Mathematische Zeitschrift

, Volume 223, Issue 3, pp 397–409 | Cite as

On the existence and non-existence of closed trajectories for some Hamiltonian flows

  • Viktor L. Ginzburg


Vector Bundle Strong Magnetic Field Contact Type Geodesic Flow Closed Trajectory 
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  1. [Ar1]
    Arnold, V.I.: Some remarks on flows of line elements and frames. Soviet Math. Dokl.2, 562–564 (1961)Google Scholar
  2. [Ar2]
    Arnold, V.I.: First steps of symplectic topology, Russian Math. Surveys41(6), 1–21 (1986)CrossRefGoogle Scholar
  3. [BR]
    Banyaga, A., Rukimbira, Ph.: Weak stability of almost regular contact flows. PreprintGoogle Scholar
  4. [Be]
    Benci, V.: Periodic solutions of Lagrangian systems on a compact manifold. J. Diff. Eq.63, 135–161 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [CZ]
    Conley, C.C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold. Invent. Math.73, 33–49 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [F1]
    Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys.120, 575–611 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [Gi1]
    Ginzburg, V.L.: New generalizations of Poincaré’s geometric theorem. Funct. Anal. Appl.21(2), 100–106 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Gi2]
    Ginzburg, V.L.: On closed characteristics of 2-forms. Russian Math. Surveys43(2), 225–226 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [Gi3]
    Ginzburg, V.L.: On closed characteristics of 2-forms. PhD Thesis, Berkeley, 1990Google Scholar
  10. [Gi4]
    Ginzburg, V.L.: Accessible points and closed trajectories of mechanical systems. Preprint, 1993. To appear as an appendix to Global Analysis in Mathematical Physics by Yu. Gliklikh to be published by Springer-VerlagGoogle Scholar
  11. [Gi5]
    Ginzburg, V.L.: An embeddingS 2n−1→ℝ2n, 2n−1≧7, whose Hamiltonian flow has no periodic trajectories. IMRN 83–96 (1995) no. 2Google Scholar
  12. [Hed]
    Hedlund, G.A.: Fuchsian groups and transitive horocycles. Duke Math. J.2, 530–542 (1936)CrossRefMathSciNetGoogle Scholar
  13. [Her]
    Herman, M.-R.: Examples de flots hamiltoniens dont aucune perturbations en topologieC n’a d’orbites périodiques sur ouvert de surfaces d’énergies. C.R. Acad. Sci. Paris Sér. I Math.312, 989–994 (1991)zbMATHMathSciNetGoogle Scholar
  14. [HS]
    Hofer, H., Salamon, D.: Floer homology and Novikov rings, in the Floer Memorial Volume, H. Hofer, et al. (editors), Birkhäuser, Boston, 1995Google Scholar
  15. [HV]
    Hofer, H., Viterbo, C.: The Weinstein conjecture for cotangent bundles and related results. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)15 (1988) no 3 411–445 (1989)Google Scholar
  16. [HZ]
    Hofer, H., Zehnder, E.: Periodic solution on hypersurfaces and a result by C. Viterbo. Invent. Math.90, 1–9 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Ku]
    Kuperberg, G.: A volume-preserving counterexample to the Seifert conjecture. Comment. Math. Helvetici71, 70–97 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [MO]
    McCord, C., Oprea, J.: Rational Ljusternik-Schnirelman category and the Arnold’s conjecture for nilmanifolds. Topology32, 701–717 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  19. [McD]
    McDuff, D.: Symplectic manifolds with contact type boundaries. Invent. Math.103, 651–671 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [No]
    Novikov, S.P.: The Hamiltonian formalism and many-valued analogue of Morse theory. Russian Math. Surveys37(5), 1–56 (1982)zbMATHCrossRefGoogle Scholar
  21. [NT]
    Novikov, S.P., Taimanov, I.A.: Periodic extremals of many-valued or not-every-where-positive functionals. Sov. Math. PDokl.29(1), 18–20 (1984)zbMATHMathSciNetGoogle Scholar
  22. [Tab]
    Tabachnikov, S.L.: Characteristic classes of Lagrangian foliations. Funct. Anal. Appl.23(2), 162–163 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  23. [Tai]
    Taimanov, I.A.: Closed extremals on two-dimensional manifolds. Russian Math. Surveys,47(2), 163–211 (1992)CrossRefMathSciNetGoogle Scholar
  24. [Vi]
    Viterbo, C.: A proof of Weinstein’s conjecture in IR2n. Ann. Inst. H. Poincaré, Anal. Non Linéaire4, 337–356 (1987)zbMATHMathSciNetGoogle Scholar
  25. [We1]
    Weinstein, A.: On the hypothesis of Rabinowitz’ periodic orbit theorem. J. Diff. Eq.33, 353–358 (1979)zbMATHCrossRefGoogle Scholar
  26. [We2]
    Weinstein, A.:C 0-perturbation theorem for symplectic fixed points and Lagrangian intersections. Seminaire Sud-Rhodanien de géométrie, III (Lyon, 1983), Traveaux en cours, Paris, Hermann, 140–144 (1984)Google Scholar
  27. [Ze]
    Zehnder, E.: Remarks on periodic solutions on hypersurfaces, in Periodic solutions of hamiltonian systems and related topics by Rabinowitz et al., Reidel Publishing Co. 267–279 (1987)Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Viktor L. Ginzburg
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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