Geodesic rays, Busemann functions and monotone twist maps

  • Victor Bangert


We study the action-minimizing half-orbits of an area-preserving monotone twist map of an annulus. We show that these so-called rays are always asymptotic to action-minimizing orbits. In the spirit of Aubry-Mather theory which analyses the set of action-minimizing orbits we investigate existence and properties of rays. By analogy with the geometry of the geodesics on a Riemannian 2-torus we define a Busemann function for every ray. We use this concept to prove that the minimal average action A(α) is differentiable at irrational rotation numbersα while it is generically non-differentiable at rational rotation numbers (cf. also [18]). As an application of our results in the geometric framework we prove that a Riemannian 2-torus which has the same marked length spectrum as a flat 2-torus is actually isometric to this flat torus.

Mathematics subject classification

58E10 58F05 53C22 


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  1. 1.
    Aubry, S.: The devil's staircase transformation in incommensurable lattices. Lect. Notes Math., Vol. 925, pp. 221–245) Berlin, Heidelberg, New York: Springer 1982Google Scholar
  2. 2.
    Ballmann, W., Gromov, M., Schroeder, V.: Manifolds of nonpositive curvature. Basel-Boston: Birkhäuser 1985Google Scholar
  3. 3.
    Bangert, V.: Mather sets for twist maps and geodesics on tori. In: Kirchgraber, U., Walther, H.O. (eds.) Dynamics reported, Vol. 1, pp. 1–56. Stuttgart-Chichester: B.G. Teubner-John Wiley 1988Google Scholar
  4. 4.
    Bangert, V.: Minimal geodesics. Ergodic Theory Dyn. Syst.10, 263–286 (1990)Google Scholar
  5. 5.
    Busemann, H.: The geometry of geodesics. New York: Academic Press 1955Google Scholar
  6. 6.
    Bylayi, M.L., Polterovich, L.V.: Geodesic flow on the two-dimensional torus and phase transitions “commensurability-noncommensurability”. Funct. Anal. Appl.20, 260–266 (1986)Google Scholar
  7. 7.
    Croke, C.: Rigidity for surfaces of non-positive curvature. Comment. Math. Helv.65, 150–169 (1990)Google Scholar
  8. 8.
    Croke, C., Fathi, A., Feldman, J.: The marked length-spectrum of a surface of nonpositive curvature. Topology31, 847–855 (1992)Google Scholar
  9. 9.
    Hedlund, G.A.: Geodesics on a two-dimensional Riemannian manifold with periodic coefficients. Ann. Math.33, 719–739 (1932)Google Scholar
  10. 10.
    Hopf, E.: Closed surfaces without conjugate points. Proc. Natl. Acad. Sci.34, 47–51 (1948)Google Scholar
  11. 11.
    Innami, N.: Differentiability of Busemann functions and total excess. Math. Z.180, 235–247 (1982)Google Scholar
  12. 12.
    Innami, N.: On the terminal points of co-rays and rays. Arch. Math.45, 468–470 (1985)Google Scholar
  13. 13.
    Innami, N.: Families of geodesics which distinguish flat tori. Math. J. Okayama Univ.28, 207217 (1986)Google Scholar
  14. 14.
    Katok, A.: Four applications of conformal equivalence to geometry and dynamics. Ergodic Theory Dyn. Syst.8, 129–152 (1988)Google Scholar
  15. 15.
    Mather, J.N.: More Denjoy minimal sets for area preserving diffeomorphismus. Comment. Math. Helv.60, 508–557 (1985)Google Scholar
  16. 16.
    Mather, J.N.: Modulus of continuity for Peierls' barrier. In: Rabinowitz, P.H., et al. (eds.) Periodic solutions of Hamiltonian systems and related topics. (NATO ASI Series C, Vol. 209, pp. 177–202) Dordrecht: D. Reidel 1987Google Scholar
  17. 17.
    Mather, J.N.: Minimal measures. Comment. Math. Helv.64, 375–394 (1989)Google Scholar
  18. 18.
    Mather, J.N.: Differentiability of the minimal average action as a function of the rotation number. Bol. Soc. Bras. Mat.21, 59–70 (1990)Google Scholar
  19. 19.
    Morse, M.: A fundamental class of geodesics on any closed surface of genus greater than one. Trans. Am. Math. Soc.26, 25–60 (1924)Google Scholar
  20. 20.
    Otal, J.-P.: Le spectre marqué des longueurs des surfaces à courbure négative. Ann. Math.131, 151–162 (1990)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Victor Bangert
    • 1
  1. 1.Mathematisches InstitutUniversität FreiburgFreiburgGermany

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