Multiple brake orbits on compact convex symmetric reversible hypersurfaces in 𝐑 2n
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 3, pp. 531-554.

In this paper, we prove that there exist at least [n+1 2]+1 geometrically distinct brake orbits on every C 2 compact convex symmetric hypersurface Σ in 𝐑 2n for n2 satisfying the reversible condition NΣ=Σ with N= diag (-I n ,I n ). As a consequence, we show that there exist at least [n+1 2]+1 geometrically distinct brake orbits in every bounded convex symmetric domain in 𝐑 n with n2 which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for n=3. As an application, for n=4and5, we prove that if there are exactly n geometrically distinct closed characteristics on Σ, then all of them are symmetric brake orbits after suitable time translation.

DOI : 10.1016/j.anihpc.2013.03.010
Classification : 58E05, 70H05, 34C25
Mots clés : Brake orbit, Maslov-type index, Seifert conjecture, Convex symmetric
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     title = {Multiple brake orbits on compact convex symmetric reversible hypersurfaces in $ {\mathbf{R}}^{2n}$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     publisher = {Elsevier},
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Zhang, Duanzhi; Liu, Chungen. Multiple brake orbits on compact convex symmetric reversible hypersurfaces in $ {\mathbf{R}}^{2n}$. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 3, pp. 531-554. doi : 10.1016/j.anihpc.2013.03.010. http://www.numdam.org/articles/10.1016/j.anihpc.2013.03.010/

[1] A. Ambrosetti, V. Benci, Y. Long, A note on the existence of multiple brake orbits, Nonlinear Anal. 21 (1993), 643 -649 | MR | Zbl

[2] V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 401 -412 | EuDML | Numdam | MR | Zbl

[3] V. Benci, F. Giannoni, A new proof of the existence of a brake orbit, Advanced Topics in the Theory of Dynamical Systems, Notes Rep. Math. Sci. Eng. vol. 6 (1989), 37 -49 | Zbl

[4] S. Bolotin, Libration motions of natural dynamical systems, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1978), 72 -77 | MR | Zbl

[5] S. Bolotin, V.V. Kozlov, Librations with many degrees of freedom, J. Appl. Math. Mech. 42 (1978), 245 -250 | MR

[6] S.E. Cappell, R. Lee, E.Y. Miller, On the Maslov-type index, Comm. Pure Appl. Math. 47 (1994), 121 -186 | MR | Zbl

[7] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin (1990) | MR | Zbl

[8] H. Gluck, W. Ziller, Existence of periodic solutions of conservative systems, Seminar on Minimal Submanifolds, Princeton University Press (1983), 65 -98 | MR

[9] E.W.C. Van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy, J. Math. Anal. Appl. 132 (1988), 1 -12 | MR | Zbl

[10] K. Hayashi, Periodic solution of classical Hamiltonian systems, Tokyo J. Math. 6 (1983), 473 -486 | MR | Zbl

[11] C. Liu, Maslov-type index theory for symplectic paths with Lagrangian boundary conditions, Adv. Nonlinear Stud. 7 no. 1 (2007), 131 -161 | MR | Zbl

[12] C. Liu, Asymptotically linear Hamiltonian systems with Lagrangian boundary conditions, Pacific J. Math. 232 no. 1 (2007), 233 -255 | MR | Zbl

[13] C. Liu, Y. Long, C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in 𝐑 2n , Math. Ann. 323 no. 2 (2002), 201 -215 | MR | Zbl

[14] C. Liu, D. Zhang, Iteration theory of L-index and multiplicity of brake orbits, arXiv:0908.0021v1 [math.SG] | MR | Zbl

[15] Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999), 113 -149 | MR | Zbl

[16] Y. Long, Index Theory for Symplectic Paths with Applications, Birkhäuser, Basel (2002) | MR | Zbl

[17] Y. Long, D. Zhang, C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Adv. Math. 203 (2006), 568 -635 | MR | Zbl

[18] Y. Long, C. Zhu, Closed characteristics on compact convex hypersurfaces in 𝐑 2n , Ann. of Math. 155 (2002), 317 -368 | MR | Zbl

[19] P.H. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems, Nonlinear Anal. 11 (1987), 599 -611 | MR

[20] H. Seifert, Periodische Bewegungen mechanischer Systeme, Math. Z. 51 (1948), 197 -216 | EuDML | MR | Zbl

[21] A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann. 283 (1989), 241 -255 | EuDML | MR | Zbl

[22] D. Zhang, Brake type closed characteristics on reversible compact convex hypersurfaces in 𝐑 2n , Nonlinear Anal. 74 (2011), 3149 -3158 | MR | Zbl

[23] D. Zhang, Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems, Discrete Contin. Dyn. Syst. (2013), arXiv:1110.6915v1 [math.SG] | MR

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