Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces. (English) Zbl 1357.37083
Summary: So far, it is still unknown whether all the closed characteristics on a symmetric compact star-shaped hypersurface \(\Sigma\) in \(\mathbb{R}^{2n}\) are symmetric. In order to understand behaviors of such orbits, in this paper we establish first two new resonance identities for symmetric closed characteristics on symmetric compact star-shaped hypersurface \(\Sigma\) in \(\mathbb{R}^{2n}\) when there exist only finitely many geometrically distinct symmetric closed characteristics on \(\Sigma\), which extend the identity established by H. Liu and Y. Long [J. Differ. Equations 255, No. 9, 2952–2980 (2013; Zbl 1321.37060)] for symmetric strictly convex hypersurfaces. Then as an application of these identities and the identities established by H. Liu et al. [J. Funct. Anal. 266, No. 9, 5598–5638 (2014; Zbl 1347.37099)] for all closed characteristics on the same hypersurface, we prove that if there exist exactly two geometrically distinct closed characteristics on a symmetric compact star-shaped hypersuface in \(\mathbb{R}^4\), then both of them must be elliptic.
MSC:
| 37J45 | Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) |
| 34C25 | Periodic solutions to ordinary differential equations |
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
Keywords:
compact star-shaped hypersurface; closed characteristic; Hamiltonian systems; resonance identity; symmetric; stabilityReferences:
| [1] | Bangert, V., Long, Y.: The existence of two closed geodesics on every Finsler 2-sphere. Math. Ann. 346, 335-366 (2010) · Zbl 1187.53040 · doi:10.1007/s00208-009-0401-1 |
| [2] | Berestycki, H., Lasry, J.M., Mancini, G., Ruf, B.: Existence of multiple periodic orbits on starshaped Hamiltonian systems. Commun. Pure Appl. Math. 38, 253-289 (1985) · Zbl 0569.58027 · doi:10.1002/cpa.3160380302 |
| [3] | Cristofaro-Gardiner, D., Hutchings, M.: From one Reeb orbit to two. J. Differ. Geom. (to appear) arXiv:1202.4839v2 (2012) · Zbl 1338.53108 |
| [4] | Dell’Antoio, G., D’Onofrio, B., Ekeland, I.: Les systèmes hamiltoniens convexes et pairs ne sont pas ergodiques en général. C. R. Acad. Sci. Paris Sér. I(315), 1413-1415 (1992) · Zbl 0768.70014 |
| [5] | Ekeland, I.: An index theory for periodic solutions of convex Hamiltonian systems. Proc. Symp. Pure Math. 45, 395-423 (1986) · Zbl 0596.34023 · doi:10.1090/pspum/045.1/843575 |
| [6] | Ekeland, I.: Convexity Methods in Hamiltonian Mechanics. Springer-Verlag, Berlin (1990) · Zbl 0707.70003 · doi:10.1007/978-3-642-74331-3 |
| [7] | Ekeland, I., Hofer, H.: Convex Hamiltonian energy surfaces and their closed trajectories. Commun. Math. Phys. 113, 419-467 (1987) · Zbl 0641.58038 · doi:10.1007/BF01221255 |
| [8] | Ekeland, I., Lassoued, L.: Multiplicité des trajectoires fermées d’un systéme hamiltonien sur une hypersurface d’energie convexe. Ann. IHP. Anal. non Linéaire. 4, 1-29 (1987) |
| [9] | Ginzburg, V.L., Goren, Y.: Iterated index and the mean Euler characteristic. J. Topol. Anal. 7, 453-481 (2015) · Zbl 1325.53113 · doi:10.1142/S179352531550017X |
| [10] | Ginzburg, V.L., Hein, D., Hryniewicz, U.L., Macarini, L.: Closed Reeb orbits on the sphere and symplectically degenerate maxima. Acta Math. Vietnam. 38, 55-78 (2013) · Zbl 1292.53057 · doi:10.1007/s40306-012-0002-z |
| [11] | Girardi, M.: Multiple orbits for Hamiltonian systems on starshaped ernergy surfaces with symmetry. Ann. IHP. Analyse non linéaire. 1, 285-294 (1984) · Zbl 0582.70019 |
| [12] | Gromoll, D., Meyer, W.: On differentiable functions with isolated critical points. Topology 8, 361-369 (1969) · Zbl 0212.28903 · doi:10.1016/0040-9383(69)90022-6 |
| [13] | Hofer, H., Wysocki, K., Zehnder, E.: The dynamics on three-dimensional strictly convex energy surfaces. Ann. Math. 148, 197-289 (1998) · Zbl 0944.37031 · doi:10.2307/120994 |
| [14] | Hu, X., Long, Y.: Closed characteristics on non-degenerate star-shaped hypersurfaces in \[\mathbf{R}^{2n}\] R2n. Sci. China Ser. A 45, 1038-1052 (2002) · Zbl 1099.37047 · doi:10.1007/BF02879987 |
| [15] | Liu, H., Long, Y.: Resonance identity for symmetric closed characteristics on symmetric convex Hamiltonian energy hypersurfaces and its applications. J. Differ. Equ. 255, 2952-2980 (2013) · Zbl 1321.37060 · doi:10.1016/j.jde.2013.07.034 |
| [16] | Liu, H., Long, Y.: The existence of two closed characteristics on every compact star-shaped hypersurface in \[\mathbf{R}^4\] R4. Acta Math. Sin. (2014). doi:10.1007/s10114-014-4108-1 · Zbl 0582.70019 |
| [17] | Liu, H., Long, Y., Wang, W.: Resonance identities for closed characteristics on compact star-shaped hypersurfaces in \[\mathbf{R}^{2n}\] R2n. J. Funct. Anal. 166, 5598-5638 (2014) · Zbl 1347.37099 |
| [18] | Liu, H., Long, Y., Wang, W., Zhang, P.: Symmetric closed characteristics on symmetric compact convex hypersurfaces in \[\mathbf{R}^8\] R8. Commun. Math. Stat. 2, 393-411 (2014) · Zbl 1329.52004 · doi:10.1007/s40304-015-0047-0 |
| [19] | Liu, C., Long, Y., Zhu, C.: Multiplicity of closed characteristics on symmetric convex hypersurfaces in \[\mathbf{R}^{2n}\] R2n. Math. Ann. 323, 201-215 (2002) · Zbl 1005.37030 · doi:10.1007/s002089100257 |
| [20] | Long, Y.: Hyperbolic closed characteristics on compact convex smooth hypersurfaces in \[\mathbf{R}^{2n}\] R2n. J. Differ. Equ. 150, 227-249 (1998) · Zbl 0915.58032 · doi:10.1006/jdeq.1998.3486 |
| [21] | Long, Y.: Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics. Adv. Math. 154, 76-131 (2000) · Zbl 0970.37013 · doi:10.1006/aima.2000.1914 |
| [22] | Long, Y.: Index Theory for Symplectic Paths with Applications. Progress in Math. 207, Birkhäuser, Basel (2002) · Zbl 1012.37012 |
| [23] | Long, Y., Zhu, C.: Closed characteristics on compact convex hypersurfaces in \[\mathbf{R}^{2n}\] R2n. Ann. Math. 155, 317-368 (2002) · Zbl 1028.53003 · doi:10.2307/3062120 |
| [24] | Rabinowitz, P.: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math. 31, 157-184 (1978) · Zbl 0341.35051 · doi:10.1002/cpa.3160310203 |
| [25] | Rademacher, H.-B.: Morse Theorie und geschlossene Geodatische, Bonner Math. Schriften Nr. 229 (1992) · Zbl 0826.58012 |
| [26] | Szulkin, A.: Morse theory and existence of periodic solutions of convex Hamiltonian systems. Bull. Soc. Math. Fr. 116, 171-197 (1988) · Zbl 0669.58004 |
| [27] | Viterbo, C.: Une théorie de Morse pour les systèmes hamiltoniens étoilés. C. R. Acad. Sci. Paris Ser. I Math. 301, 487-489 (1985) · Zbl 0608.58037 |
| [28] | Viterbo, C.: Equivariant Morse theory for starshaped Hamiltonian systems. Trans. Am. Math. Soc. 311, 621-655 (1989) · Zbl 0676.58030 · doi:10.1090/S0002-9947-1989-0978370-5 |
| [29] | Wang, W.: Stability of closed characteristics on compact convex hypersurfaces in \[\mathbf{R}^6\] R6. J. Eur. Math. Soc. 11, 575-596 (2009) · Zbl 1167.52009 · doi:10.4171/JEMS/161 |
| [30] | Wang, W.: Existence of closed characteristics on compact convex hypersurfaces in \[{\bf R}^{2n}\] R2n. (2011) arXiv:1112.5501v3 |
| [31] | Wang, W.: Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete Continuous Dyn. Syst. 32(2), 679-701 (2012) · Zbl 1237.58015 · doi:10.3934/dcds.2012.32.679 |
| [32] | Wang, W., Hu, X., Long, Y.: Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces. Duke Math. J. 139, 411-462 (2007) · Zbl 1139.58007 · doi:10.1215/S0012-7094-07-13931-0 |
| [33] | Weinstein, A.: Periodic orbits for convex Hamiltonian systems. Ann. Math. 108, 507-518 (1978) · Zbl 0403.58001 · doi:10.2307/1971185 |
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