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Communications in Mathematical Physics

, Volume 125, Issue 3, pp 439–457 | Cite as

Dispersing billiards without focal points on surfaces are ergodic

  • A. Krámli
  • N. Simányi
  • D. Szász
Article

Abstract

Billiards are considered on two-dimensional, smooth, compact Riemannian manifolds with dispersing scatterers. We prove that these billiards are ergodic if only Vetier's conditions for the absence of focal points hold.

Keywords

Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics 
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.

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References

  1. [Bu-Si (1973)]
    Bunimovich, L.A., Sinai, Ya.G.: On the fundamental theorem in the theory of dispersing billiards. Matem. Sbornik90, 407–423 (1973)Google Scholar
  2. [G (1975)]
    Gallavotti, G.: Lectures on the billiard. Lecture Notes in Physics, Vol. 38. Moser, J. (ed.) pp. 236–295. Berlin, Heidelberg, New York: SpringerGoogle Scholar
  3. [G-O (1974)]
    Gallavotti, G., Ornstein, D.: Billiards and Bernoulli Schemes. Commun. Math. Phys.38, 83–101 (1974)Google Scholar
  4. [K-S (1986)]
    Katok, A., Strelcyn, J.-M. (with the collaboration of Ledrappier, F. and Przytycki, F.): Invariant manifolds, entropy and billiards; smooth maps with singularities. Lecture Notes in Mathematics, Vol. 1222. (Berlin, Heidelberg, New York: Springer 1986)Google Scholar
  5. [P (1977)]
    Pesin, A.Ya.: Characteristic exponents of Lyapunov and smooth ergodic theory. Russ. Math. Surv.32, 55–112 (1977)Google Scholar
  6. [S (1970)]
    Sinai, Ya.G.: Dynamical systems with elastic reflections. Russ. Math. Surv.25, 137–189 (1970)Google Scholar
  7. [S (1979)]
    Sinai, Ya.G.: Ergodic properties of the Lorentz gas (in Russian). Funk. Anal. i. Prilož.13, 46–59 (1979)Google Scholar
  8. [V1 (1982)]
    Vetier, A.: Sinai billiard in potential field (construction of fibers). Révész, P. (ed.): Coll. Math. Soc. J. Bolyai36, 1079–1146 (1982)Google Scholar
  9. [V2 (1982)]
    Vetier, A.: Sinai billiard in potential field (absolute continuity). Proc. 3rd Pann. Symp. ed. J. Mogyoródi, I. Vincze, W. Wertz. pp. 341–351 (1982)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • A. Krámli
    • 1
  • N. Simányi
    • 2
  • D. Szász
    • 2
  1. 1.Computer and Automation InstituteHungarian Academy of SciencesBudapest
  2. 2.Mathematical Institute, HASBudapestHungary

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