American Institute of Mathematical Sciences

February  2016, 36(2): 877-893. doi: 10.3934/dcds.2016.36.877

Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces

 1 Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China 2 School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071

Received  February 2014 Published  August 2015

In this paper, let $n\geq2$ be an integer, $P=diag(-I_{n-\kappa},I_\kappa,-I_{n-\kappa}, I_\kappa)$ for some integer $\kappa\in[0, n-1)$, and $\Sigma \subset {\bf R}^{2n}$ be a partially symmetric compact convex hypersurface, i.e., $x\in \Sigma$ implies $Px\in\Sigma$. We prove that if $\Sigma$ is $(r,R)$-pinched with $\frac{R}{r}<\sqrt{\frac{5}{3}}$, then $\Sigma$ carries at least two geometrically distinct P-symmetric closed characteristics which possess at least $2n-4\kappa$ Floquet multipliers on the unit circle of the complex plane.
Citation: Hui Liu, Duanzhi Zhang. Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 877-893. doi: 10.3934/dcds.2016.36.877
References:
 [1] Y. Dong and Y. Long, Closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, J. Differential Equations, 196 (2004), 226-248. doi: 10.1016/S0022-0396(03)00168-2. [2] Y. Dong and Y. Long, Stable closed characteristics on partially symmetric compact convex hypersurfaces, J. Differential Equations, 206 (2004), 265-279. doi: 10.1016/j.jde.2004.03.004. [3] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag. Berlin. 1990. doi: 10.1007/978-3-642-74331-3. [4] I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories, Comm. Math. Phys., 113 (1987), 419-469. doi: 10.1007/BF01221255. [5] I. Ekeland and J. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math., 112 (1980), 283-319. doi: 10.2307/1971148. [6] I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées d'un systéme hamiltonien sur une hypersurface d'energie convexe, Ann. IHP. Anal. non Linéaire., 4 (1987), 307-335. [7] E. Fadell and P. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation equations for Hamiltonian systems, Invent. Math., 45 (1978), 139-174. doi: 10.1007/BF01390270. [8] M. Girardi, Multiple orbits for Hamiltonian systems on starshaped ernergy surfaces with symmetry, Ann. IHP. Analyse non linéaire., 1 (1984), 285-294. [9] X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltoinan systems with application to figure-eight orbits, Commun. Math. Phys., 290 (2009), 737-777. doi: 10.1007/s00220-009-0860-y. [10] H. Liu, Stability of symmetric closed characteristics on symmetric compact convex hypersurfaces in $R^{2n}$ under a pinching condition, Acta Mathematica Sinica, English Series, 28 (2012), 885-900. doi: 10.1007/s10114-011-0494-9. [11] H. Liu, Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, Cal. Variations and PDEs, 49 (2014), 1121-1147. doi: 10.1007/s00526-013-0614-8. [12] H. Liu and D. Zhang, On the number of P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, Science China Mathematics, 58 (2015), 1771-1778. [13] C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $R^{2n}$, Math. Ann., 323 (2002), 201-215. doi: 10.1007/s002089100257. [14] Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkhäuser. Basel. 2002. doi: 10.1007/978-3-0348-8175-3. [15] Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $R^{2n}$, Ann. of Math., 155 (2002), 317-368. doi: 10.2307/3062120. [16] P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203. [17] A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France., 116 (1988), 171-197. [18] W. Wang, Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete Contin. Dyn. Syst., 32 (2012), 679-701. doi: 10.3934/dcds.2012.32.679. [19] A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518. doi: 10.2307/1971185. [20] W. Wang, X. Hu and Y. Long, Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462. doi: 10.1215/S0012-7094-07-13931-0. [21] D. Zhang, P-cyclic symmetric closed characteristics on compact convex P-cyclic symmetric hypersurface in $R^{2n}$, Discrete Contin. Dyn. Syst., 33 (2013), 947-964. doi: 10.3934/dcds.2013.33.947.

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References:
 [1] Y. Dong and Y. Long, Closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, J. Differential Equations, 196 (2004), 226-248. doi: 10.1016/S0022-0396(03)00168-2. [2] Y. Dong and Y. Long, Stable closed characteristics on partially symmetric compact convex hypersurfaces, J. Differential Equations, 206 (2004), 265-279. doi: 10.1016/j.jde.2004.03.004. [3] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag. Berlin. 1990. doi: 10.1007/978-3-642-74331-3. [4] I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories, Comm. Math. Phys., 113 (1987), 419-469. doi: 10.1007/BF01221255. [5] I. Ekeland and J. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math., 112 (1980), 283-319. doi: 10.2307/1971148. [6] I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées d'un systéme hamiltonien sur une hypersurface d'energie convexe, Ann. IHP. Anal. non Linéaire., 4 (1987), 307-335. [7] E. Fadell and P. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation equations for Hamiltonian systems, Invent. Math., 45 (1978), 139-174. doi: 10.1007/BF01390270. [8] M. Girardi, Multiple orbits for Hamiltonian systems on starshaped ernergy surfaces with symmetry, Ann. IHP. Analyse non linéaire., 1 (1984), 285-294. [9] X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltoinan systems with application to figure-eight orbits, Commun. Math. Phys., 290 (2009), 737-777. doi: 10.1007/s00220-009-0860-y. [10] H. Liu, Stability of symmetric closed characteristics on symmetric compact convex hypersurfaces in $R^{2n}$ under a pinching condition, Acta Mathematica Sinica, English Series, 28 (2012), 885-900. doi: 10.1007/s10114-011-0494-9. [11] H. Liu, Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, Cal. Variations and PDEs, 49 (2014), 1121-1147. doi: 10.1007/s00526-013-0614-8. [12] H. Liu and D. Zhang, On the number of P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, Science China Mathematics, 58 (2015), 1771-1778. [13] C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $R^{2n}$, Math. Ann., 323 (2002), 201-215. doi: 10.1007/s002089100257. [14] Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkhäuser. Basel. 2002. doi: 10.1007/978-3-0348-8175-3. [15] Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $R^{2n}$, Ann. of Math., 155 (2002), 317-368. doi: 10.2307/3062120. [16] P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203. [17] A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France., 116 (1988), 171-197. [18] W. Wang, Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete Contin. Dyn. Syst., 32 (2012), 679-701. doi: 10.3934/dcds.2012.32.679. [19] A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518. doi: 10.2307/1971185. [20] W. Wang, X. Hu and Y. Long, Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462. doi: 10.1215/S0012-7094-07-13931-0. [21] D. Zhang, P-cyclic symmetric closed characteristics on compact convex P-cyclic symmetric hypersurface in $R^{2n}$, Discrete Contin. Dyn. Syst., 33 (2013), 947-964. doi: 10.3934/dcds.2013.33.947.
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