Nonuniformly hyperbolic K-systems are Bernoulli

We prove that those non-uniformly hyperbolic maps and flows (with singularities) that enjoy the K-property are also Bernoulli. In particular, many billiard systems, including those systems of hard balls and stadia that have the K-property, and hyperbolic billiards, such as the Lorentz gas in any dimension, are Bernoulli. We obtain the Bernoulli property for both the billiard flows and the associated maps on the boundary of the phase space.

[1]  Ja B Pesin.  GEODESIC FLOWS ON CLOSED RIEMANNIAN MANIFOLDS WITHOUT FOCAL POINTS , 1977 .

[2]  D. Ornstein.  Bernoulli shifts with the same entropy are isomorphic , 1970 .

[3]  Anatole Katok,et al.  Invariant Manifolds, Entropy and Billiards: Smooth Maps With Singularities , 1986 .

[4]  Yitzhak Katzenlson.  Ergodic automorphisms ofTn are Bernoulli shifts , 1971 .

[5]  Y. Pesin.  CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY , 1977 .

[6]  V. I. Oseledec.  A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[7]  Imbedding Bernoulli shifts in flows , 1970 .

[8]  D. Ornstein.  Two Bernoulli shifts with infinite entropy are isomorphic , 1970 .

[9]  R. Markarian.  Billiards with Pesin region of measure one , 1988 .

[10]  L. Bunimovich,et al.  Markov Partitions for dispersed billiards , 1980 .

[11]  Maciej P. Wojtkowski,et al.  Principles for the design of billiards with nonvanishing Lyapunov exponents , 1986, Hamiltonian Dynamical Systems.

[12]  N. Chernov,et al.  Limit theorems and Markov approximations for chaotic dynamical systems , 1995 .

[13]  Rufus Bowen,et al.  Bernoulli maps of the interval , 1977 .

[14]  Leonid A. Bunimovich,et al.  Statistical properties of two-dimensional hyperbolic billiards , 1991 .

[15]  Halim Doss.  Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques , 1980 .

[16]  D. Ornstein,et al.  Statistical properties of chaotic systems , 1991 .

[17]  D. Szász.  Ergodicity of classical billiard balls , 1993 .

[18]  Haruo Totoki.  On a class of special flows , 1970 .

[19]  F. Ledrappier.  Some properties of absolutely continuous invariant measures on an interval , 1981, Ergodic Theory and Dynamical Systems.

[20]  L. Sucheston.  Contributions to Ergodic Theory and Probability , 1970 .

[21]  Yakov G. Sinai,et al.  Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls , 1987 .

[22]  L. A. Bunimovic.  ON A CLASS OF SPECIAL FLOWS , 1974 .

[23]  Perturbed billiard systems. II. Bernoulli properties , 1981 .

[24]  M. Ratner,et al.  Anosov flows with gibbs measures are also Bernoullian , 1974 .

[25]  S. Kakutani,et al.  Structure and continuity of measurable flows , 1942 .

[26]  Domokos Szász.  On theK-property of some planar hyperbolic billiards , 1992 .

[27]  M. Denker.  The central limit theorem for dynamical systems , 1989 .

[28]  Y. Sinai,et al.  Dynamical systems with elastic reflections , 1970 .

[29]  N. Chernov.  Statistical properties of the periodic Lorentz gas. Multidimensional case , 1994 .

[30]  N. Chernov.  New proof of Sinai's formula for the entropy of hyperbolic billiard systems. Application to Lorentz gases and Bunimovich stadiums , 1991 .

[31]  M. Ratner.  Bernoulliflows over maps of the interval , 1978 .

[32]  Benjamin Weiss,et al.  Geodesic flows are Bernoullian , 1973 .

[33]  β-automorphisms are Bernoulli shifts , 1973 .

[34]  Ergodic automorphisms of the infinite torus are bernoulli , 1974 .

[35]  Giovanni Gallavotti,et al.  Billiards and Bernoulli schemes , 1974 .

[36]  L. Bunimovich.  On the ergodic properties of nowhere dispersing billiards , 1979 .