory of $L$-index and multiplicity of brake orbits,, preprint., ().   Google Scholar

[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.  Google Scholar

[20]

L. Macarini, Hofer-Zehnder capacity and Hamiltonian circle actions, Communications in Contemporary Mathematics, 6 (2004), 913-945. doi: 10.1142/S0219199704001550.  Google Scholar

[21]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology," Second edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[22]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

C. Viterbo, A proof of the Weinstein conjecture in $\mathbbR^{2n}$, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 4 (1987), 337-356.  Google Scholar

[28]

C. Viterbo, Capacités symplectiques et applications (d'aprés Ekeland- Hofer, Gromov), Astérisque No., 177-178 (1989), 345-362.  Google Scholar

[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.  Google Scholar

[30]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. Math. (2), 108 (1978), 507-518. doi: 10.2307/1971185.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378. doi: 10.1007/BF01215653.  Google Scholar

[3]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics. II, Math. Z., 203 (1990), 553-567. doi: 10.1007/BF02570756.  Google Scholar

[4]

M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

H. Hofer, On the topolgical properties of symplectic maps, Proc. Roy. Soc. Edinburgh, 115 (1990), 25-38. doi: 10.1017/S0308210500024549.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Hofer and E. Zehnder, "Symelectic Invariants and Hamiltonian Dynamics," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994.  Google Scholar

[12]

M.-Y. Jiang, Hofer-Zehnder symplectic capatcity for two dimensional manifolds, Proc. Royal Soc. Edinb., 123 (1993), 945-950. doi: 10.1017/S0308210500029590.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

G. Lu, Finiteness of the Hofer-Zehnder capacity of neighborhoods of syplectic submanifolds, IMRN, 2006, Art. ID 76520, 33 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

C. Liu and D. Zhang, Iteration theory of $L$-index and multiplicity of brake orbits,, preprint., ().   Google Scholar

[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.  Google Scholar

[20]

L. Macarini, Hofer-Zehnder capacity and Hamiltonian circle actions, Communications in Contemporary Mathematics, 6 (2004), 913-945. doi: 10.1142/S0219199704001550.  Google Scholar

[21]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology," Second edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[22]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

C. Viterbo, A proof of the Weinstein conjecture in $\mathbbR^{2n}$, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 4 (1987), 337-356.  Google Scholar

[28]

C. Viterbo, Capacités symplectiques et applications (d'aprés Ekeland- Hofer, Gromov), Astérisque No., 177-178 (1989), 345-362.  Google Scholar

[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.  Google Scholar

[30]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. Math. (2), 108 (1978), 507-518. doi: 10.2307/1971185.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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