Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T03:41:12.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Alpha-congruence for dispersive billiards

Published online by Cambridge University Press:  19 September 2008

Kari Eloranta
Kari Eloranta
Affiliation:
Institute of Mathematics, Helsinki University of Technology, 02150 Espoo, Finland
Get access Rights & Permissions [Opens in a new window]

Abstract

We show the stability in the sense of α-congruence of dispersive (Sinai) planar billiards that are Bernoulli flows. The perturbations are either billiards on slightly altered tables or geodesic flows on nearby manifolds.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 2 , June 1991 , pp. 309 - 331
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[AA]Arnold, V. I. & Avez, A.. Ergodic Problems of Classical Mechanics W. A. Benjamin (1968).Google Scholar
[BS]Bunimovich, L. A. & Sinai, Ya. G.. On a fundamental theorem in the theory of dispersive billiards. Math. USSR Sb. 19 (1973), 407–424.Google Scholar
[BSt]Bennetin, G. & Strelcyn, J-M.. Numerical experiments on the free motion of a point mass moving in a plane convex region: Stochastic transition and entropy. Phys. Rev. A 17 (1978), 773–785.Google Scholar
[Eb]Eberlein, P.. When is a geodesic flow of Anosov type I? J. Diff. Geom. 8 (1973), 437–463.Google Scholar
[El]Eloranta, K.. Alpha-congruence for Markov processes. Ann. Prob. 18(4) (1990), 1583–1601.CrossRefGoogle Scholar
[GO]Gallavotti, G. & Ornstein, D. S.. The billiard flow with a convex scatterer is Bernoulli. Comm. Math. Phys. 38 (1974), 83–101.CrossRefGoogle Scholar
[Ke]Keller, G.. Ergodizität und K-eigenshaft der Kollisions Transformation beim Sinai-Billiard mil Endlichem Horizont, Thesis, Mathematische Institut, University of Erlangen (1977).Google Scholar
[Kh]Khovanskii, A. N.. The Application of Continued Fractions. P. Noordhoff (1963).Google Scholar
[KS]Katok, A. & Strelcyn, J-M.. Invariant manifolds, entropy and billiards: Smooth maps with singularities. Lecture Notes in Mathematics 1222, Springer (1986).Google Scholar
[M]Manning, A.. Curvature bounds for the entropy of the geodesic flow on a surface. J. London Math. Soc. (2), 24 (1981), 351–7.Google Scholar
[O]Ornstein, D. S.. Ergodic Theory, Randomness and Dynamical Systems. Yale (1974).Google Scholar
[OW]Ornstein, D. S. & Weiss, B.. Statistical properties of chaotic systems. Bull. Amer. Math. Soc. 24 (1) (1991), 11–16.CrossRefGoogle Scholar
[P]Pesin, Ya. B.. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys 32(4) (1977), 55–114.CrossRefGoogle Scholar
[S1]Sinai, Ya. G.. Dynamical systems with elastic reflections. Russian Math. Surveys 25(2) (1970), 137–189.Google Scholar
[S2]Sinai, Ya. G.. Classical dynamical systems with countable Lebesque spectrum II. Amer. Math. Soc. Transl. (2) 68 (1968), 34–88.Google Scholar
[Sp]Spivak, M.. A Comprehensive Introduction to Differential Geometry, Vol. 3, Publish or Perish (1979).Google Scholar
[W]Wojtkowski, M.. Principles for the design of billiards with nonvanishing Lyapunov exponents. Comm. Math. Phys. 105 (1986), 391–414.Google Scholar