Abstract
In this paper, let \(\Sigma \) be a reversible symmetric compact convex hypersurface in \(R^{2n}\). We proved that if \(\Sigma \) possesses exactly n brake orbits with \(n\ge 3\), there are at least \(n-2\) of them have irrational mean indices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cappel, S.E., Lee, R., Miller, E.Y.: On the maslov-type index. Comm. Pual Appl. Math. 47, 121–186 (1994)
Ekeland, I.: An index theory for periodic solutions of convex Hamiltonion systems. Proc. Symp. Pure Math. 45, 395–423 (1986)
Ekeland, I.: Convex Methods In Hamiltonian Mechnics. Springer-Verlag, Berlin (1990)
Ekeland, I., Hofer, H.: Convex Hamiltonian energy surfaces and their closed trajectories. Comm. Math. Phys. 113, 419–467 (1987)
Fadell, E., Rabinowitz, P.: Generalized cohomological index theories for Lie group actions with an applications to bifurcation questions for Hamiltonian systems. Invent. Math. 45, 139–174 (1978)
Giambò, R., Giannoni, F., Piccione, P.: Multiple brake orbits in \(m\)-dimensional discs. Calc. Var. Partial Differ. Equ. 54, 2553–2580 (2015)
Hu, X., Ou, Y.: Stability of closed characteristics on compact convex hypersurfaces in \({\bf R}^{2n}\). arXiv:1405.4057
Liu, C., Zhang, D.: Iteration theory of L-index and multiplicity of brake orbits. J. Diff. Equ. 257, 1194–1245 (2014)
Liu, C., Zhang, D.: Seifert conjecture in the even convex case. Comm. Pure Appl. Math. 67, 1563–1604 (2014)
Liu, H., Long, Y., Wang, W.: Non-hyperbolic closed characteristics on symmetric compact convex hypersurfaces in \({ R}^{2n}\). Adv. Non. 14, 531–546 (2014)
Long, Y.: Index theory for symplectic paths with application. Progress in Mathematics, vol. 207, Birkh\(\ddot{a}\)user Verlag (2002)
Long, Y., Zhang, D., Zhu, C.: Multiple brake orbits in bounded convex symmetric domains. Adv. Math. 203, 568–635 (2006)
Long, Y., Zhu, C.: Closed characteristics on compact convex hypersurfaces in \({ R}^{2n}\). Ann. Math. 155, 317–368 (2002)
Robin, J., Salamon, D.: The Maslov indices for paths. Topology 32, 827–844 (1993)
Rabinowitz, P.: On the existence of periodic solutions for a class of symmetric Hamiltonian systems. Nonlinear Anal. 11, 599–611 (1987)
Szulkin, A.: An index theory and existenceof multiple brake orbits for star-shaped Hamiltonion systems. Math. Ann. 283, 241–255 (1989)
Zhang, D.: \(P\)-cyclic symmetric closed characteristics on compact convex \(P\)-cyclic symmetric hypersurface in \({ R}^{2n}\). Discrete Contin. Dyn. Syst. 33, 947–964 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Paul Rabinowitz.
Partially supported by NSF of China (11422103, 11271200) and LPMC of Nankai University.
Rights and permissions
About this article
Cite this article
Fan, Z., Zhang, D. Stability of the brake orbits on reversible symmetric compact convex hypersurfaces in \(\mathbf{R}^{2n}\) . J. Fixed Point Theory Appl. 19, 503–527 (2017). https://doi.org/10.1007/s11784-016-0363-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-016-0363-3