On smooth Hamiltonian flows limited to ergodic billiards

  • Dmitry TuraevEmail author
  • Vered Rom-Kedar
Dynamics And Chaos
Part of the Lecture Notes in Physics book series (LNP, volume 511)


Sufficient conditions are found so that a family of smooth Hamiltonian flows limits to a billiard flow as a parameter ɛ→0. This limit is proved to be C 1 near non-singular orbits and C 0 near orbits tangent to the billiard boundary. These results are used to prove that scattering (thus ergodic) billiards with tangent periodic orbits or tangent homoclinic orbits produce nearby Hamiltonian flows with elliptic islands. This implies that ergodicity may be lost for smooth potentials which are arbitrarily close to ergodic billiards. Thus, in some cases, anomoulous transport associated with stickiness to stability islands is expected


58F15 82C05 34C37 58F05 58F13 58F14 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Bunimovich and Y. Sinai. The fundamental theorem of the theory of scattering billiards. (Russian) Mat. Sb. (N.S.), 90(132):415–431, 1973.Google Scholar
  2. 2.
    L. Bunimovich and Y. Sinai. Markov partitions for dispersed billiards. Comm. Math. Phys., 78(2):247–280, 1980/81.zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    L. Bunimovich, Y. Sinai, and N. Chernov. Markov partitions for two-dimensional hyperbolic billiards. Russian Math. Surveys, 45(3):105–152, 1990. Translation of Uspekhi Mat. Nauk, 45, 3(273), 97–134, 221.zbMATHCrossRefMathSciNetADSGoogle Scholar
  4. 4.
    L. Bunimovich, Y. Sinai, and N. Chernov. Statistical properties of two-dimensional hyperbolic billiards. Uspekhi Mat. Nauk, 46(4(280)):43–92, 192, 1991. In Russian. Translation in Russian Math. Surveys 46(4) (1991) 47–106.zbMATHMathSciNetGoogle Scholar
  5. 5.
    L. A. Bunimovich. Decay of correlations in dynamical systems with chaotic behavior. Zh. Eksp. Teor. Fiz, 89:1452–1471, 1985. In Russian. Translation in Sov Phys. JETP 62 (4), 842–852.Google Scholar
  6. 6.
    P. Constantin, E. Grossman, and M. Mungan. Inelastic collisions of three particles on a line as a two-dimensional billiard. Physica D, 83(4):409–420, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    I. P. Cornfeld, S. V. Fomin, and Y. G. Sinai. Ergodic theory. Number 245 in Fundamental Principles of Mathematical Scienc. Springer-Verlag, New York-Berlin, 1982. Translated from the Russian by A. B. Sosinskiĭ.zbMATHGoogle Scholar
  8. 8.
    A. Delshams and R. Ramirez-Ros. Poincaré-melnikov-arnold method for analytic planar maps. Nonlinearity, 9:1–26, 1996.zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    V. Donnay. Elliptic islands in generalized sinai billiards. Ergod. Th. & Dynam. Sys., 16:975–1010, 1996.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    DSTOOLS. Computer program. Cornell university, center of applied mathematics, Ithaca, NY 14853.Google Scholar
  11. 11.
    G. Gallavotti and D. Ornstein. Billiards and Bernoulli schemes. Comm. Math. Phys., 38:83–101, 1974.zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    N. Gavrilov and L. Shilnikov. On three dimensional dynamical systems close to systems with a structurally unstable homoclinic curve i. Math. USSR Sb., 88(4):467–485, 1972.CrossRefGoogle Scholar
  13. 13.
    S. V. Gonchenko, L. P. Shilnikov, and D. V. Turaev. Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits. Chaos, 6(1):15–31, 1996.zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    M. Gutzwiller. Chaos in Classical and Quantum mechanic. Springer-Verlag, New York, NY, 1990.Google Scholar
  15. 15.
    K. Hansen. Bifurcations and complete chaos for the diamagnetic kepler problem. Phys. Rev. E., 51(3):1838–1844, 1995.CrossRefADSGoogle Scholar
  16. 16.
    M. Jakobson. Absolutely continuous invariant measures for one parameter families of one-dimensional maps. Comm. Math. Phys., 81:39–88, 1981.zbMATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Katok, A.B. and Strelcyn, J.M. and Ledrappier, F. and F. Przytycki. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, volume 1222. Springer-Verlag, 1986.Google Scholar
  18. 18.
    L.P. Shilnikov. On a Poincaré-Birkhoff problem. Math. USSR Sbornik, 74, 1967.Google Scholar
  19. 19.
    J. Marsden. Generalized Hamiltonian mechanics; a mathematical exposition of non-smooth dynamical systems and classical Hamiltonian mechanic. Arch. for Rational Mech. and Anal., 28(5):323–361, 1968.zbMATHADSMathSciNetGoogle Scholar
  20. 20.
    S. Newhouse. Quasi-elliptic periodic points in conservative dynamical systems. Amer. J. Of Math., 99(5):1061–1087, 1977.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Y. Sinai. Dynamical systems with elastic reflections: Ergodic properties of scattering billiards. Russian Math. Sur., 25(1):137–189, 1970.CrossRefzbMATHADSGoogle Scholar
  22. 22.
    Y. Sinai and N. Chernov. Ergodic properties of some systems of two-dimensional disks and three-dimensional balls. Uspekhi Mat. Nauk, 42(3(255)):153–174, 256, 1987. In Russian.MathSciNetGoogle Scholar
  23. 23.
    S. Smale. Diffeomorphisms with many periodic points. In C. Cairns, editor, Differential and combinatorial topology, pages 63–80. Princeton University Press, Princeton, 1963.Google Scholar
  24. 24.
    D. Szász. Boltzmann's ergodic hypothesis, a conjecture for centuries? Studia Sci. Math. Hungar., 31(1–3):299–322, 1996.zbMATHMathSciNetGoogle Scholar
  25. 25.
    D. Turaev and V. Rom-Kedar. Islands appearing in near-ergodic flows. Nonlinearity, 11(3):575–600, 1998. to appear.zbMATHCrossRefADSMathSciNetGoogle Scholar
  26. 26.
    M. Wojtkowski. Principles for the design of billiards with nonvanishing lyapunov exponents. Comm. Math. Phys., 105(3):391–414, 1986.zbMATHCrossRefADSMathSciNetGoogle Scholar
  27. 27.
    G. Zaslavsky and M. Edelman. Maxwell's demon as a dynamical model. Phys. Rev. E, 56(5):5310–5320, 1997.CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  1. 1.The Department of Applied Mathematics and Computer ScienceThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations