P-cyclic symmetric closed characteristics on compact convex P-cyclic symmetric hypersurface in <I>R</I><SUP><I>2n</I></SUP>

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February 2013, 33(2): 947-964. doi: 10.3934/dcds.2013.33.947

$P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R²ⁿ

Duanzhi Zhang ^1,

1.	School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received July 2011 Revised November 2011 Published September 2012

Abstract
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Let Σ be a $C^2$ compact strictly convex hypersurface in R²ⁿ with $n\ge 2$. Suppose $PΣ=Σ$ with $P$ being a $2n\times 2n$ symplectic and orthogonal matrix and $P^r=I_{2n}$. We prove that there are at least two geometrically distinct $P$-cyclic symmetric closed characteristics $(\tau_j,x_j)$ on Σ in the sense that $x_j(t+\frac{\tau_j}{r})=Px_j(t)$ for all $t∈R$ with $j=1,2$. As a corollary we obtain the existence of two geometrically distinct central symmetric closed characteristics on any $C^2$ central symmetric compact convex hypersurface in R²ⁿ with $n\ge 2$.

Keywords: dual variation., Maslov-type index, convex Hamiltonian systems, closed characteristics, $P$-cyclic symmetric.

Mathematics Subject Classification: Primary: 58E05, 70H05; Secondary: 34C2.

Citation: Duanzhi Zhang. $P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R²ⁿ. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 947-964. doi: 10.3934/dcds.2013.33.947

References:

[1]	S. E. Cappell, R. Lee and E. Y. Miller, On the Maslov-type index, Comm. Pure Appl. Math, 47 (1994), 121-186.
[2]	C. Conley and E. Zehnder, Maslov-type index theory for flows and periodix solutions for Hamiltonian equations, Commu. Pure. Appl. Math, 37 (1984), 207-253. doi: 10.1002/cpa.3160370204.
[3]	Y. Dong and Y. Long, Closed characteristics on partially symmetric compact convex hyper surfaces in R²ⁿ, J. Differential Equations, 196 (2004), 226-248. doi: 10.1016/S0022-0396(03)00168-2.
[4]	Y. Dong, P-index theory for linear Hamiltonian systems and multiple solutions for nonlinear Hamiltonian systems, Nonlinearity, 19 (2006), 1275-1294. doi: 10.1088/0951-7715/19/6/004.
[5]	J. J. Duistermaat, "Fourier Integral Operators," Birkhäauser, Basel, 1996.
[6]	I. Ekeland, "Convexity Methods in Hamiltonian Mechanics," Spring-Verlag, Berlin, 1990.
[7]	I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and there closed trajectories, Comm. Math. Phys., 113 (1987), 419-467. doi: 10.1007/BF01221255.
[8]	I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées de systèmes hamiltoniens convexes, Ann. Inst. H. Poincaré Anal. NonLinéaire, 4 (1987), 307-335.
[9]	E. Fadell and P. H. Rabinowitz, Generalized cohomologyical index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Ivent. Math, 45 (1978), 139-174.
[10]	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.
[11]	C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hyper surfaces in R²ⁿ, Math. Ann., 323 (2002), 201-215.
[12]	C. Liu and D. Zhang, Iteration theory of L-index and Multiplicity of brake orbits, arXiv:0908.0021.
[13]	H. Liu, P-invariant closed characteristics on partially symmetric compact convex hypersurfsces in R²ⁿ. preprint
[14]	Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Advances in Math, 203 (2006), 568-635.
[15]	Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math, 187 (1999), 113-149.
[16]	Y. Long, "Index Theory for Symplectic Paths with Applications," Birkhäuser, Basel, 2002.
[17]	Y. Long and C. Zhu, Maslov-type index theory for symplectic paths and spectral flow(II), Chinese Ann. of Math, 21 (2000), 89-108. doi: 10.1142/S0252959900000133.
[18]	Y. Long and C. Zhu, Closed characteristics on compact convex hyper surface in R²ⁿ, Ann. of Math, 155 (2002), 317-368.
[19]	P. H. Rabinowitz, Periodic solution of Hamiltonian systems, Commu. Pure Appl. Math, 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.
[20]	J. Robbin and D. Salamon, The Maslov indices for paths, Topology, 32 (1993), 827-844. doi: 10.1016/0040-9383(93)90052-W.
[21]	J. Robbin and D. Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc., 27 (1995), 1-33. doi: 10.1112/blms/27.1.1.
[22]	A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France, 116 (1988), 171-197.
[23]	K. Uhlenbeck, The Morse index theorem in Hilbert space, J. Differential Geom., 8 (1973), 555-564.
[24]	W. Wang, X. Hu and Y. Long, Resonance identity, stability, and multiplicity of closed characteristics on compact convex hyper surfaces, 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. of Math, 108 (1978), 507-518. doi: 10.2307/1971185.
[26]	D. Zhang, Multiple symmetric brake orbits in bounded convex symmetric domains, Advanced Nonl. Studies, 6 (2006), 643-653.
[27]	C. Zhu and Y. Long, Maslov index theory for symplectic paths and spectral flow(I), Chinese Ann. of Math, 208 (1999), 413-424. doi: 10.1142/S0252959999000485.

show all references

References:

[1]	S. E. Cappell, R. Lee and E. Y. Miller, On the Maslov-type index, Comm. Pure Appl. Math, 47 (1994), 121-186.
[2]	C. Conley and E. Zehnder, Maslov-type index theory for flows and periodix solutions for Hamiltonian equations, Commu. Pure. Appl. Math, 37 (1984), 207-253. doi: 10.1002/cpa.3160370204.
[3]	Y. Dong and Y. Long, Closed characteristics on partially symmetric compact convex hyper surfaces in R²ⁿ, J. Differential Equations, 196 (2004), 226-248. doi: 10.1016/S0022-0396(03)00168-2.
[4]	Y. Dong, P-index theory for linear Hamiltonian systems and multiple solutions for nonlinear Hamiltonian systems, Nonlinearity, 19 (2006), 1275-1294. doi: 10.1088/0951-7715/19/6/004.
[5]	J. J. Duistermaat, "Fourier Integral Operators," Birkhäauser, Basel, 1996.
[6]	I. Ekeland, "Convexity Methods in Hamiltonian Mechanics," Spring-Verlag, Berlin, 1990.
[7]	I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and there closed trajectories, Comm. Math. Phys., 113 (1987), 419-467. doi: 10.1007/BF01221255.
[8]	I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées de systèmes hamiltoniens convexes, Ann. Inst. H. Poincaré Anal. NonLinéaire, 4 (1987), 307-335.
[9]	E. Fadell and P. H. Rabinowitz, Generalized cohomologyical index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Ivent. Math, 45 (1978), 139-174.
[10]	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.
[11]	C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hyper surfaces in R²ⁿ, Math. Ann., 323 (2002), 201-215.
[12]	C. Liu and D. Zhang, Iteration theory of L-index and Multiplicity of brake orbits, arXiv:0908.0021.
[13]	H. Liu, P-invariant closed characteristics on partially symmetric compact convex hypersurfsces in R²ⁿ. preprint
[14]	Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Advances in Math, 203 (2006), 568-635.
[15]	Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math, 187 (1999), 113-149.
[16]	Y. Long, "Index Theory for Symplectic Paths with Applications," Birkhäuser, Basel, 2002.
[17]	Y. Long and C. Zhu, Maslov-type index theory for symplectic paths and spectral flow(II), Chinese Ann. of Math, 21 (2000), 89-108. doi: 10.1142/S0252959900000133.
[18]	Y. Long and C. Zhu, Closed characteristics on compact convex hyper surface in R²ⁿ, Ann. of Math, 155 (2002), 317-368.
[19]	P. H. Rabinowitz, Periodic solution of Hamiltonian systems, Commu. Pure Appl. Math, 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.
[20]	J. Robbin and D. Salamon, The Maslov indices for paths, Topology, 32 (1993), 827-844. doi: 10.1016/0040-9383(93)90052-W.
[21]	J. Robbin and D. Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc., 27 (1995), 1-33. doi: 10.1112/blms/27.1.1.
[22]	A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France, 116 (1988), 171-197.
[23]	K. Uhlenbeck, The Morse index theorem in Hilbert space, J. Differential Geom., 8 (1973), 555-564.
[24]	W. Wang, X. Hu and Y. Long, Resonance identity, stability, and multiplicity of closed characteristics on compact convex hyper surfaces, 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. of Math, 108 (1978), 507-518. doi: 10.2307/1971185.
[26]	D. Zhang, Multiple symmetric brake orbits in bounded convex symmetric domains, Advanced Nonl. Studies, 6 (2006), 643-653.
[27]	C. Zhu and Y. Long, Maslov index theory for symplectic paths and spectral flow(I), Chinese Ann. of Math, 208 (1999), 413-424. doi: 10.1142/S0252959999000485.

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