Mather Sets for Twist Maps and Geodesics on Tori

  • V. Bangert
Part of the Dynamics Reported book series (DYNAMICS, volume 1)

Abstract

The title refers to a theory which is based on independent research in three different fields—differential geometry, dynamical systems and solid state physics—and which has attracted growing interest and research activity in the last few years. The objects of this theory are respectively:
  1. (1)

    Geodesics on a 2-dimensional torus with Riemannian (or symmetric Finsler) metric.

     
  2. (2)

    The dynamics of monotone twist maps of an annulus.

     
  3. (3)

    The discrete Frenkel-Kontorova model.

     

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren der math. Wissenschaften 250. New York–HeidelbergBerlin: Springer, 1983.Google Scholar
  2. [2]
    V. I. Arnold and A. Avez, Problèmes Ergodiques de la Mécanique Classique. Paris: Gauthiers-Villars, 1967.Google Scholar
  3. [3]
    S. Aubry, The twist map, the extended Frenkel–Kontorova model and the devil’s staircase. Physica, 7D (1983), 240–58.MathSciNetGoogle Scholar
  4. [4]
    S. Aubry and P. Y. LeDaeron, The discrete Frenkel–Kontorova model and its extensions I: exact results for the ground states. Physica, 8D (1983), 381–422.MathSciNetGoogle Scholar
  5. [5]
    V. Bangert, A uniqueness theorem for za-periodic variational problems. Comment. Math. Helv., 62 (1987), 511–31.MathSciNetCrossRefGoogle Scholar
  6. [6]
    D. Bernstein, Birkhoff periodic points for twist maps with the graph intersection property. Erg. Th. Dynam. Sys., 5 (1985), 531–7.Google Scholar
  7. [7]
    D. Bernstein and A. Katok, Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians. Invent. math., 88 (1987), 225–41.MathSciNetCrossRefGoogle Scholar
  8. [8]
    G. D. Birkhoff, Surface transformations and their dynamical applications. Acta Math., 43 (1922), 1–119.Google Scholar
  9. [9]
    G. D. Birkhoff, Dynamical Systems. Am. Math. Soc. Colloq. Publ. IX. Providence RI: Am. Math. Soc., 1927.Google Scholar
  10. [10]
    G. D. Birkhoff, Sur quelques courbes fermées remarquables. Bull. Soc. Math. France, 60 (1932), 1–26.Google Scholar
  11. [11]
    J. S. Birman and C. Series, Geodesics with bounded intersection number on surfaces are sparsely distributed. Topology, 24 (1985), 217–25.MathSciNetCrossRefGoogle Scholar
  12. [12]
    P. L. Boyland and G. R. Hall, Invariant circles and the order structure of periodic orbits in monotone twist maps. Topology, 26 (1987), 21–35.MathSciNetCrossRefGoogle Scholar
  13. [13] H. Busemann, The Geometry of Geodesics. New York: Academic Press, 1955. [14]
    H. Busemann and F. P. Pedersen, Tori with one-parameter groups of motions. Math. Scand., 3 (1955), 209–20.MathSciNetGoogle Scholar
  14. [15]
    P. Buser, Riemannsche Flächen und Längenspektrum vom trigonometrischen Standpunkt aus. Habilitationsschrift. Bonn, 1980.Google Scholar
  15. [16]
    C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order. Part II: Calculus of Variations. San Francisco: Holden Day, 1967.Google Scholar
  16. [17]
    A Chenciner, La dynamique au voisinage d’un point fixe elliptic conservatif; de Poincaré et Birkhoff à Aubry et Mather. Sém. Bourbaki, Exposé 622, Vol. 1983,84. Astérisque, 121, 122 (1985), 147–70.Google Scholar
  17. [18]
    I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory. Grundlehren der math. Wissenschaften 245. New York–Heidelberg–Berlin: Springer, 1982.Google Scholar
  18. [19]
    A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore. J. de Math. Pure et Appl., sér. 9, 11 (1932), 333–75.Google Scholar
  19. [20]
    R. Douady, Applications du théorème des tores invariants. Thèse Sème Cycle. Université Paris V II, 1982.Google Scholar
  20. [21]
    B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry—Methods and Applications, Part IL Graduate Texts in Math. 104. New York: Springer, 1984.Google Scholar
  21. [22]
    A Duschek and W. Mayer, Lehrbuch der Differentialgeometrie. Leipzig–Berlin: Teubner, 1930.Google Scholar
  22. [23]
    M. Freedman, J. Hass and P. Scott, Closed geodesics on surfaces. Bull. London Math. Soc., 14 (1982), 385–91.CrossRefGoogle Scholar
  23. [24]
    M. Gromov, Structures métriques pour les variétés Riemanniennes. Rédigé par J. Lafontaine et P. Pansu. Paris: CEDIC, 1981.Google Scholar
  24. [25]
    G. R. Hall, A topological version of a theorem of Mather on twist maps. Erg. Th. Dynam. Sys., 4 (1984), 585_-603.Google Scholar
  25. [26]
    G. A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients. Ann. of Math., 33 (1932), 719–39.MathSciNetCrossRefGoogle Scholar
  26. [27]
    G. A. Hedlund, The dynamics of geodesic flows. Bull. Am. Math. Soc., 45 (1939), 241–60.CrossRefGoogle Scholar
  27. [28]
    R. Herman, Introduction à l’étude des courbes invariantes par les difféomorphismes de l’anneau. Astérisque, 103104 (1983).Google Scholar
  28. [29]
    E. Hopf, Statistik der Lösungen geodätischer Probleme vom instabilen Typus II. Math. Ann., 116 (1940), 590–608.Google Scholar
  29. [30]
    E. Hopf, Closed surfaces without conjugate points. Proc. Nat. Acad. Sci., 34 (1948), 47–51.CrossRefGoogle Scholar
  30. [31]
    A. Katok, Some remarks on the Birkhoff and Mather twist theorems. Erg. Th. Dynam. Sys., 2 (1982), 183–94.Google Scholar
  31. [32]
    B. F. Kimball, Geodesics on a toroid. Am. J. Math., 52 (1932), 29–52.Google Scholar
  32. [33]
    R. S. MacKay and J. Stark, Lectures on Orbits of Minimal Action for Area-Preserving Maps. Preprint, University of Warwick, May 1985.Google Scholar
  33. [34]
    R. S. MacKay and I. C. Percival, Converse KAM: theory and practice. Comm. Math. Phys., 98 (1985), 469–512.MathSciNetCrossRefGoogle Scholar
  34. [35]
    J. N. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology, 21 (1982), 457–67.MathSciNetCrossRefGoogle Scholar
  35. [36]
    J. N. Mather, Non=uniqueness of solutions of Percival’s Euler–Lagrange equation. Comm. Math. Phys., 86 (1982), 465–76.CrossRefGoogle Scholar
  36. [37]
    J. N. Mather, Glancing billiards. Erg. Th. Dynam. Sys., 2 (1982), 397–403.Google Scholar
  37. [38]
    J. N. Mather, A criterion for the non-existence of invariant circles. Publ. Math. IRES, 63 (1986), 153–204.Google Scholar
  38. [39]
    J. N. Mather, Non-existence of invariant circles. Erg. Th. Dynam. Sys., 4 (1984), 301–9.Google Scholar
  39. [40]
    J. N. Mather, More Denjoy minimal sets for area preserving diffeomorphisms. Comment. Math. Helv., 60 (1985), 508–57.CrossRefGoogle Scholar
  40. [41]
    J. N. Mather, Existence of asymptotic orbits for area-preserving monotone twist diffeomorphisms. Manuscript, 1985.Google Scholar
  41. [42]
    M. Morse, Recurrent geodesics on a surface of negative curvature. Trans. Am. Math. Soc., 22 (1921), 84–100.CrossRefGoogle Scholar
  42. [43]
    M. Morse, A fundamental class of geodesics on any closed surface of genus greater than one. Trans. Am. Math. Soc., 26 (1924), 25–60.CrossRefGoogle Scholar
  43. [44]
    J. Moser, Stable and Random Motions in Dynamical Systems. Ann. of Math. Studies 77. Princeton NJ: Princeton Univ. Press, 1973.Google Scholar
  44. [45]
    J. Moser, Monotone twist mappings and the calculus of variations. Erg. Th. Dynam. Sys., 6 (1986), 325–33.Google Scholar
  45. [46]
    J. Moser, Break-down of Stability. In J. M. Jowett, M. Month, S. Turner (eds), Nonlinear Dynamics Aspects of Particle Accelerators. Lect. Notes in Physics 247, 492–518. Berlin–New York: Springer, 1986.Google Scholar
  46. [47]
    J. Moser, Minimal solutions of variational problems on a torus. Ann. Inst. Henri Poincaré—Analyse non linéaire, 3 (1986), 229–72.Google Scholar
  47. [48]
    J. Moser, Recent developments in the theory of Hamiltonian systems. SIAM Review, 28 (1986), 459–85.MathSciNetCrossRefGoogle Scholar
  48. [49]
    I. C. Percival, Variational principles for invariant tori and cantori. In M. Month, J. C. Herrara (eds), Non-Linear Dynamics and the Beam-Beam Interaction. Am. Inst. Phys. Conf. Proc., 57 (1980), 310–20.Google Scholar
  49. [50]
    J. Stillwell, Classical Topology and Combinatorial Group Theory. Graduate Texts in Math. 72. New York: Springer, 1980.Google Scholar
  50. [51]
    E. M. Zaustinsky, Extremals on compact E-surfaces. Trans. Am. Math. Soc., 102 (1962), 433–45.Google Scholar

Copyright information

© John Wiley & Sons and B. G. Teubner 1988

Authors and Affiliations

  • V. Bangert
    • 1
  1. 1.Mathematisches InstitutUniversität BernSwitzerland

Personalised recommendations