Communications in Mathematical Physics

, Volume 146, Issue 2, pp 357–396 | Cite as

Ergodic systems ofn balls in a billiard table

  • Leonid Bunimovich
  • Carlangelo Liverani
  • Alessandro Pellegrinotti
  • Yurii Suhov


We consider the motion ofn balls in billiard tables of a special form and we prove that the resulting dynamical systems are ergodic on a constant energy surface; in fact, they enjoy theK-property. These are the first systems of interacting particles proven to be ergodic for an arbitrary number of particles.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [B] Boltzmann, L.: Lectures on gas theory (translation from german). Berkeley, CA: University of California Press, 1964Google Scholar
  2. [BG1] Burns, K., Gerber, M.: Continuous invariant cone families and ergodicity of flows in dimension three. Ergodic Th. Dynam. Syst.9, 19–25 (1989)Google Scholar
  3. [BG2] Burns, K., Gerber, M.: Real analytic Bernoulli geodesic flows onS 2. Ergodic Theoret. Dynam. Syst.9, 27–45 (1989)Google Scholar
  4. [BS] Bunimovich, L. A., Sinai, Ya. G.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys.78, 479–497 (1981)Google Scholar
  5. [Bu1] Bunimovich, L. A.: Decay of correlation in dynamical systems with chaotic behaviour. Sov. Phys. JETPh62, 1452–1471 (1985)Google Scholar
  6. [Bu2] Bunimovich, L.A.: A theorem on ergodicity of two-dimensional hyperbolic billiards. Commun. Math. Phys.130, 599–621 (1990)Google Scholar
  7. [Bu3] Bunimovich, L. A.: On billiards close to dispersing. Matem. Sbornik95, 40–73 (1974)Google Scholar
  8. [C] Chernov, N. I.: Ergodic Hamiltonian system of two particles in an external field. PreprintGoogle Scholar
  9. [DL] Donnay, V., Liverani, C.: Potentials on the two-torus for which the Hamiltonian flow is ergodic. Commun. Math. Phys.135, 267–302 (1991)CrossRefGoogle Scholar
  10. [E] Engelking, R.: Dimension theory. Amsterdam: North Holland 1978Google Scholar
  11. [Gi] Gibbs, J. W.: Elementary principles in statistical mechanics. New York, 1902Google Scholar
  12. [G] Gallavotti, G.: Lectures on the billiard. Lect. Notes in Phys. Vol.38, pp. 236–295. Berlin, Heidelberg, New York: Springer 1975Google Scholar
  13. [H] Hopf, E.: Statistik der Geodatischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. akad. wiss., Leipzig91, 261–304 (1939)Google Scholar
  14. [K] Katok, A.: Invariant cone families and stochastic properties of a smooth dynamical systems. PreprintGoogle Scholar
  15. [Kr] Krylov, N. S.: Works on the foundation of statistical physics. Princeton, NJ: Princeton University Press 1979Google Scholar
  16. [KS] Katok, A., Strelcyn, J. M. with collaboration of Ledrappier F. and Przytycki F.: Invariant manifolds, entropy and billiards, smooth maps with singularities. Lect. Notes in Math. Vol.1222. Berlin, Heidelberg, New York: Springer 1986Google Scholar
  17. [KSS1] Krámli, A., Simányi, N., Szász, D.: Three billiard balls on thev-dimensional torus is aK-flow. Ann. Math.133, 37–72 (1991)Google Scholar
  18. [KSS2] Krámli, A., Simányi, N., Szász, D.: TheK-property of four billiard balls. Commun. Math. Phys.144, 107–148 (1992)CrossRefGoogle Scholar
  19. [KSS3] Krámli, A., Simźnyi, N., Szász, D.: A “transversal” fundamental theorem for semidispersing billiards. Commun. Math. Phys.129, 535–560 (1990); Erratum. Commun. Math. Phys.138, 207–208 (1991)CrossRefGoogle Scholar
  20. [KSS4] Krámli, A., Simźnyi, N., Szász, D.: Ergodic properties of semi-dispersing billiards. I two cylindric scatterers in the 3-D torus. Nonlinearity2, 311–326 (1989)CrossRefGoogle Scholar
  21. [LW] Liverani, C., Wojtkowski, M. P.: Ergodicity of Hamiltonian systems. PreprintGoogle Scholar
  22. [O] Osceledec, V. I.: A multiplicative ergodic theorem: characteristic Lyapunov exponents of dynamical systems. Trans. Mosc. Math. Soc.19, 197–231 (1968)Google Scholar
  23. [P] Besin, Ya. B.: Lyapunov characteristic exponents and smooth ergodic theory. Russ. Math. Surv.32(4), 55–114 (1977)Google Scholar
  24. [SC] Sinai, Ya. G., Chernov, N. I.: Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls. Usp. Math. Nauk42, 153–174 (1987)Google Scholar
  25. [S1] Sinai, Ya. G.: On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics. Dokl. Akad. Nauk. SSSR153, 1261–1264 (1963)Google Scholar
  26. [S2] Sinai, Ya. G.: Dynamical systems with elastic reflections. Russ. Math. Surv.25(1), 137–189 (1970)Google Scholar
  27. [S3] Sinai, Ya. G.: The development of Krylov's ideas. In: Works on the foundations of statistical physics, by Krylov Princeton, NJ: Princeton University Press 1979, pp. 239–281Google Scholar
  28. [W1] Wojtkowski, M.: Invariant families of cones and Lyapunov exponents. Ergodic Theoret. Dynam. Syst.5, 145–161 (1985)Google Scholar
  29. [W2] Wojtkowski, M.: A systems of one dimensional balls with gravity. Commun. Math. Phys.126, 507–533 (1990)Google Scholar
  30. [W3] Wojtkowski, M.: The system of one dimensional balls in an external field II. Commun. Math. Phys.127, 425–432 (1990)CrossRefGoogle Scholar
  31. [W4] Wojtkowski, M.: Measure theoretic entropy of the system of hard spheres. Ergodic Theoret. Dynam. Syst.8, 133–153 (1988)Google Scholar
  32. [W5] Wojtkowski, M.: Systems of classical interacting particles with nonvanishing Lyapunov exponents. PreprintGoogle Scholar
  33. [W6] Wojtkowski, M.: Private communicationGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Leonid Bunimovich
    • 1
    • 2
  • Carlangelo Liverani
    • 3
  • Alessandro Pellegrinotti
    • 4
  • Yurii Suhov
    • 5
    • 6
  1. 1.Shirshov Institute of OceanologyRussian Academy of SciencesMoscowRussia
  2. 2.Fakultät für PhysikUniversität BielefeldBielefeld 1FRG
  3. 3.Mathematics DepartmentUniversity of Rome IIRomeItaly
  4. 4.Mathematics DepartmentUniversity of Rome IRomeItaly
  5. 5.Institute for Problems of Information TransmissionRussian Academy of SciencesMoscowRussia
  6. 6.Statistical Laboratory, DPMMSUniversity of CambridgeCambridgeEngland, UK

Personalised recommendations