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Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $
1. | School of Mathematics Department, Shandong University, Jinan 250100, China |
There is a long standing conjecture that there are at least $ n $ closed characteristics on any compact convex hypersurface $ \Sigma $ in $ \mathbb{R}^{2n} $. In this paper, we provide some new estimates and prove that there are at least $ [\frac{3n}{4}] $ closed characteristics on $ \Sigma $ for any positive integer $ n $, where $ \Sigma $ satisfies $ \Sigma = P\Sigma $ for a certain class of symplectic matrix $ P $. These results are not considered in previous papers.
References:
[1] |
S. E. Cappell, R. Lee and E. Y. Miller,
On the maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186.
doi: 10.1002/cpa.3160470202. |
[2] |
Y. Dong and Y. Long,
Closed characteristics on partically symmetric compact convex hypersurfaces in $\mathbb{R}^2n$, J. Diff. Equ., 196 (2004), 226-248.
doi: 10.1016/S0022-0396(03)00168-2. |
[3] |
H. Duan and H. Liu, Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in $\mathbb{R}^2n$,, Cal. Variations and PDEs, 56 (2017), Paper No. 65, 30 pp.
doi: 10.1007/s00526-017-1173-1. |
[4] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-74331-3. |
[5] |
I. Ekeland and H. Hofer,
Convex Hamiltonian energy surfaces and their periodic trajectories, Commun. Math. Phys., 113 (1987), 419-469.
doi: 10.1007/BF01221255. |
[6] |
I. Ekeland and J. M. Lasry,
On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Annals of Math., 112 (1980), 283-319.
doi: 10.2307/1971148. |
[7] |
I. Ekeland and L. Lassoued,
Multiplicite des trajectoires fermees d'un systeme hamiltonien sur une hypersurface d'energie convexe, Ann. IHP. Anal. Non Linéaire, 4 (1987), 307-335.
doi: 10.1016/S0294-1449(16)30362-6. |
[8] |
E. R. Fadell and P. H. Rabinowitz,
Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Inv. Math., 45 (1978), 139-174.
doi: 10.1007/BF01390270. |
[9] |
X. Hu and S. Sun,
Index and stability of symmetric periodic orbits in Hamiltonian systems with its application to figure-eight orbit, Commun. Math. Phys., 290 (2009), 737-777.
doi: 10.1007/s00220-009-0860-y. |
[10] |
C. Liu, Y. Long and C. Zhu,
Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbb{R}^2n$, Math. Ann., 323 (2002), 201-215.
doi: 10.1007/s002089100257. |
[11] |
C. Liu and S. Tang,
Maslov $(P, \omega)$-index theory for symplectic paths, Adv. Nonlinear Studies, 15 (2015), 963-990.
doi: 10.1515/ans-2015-0412. |
[12] |
C. Liu and D. Zhang,
Iteration theory of L-index and multiplicity of brake orbits, J. Diff. Equ., 257 (2014), 1194-1245.
doi: 10.1016/j.jde.2014.05.006. |
[13] |
H. Liu,
Multiple $P$-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $\mathbb{R}^2n$, Cal. Variations and PDEs, 49 (2014), 1121-1147.
doi: 10.1007/s00526-013-0614-8. |
[14] |
Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, No. 207, Birkhauser, Basel, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[15] |
Y. Long and C. Zhu,
Maslov-type index theorey for symplectic paths and spectral flow Ⅱ, Chinese. Ann. Math. Ser. B, 21 (2000), 89-108.
doi: 10.1142/S0252959900000133. |
[16] |
Y. Long and C. Zhu,
Closed characteristics on compact convex hypersurfaces in $\mathbb{R}^2n$, Ann. Math., 155 (2002), 317-368.
doi: 10.2307/3062120. |
[17] |
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. |
[18] |
P. H. Rabinowitz,
Peroidic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
[19] |
P. H. Rabinowitz,
On the existence of periodic solutions for a class of symmetric Hamiltonian system, Nonlinear Anal., 11 (1987), 599-611.
doi: 10.1016/0362-546X(87)90075-7. |
[20] |
J. Robbin and D. Salamon,
The maslov index for paths, Topology, 32 (1993), 827-844.
doi: 10.1016/0040-9383(93)90052-W. |
[21] |
A. Szulkin,
Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France, 116 (1988), 171-197.
doi: 10.24033/bsmf.2094. |
[22] |
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. |
[23] |
W. Wang,
Closed characteristics on compact convex hypersurfaces in $\mathbb{R}^8$, Adv. Math., 297 (2016), 93-148.
doi: 10.1016/j.aim.2016.03.044. |
[24] |
W. Wang, X. Hu and Y. Long,
Resonance identity, stability and multiplicity of closed characteristics on the conpact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462.
doi: 10.1215/S0012-7094-07-13931-0. |
[25] |
A. Weinstein,
Periodic orbits for convex Hamiltonian systems, Ann. Math., 108 (1978), 507-518.
doi: 10.2307/1971185. |
[26] |
D. Zhang,
P-cyclic symmetric closed characteristics on compact convex P-cyclic symmetric hypersurface in $\mathbb{R}^2n$, Discrete Continuous Dynam. Systems, 33 (2013), 947-964.
doi: 10.3934/dcds.2013.33.947. |
show all references
References:
[1] |
S. E. Cappell, R. Lee and E. Y. Miller,
On the maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186.
doi: 10.1002/cpa.3160470202. |
[2] |
Y. Dong and Y. Long,
Closed characteristics on partically symmetric compact convex hypersurfaces in $\mathbb{R}^2n$, J. Diff. Equ., 196 (2004), 226-248.
doi: 10.1016/S0022-0396(03)00168-2. |
[3] |
H. Duan and H. Liu, Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in $\mathbb{R}^2n$,, Cal. Variations and PDEs, 56 (2017), Paper No. 65, 30 pp.
doi: 10.1007/s00526-017-1173-1. |
[4] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-74331-3. |
[5] |
I. Ekeland and H. Hofer,
Convex Hamiltonian energy surfaces and their periodic trajectories, Commun. Math. Phys., 113 (1987), 419-469.
doi: 10.1007/BF01221255. |
[6] |
I. Ekeland and J. M. Lasry,
On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Annals of Math., 112 (1980), 283-319.
doi: 10.2307/1971148. |
[7] |
I. Ekeland and L. Lassoued,
Multiplicite des trajectoires fermees d'un systeme hamiltonien sur une hypersurface d'energie convexe, Ann. IHP. Anal. Non Linéaire, 4 (1987), 307-335.
doi: 10.1016/S0294-1449(16)30362-6. |
[8] |
E. R. Fadell and P. H. Rabinowitz,
Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Inv. Math., 45 (1978), 139-174.
doi: 10.1007/BF01390270. |
[9] |
X. Hu and S. Sun,
Index and stability of symmetric periodic orbits in Hamiltonian systems with its application to figure-eight orbit, Commun. Math. Phys., 290 (2009), 737-777.
doi: 10.1007/s00220-009-0860-y. |
[10] |
C. Liu, Y. Long and C. Zhu,
Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbb{R}^2n$, Math. Ann., 323 (2002), 201-215.
doi: 10.1007/s002089100257. |
[11] |
C. Liu and S. Tang,
Maslov $(P, \omega)$-index theory for symplectic paths, Adv. Nonlinear Studies, 15 (2015), 963-990.
doi: 10.1515/ans-2015-0412. |
[12] |
C. Liu and D. Zhang,
Iteration theory of L-index and multiplicity of brake orbits, J. Diff. Equ., 257 (2014), 1194-1245.
doi: 10.1016/j.jde.2014.05.006. |
[13] |
H. Liu,
Multiple $P$-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $\mathbb{R}^2n$, Cal. Variations and PDEs, 49 (2014), 1121-1147.
doi: 10.1007/s00526-013-0614-8. |
[14] |
Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, No. 207, Birkhauser, Basel, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[15] |
Y. Long and C. Zhu,
Maslov-type index theorey for symplectic paths and spectral flow Ⅱ, Chinese. Ann. Math. Ser. B, 21 (2000), 89-108.
doi: 10.1142/S0252959900000133. |
[16] |
Y. Long and C. Zhu,
Closed characteristics on compact convex hypersurfaces in $\mathbb{R}^2n$, Ann. Math., 155 (2002), 317-368.
doi: 10.2307/3062120. |
[17] |
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. |
[18] |
P. H. Rabinowitz,
Peroidic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
[19] |
P. H. Rabinowitz,
On the existence of periodic solutions for a class of symmetric Hamiltonian system, Nonlinear Anal., 11 (1987), 599-611.
doi: 10.1016/0362-546X(87)90075-7. |
[20] |
J. Robbin and D. Salamon,
The maslov index for paths, Topology, 32 (1993), 827-844.
doi: 10.1016/0040-9383(93)90052-W. |
[21] |
A. Szulkin,
Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France, 116 (1988), 171-197.
doi: 10.24033/bsmf.2094. |
[22] |
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. |
[23] |
W. Wang,
Closed characteristics on compact convex hypersurfaces in $\mathbb{R}^8$, Adv. Math., 297 (2016), 93-148.
doi: 10.1016/j.aim.2016.03.044. |
[24] |
W. Wang, X. Hu and Y. Long,
Resonance identity, stability and multiplicity of closed characteristics on the conpact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462.
doi: 10.1215/S0012-7094-07-13931-0. |
[25] |
A. Weinstein,
Periodic orbits for convex Hamiltonian systems, Ann. Math., 108 (1978), 507-518.
doi: 10.2307/1971185. |
[26] |
D. Zhang,
P-cyclic symmetric closed characteristics on compact convex P-cyclic symmetric hypersurface in $\mathbb{R}^2n$, Discrete Continuous Dynam. Systems, 33 (2013), 947-964.
doi: 10.3934/dcds.2013.33.947. |
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