Advertisement

Communications in Mathematical Physics

, Volume 126, Issue 1, pp 13–24 | Cite as

Aubry-Mather sets and Birkhoff's theorem for geodesic flows on the two-dimensional torus

  • M. L. Bialy
Article

Abstract

In this paper we state the graph property for incompressible continuouse tori invariant under goedesic flows of Riemannian metrics on the two-dimensional torus. Also our method gives a new proof of Birkhoff's theorem for twist maps of the cylinder. We prove that if there exist an invariant incompressible torus of geodesic flow with irrational rotation number then it necessarily contains the Aubry-Mather set with this rotation number.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ar] Arnold, V.I.: Mathematical methods of classical mechanics. Berlin, Heidelberg, New York: Springer c 1978Google Scholar
  2. [A] Aubry, S., Le Daeron, P.Y.: The discrete Frencel-Kontorova model and its extension. Physica8D, 381–422 (1983)Google Scholar
  3. [Ba] Bangert, V.: Mather sets for twist maps and geodesics on tori. Dynamics reported 1 (1987)Google Scholar
  4. [Be] Beck, A.: Continuous flows in the plane. Berlin, Heidelberg, New York: Springer 1974Google Scholar
  5. [B] Bialy, M.: On the number of caustics for invariant tori of hamiltonian systems with two degrees of freedom. To appear in Ergodic Theorem Dyna. Syst.Google Scholar
  6. [B-P1] Bialy, M.L., Polterovich, L.V.: Geodesic flows on the two-dimensional torus and phase transitions “commensurability-noncommensurability”. Funct Anal. Appl20, 260–266 (1986)Google Scholar
  7. [B-P2] Bialy, M.L., Polterovich, L.V.: Lagrangian singularities of invariant tori of hamiltonian systems with two degrees of freedom. To appear in Inv. Math. 1989Google Scholar
  8. [C-L] Coddington, E., Levinson, N.: Theory of ordinary differential equation. New York: McGraw-Hill 1955.Google Scholar
  9. [E] Edwards, C.H.: Concentric solid tori in the 3-sphere. Trans. Am. Math. Soc.102, 1–17 (1962)Google Scholar
  10. [Hed] Hedlund, G.: Geodesics on a two-dimensional Riemannian manifold with periodic coefficients. Ann. Math. Ser. 233, 719–739 (1932)Google Scholar
  11. [H1] Herman, M.: Sur les courbes invariantes par les diffeomorphisms de l'anneau. Asterisque103–104 (1983)Google Scholar
  12. [H2] Herman, M.: Existence et non existence de tores invariants par des diffeomorphismes symplectiques. Preprint (1988)Google Scholar
  13. [M] Mather, J.: A criterion for the non-existence of invariant circles. Publ. Math. I.H.E.S.63, 153–204 (1986)Google Scholar
  14. [Mu] Munkres, J.: Obstructions to the smoothing of piecewise-differentiable homeomorphisms. Ann. Math. Ser. 2,72, 521–554 (1960)Google Scholar
  15. [Ru] Rushing, B.: Topological Embeddings New York, London: Academic Press 1973Google Scholar
  16. [St] Sternberg, Sh.: Lectures on differential geometry. New Jersey: Prentice-Hall 1964Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • M. L. Bialy
    • 1
  1. 1.Department of Theoretical MathematicsThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations