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Symmetrical symplectic capacity with applications
1. | School of Mathematics and LPMC, Nankai University, Tianjin 300071, China |
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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