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June  2021, 41(6): 2635-3652. doi: 10.3934/dcds.2020378

Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $

Lei Liu 1,, and  Li Wu 1,

1. 

School of Mathematics Department, Shandong University, Jinan 250100, China

* Corresponding author: Lei Liu

Received  May 2020 Revised  September 2020 Published  June 2021 Early access  November 2020

Fund Project: The first author is supported by NSFC grant No.11425105

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.

Keywords: Compact convex hypersurfaces, P-symmetric closed characteristics, Iteration theory, Maslov-type index and Maslov index, Hamiltonian system.
Mathematics Subject Classification: Primary: 58E05, 70H12, 34C25.
Citation: Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2635-3652. doi: 10.3934/dcds.2020378
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

J. Robbin and D. Salamon, The maslov index for paths, Topology, 32 (1993), 827-844.  doi: 10.1016/0040-9383(93)90052-W.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

J. Robbin and D. Salamon, The maslov index for paths, Topology, 32 (1993), 827-844.  doi: 10.1016/0040-9383(93)90052-W.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

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