Geodesic rays, Busemann functions and monotone twist maps

  • Victor Bangert
Article

Abstract

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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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

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

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