The existence of two closed characteristics on every compact star-shaped hypersurface in 鈩?sup class="a-plus-plus">4

详细信息    查看全文

• 作者：Hui Liu ; Yi Ming Long
• 关键词：Compact star ; shaped hypersurface ; closed characteristic ; Hamiltonian systems ; resonance identity ; multiplicity ; 58E05 ; 37J45 ; 34C25
• 刊名：Acta Mathematica Sinica
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：32
• 期：1
• 页码：40-53
• 全文大小：247 KB
• 参考文献：[1]Bourgeois, F.: Introduction to contact homology. Lecture Notes, available at http://鈥媓omepages.鈥媣ub.鈥媋c.鈥?be/fbourgeo/
  [2]Cristofaro-Gardiner, D., Hutchings, M.: From one Reeb orbit to two, arXiv:1202.4839 (2012)
  [3]Ekeland, I.: Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990CrossRef MATH
  [4]Ekeland, I., Lassoued, L.: Multiplicit茅 des trajectoires ferm茅es d鈥檜n syst茅me hamiltonien sur une hypersurface d鈥檈nergie convexe. Ann. IHP. Anal. Non Lin茅aire, 4, 1鈥?9 (1987)MathSciNet
  [5]Ekeland, I., Hofer, H.: Convex Hamiltonian energy surfaces and their closed trajectories. Comm. Math. Phys., 113, 419鈥?67(1987)CrossRef MathSciNet MATH
  [6]Ginzburg, V. L., Hein, D., Hryniewicz, U. L., et al.: Closed Reeb orbits on the sphere and symplectically degenerate maxima. Acta Math. Vietnam., 38, 55鈥?8 (2013)CrossRef MathSciNet MATH
  [7]Gromoll, D., Meyer, W.: On differentiable functions with isolated critical points. Topology, 8, 361鈥?69 (1969)CrossRef MathSciNet MATH
  [8]Hofer, H., Wysocki, K., Zehnder, E.: The dynamics on three-dimensional strictly convex energy surfaces. Ann. of Math., 148, 197鈥?89 (1998)CrossRef MathSciNet MATH
  [9]Hryniewicz, U., Macarini, L.: Local contact homology and applications, arXiv:1202.3122 (2012)
  [10]Hu, X., Long, Y.: Closed characteristics on non-degenerate star-shaped hypersurfaces in 鈩?sup>2n . Sci. China Ser. A, 45, 1038鈥?052 (2002)CrossRef MathSciNet MATH
  [11]Liu, H., Long, Y., Wang, W.: Resonance identities for closed characteristics on compact star-shaped hypersurfaces in 鈩?sup>2n . J. Funct. Anal., 266, 5598鈥?638 (2014)CrossRef MathSciNet MATH
  [12]Long, Y.: Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics. Adv. Math., 154, 76鈥?31 (2000)CrossRef MathSciNet MATH
  [13]Long, Y.: Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkh盲user, Basel, 2002CrossRef
  [14]Long, Y.: Resonance identities and closed characteristics on compact star-shaped hypersurfaces in 鈩?sup>2n , Lecture at the Institute for Advanced Study, October 11, 2011
  [15]Long, Y., Zhu, C.: Closed characteristics on compact convex hypersurfaces in 鈩?sup>2n . Ann. of Math., 155, 317鈥?68 (2002)CrossRef MathSciNet MATH
  [16]Rabinowitz, P.: Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math., 31, 157鈥?84 (1978)CrossRef MathSciNet
  [17]Szulkin, A.: Morse theory and existence of periodic solutions of convex Hamiltonian systems. Bull. Soc. Math. France., 116, 171鈥?97 (1988)MathSciNet MATH
  [18]Viterbo, C.: Une th茅orie de Morse pour les systemes hamiltoniens 茅toil茅s. C. R. Math. Acad. Sci. Paris, Ser. I., 301, 487鈥?89 (1985)MathSciNet MATH
  [19]Viterbo, C.: Equivariant Morse theory for starshaped Hamiltonian systems. Trans. Amer. Math. Soc., 311, 621鈥?55 (1989)CrossRef MathSciNet MATH
  [20]Wang, W., Hu, X., Long, Y.: Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces. Duke Math. J., 139, 411鈥?62 (2007)CrossRef MathSciNet MATH
  [21]Wang, W.: Existence of closed characteristics on compact convex hypersurfaces in 鈩?sup>2n , arXiv:1112.5501, (2011)
  [22]Weinstein, A.: Periodic orbits for convex Hamiltonian systems. Ann. of Math., 108, 507鈥?18 (1978)CrossRef MathSciNet MATH
  
• 作者单位：Hui Liu (1)
  Yi Ming Long (2)
  
  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P. R. China
  2. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, P. R. China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Chinese Library of Science
  
• 出版者：Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, co-published
• ISSN：1439-7617

文摘

Recently, Cristofaro-Gardiner and Hutchings proved that there exist at least two closed characteristics on every compact star-shaped hypersuface in 鈩?sup>4. Then Ginzburg, Hein, Hryniewicz, and Macarini gave this result a second proof. In this paper, we give it a third proof by using index iteration theory, resonance identities of closed characteristics and a remarkable theorem of Ginzburg et al. Keywords Compact star-shaped hypersurface closed characteristic Hamiltonian systems resonance identity multiplicity