Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T11:36:36.875Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonuniformly hyperbolic K-systems are Bernoulli

Published online by Cambridge University Press:  19 September 2008

N. I. Chernov  and
C. Haskell
N. I. Chernov
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA
C. Haskell
Affiliation:
State University of New York at Stony Brook, Stony Brook, New York 11794-3660, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 1 , February 1996 , pp. 19 - 44
Copyright
Copyright © Cambridge University Press 1996

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

[1]Ambrose, W. and Kakutani, S.. Structure and continuity of measurable flows. Duke Math. J. (9) (1942), 25–12.CrossRefGoogle Scholar
[2]Bowen, R.. Bernoulli equilibrium states for Axiom A diffeomorphisms. Math. Sys. Theory 8 (1975), 289–294.CrossRefGoogle Scholar
[3]Bowen, R.. Bernoulli maps of the interval. Israel J. Math. (28) (1977), 161–168.CrossRefGoogle Scholar
[4]Bunimovich, L. A.. On a class of special flows. Math. USSR Izy. (8) (1974), 219–232.CrossRefGoogle Scholar
[5]Bunimovich, L. A.. On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. (65) (1979), 295–312.CrossRefGoogle Scholar
[6]Bunimovich, L. A. and Sinai, Ya. G.. Markov partitions for dispersed billiards. Commun. Math. Phys. (73) (1980), 247–280.CrossRefGoogle Scholar
[7]Bunimovich, L. A., Sinai, Ya. G. and Chernov, N. I.. Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. (46) (1991), 47–106.CrossRefGoogle Scholar
[8]Chernov, N. I.. A new proof of the Sinai formula for computing the entropy of hyperbolic billiards. Applications to a Lorentz gas and to the Bunimovich stadium. Funct. Anal. Appl. 25 (1991), 204–219.CrossRefGoogle Scholar
[9]Chernov, N. I.. Statistical properties of the periodic Lorentz gas. Multidimensional case. J. Stat. Phys. 74 (1994), 11–53.CrossRefGoogle Scholar
[10]Chernov, N. I.. Limit theorems and Markov approximations for chaotic dynamical systems. Prob. Theory Rel. Fields 101 (1995), 321–362.CrossRefGoogle Scholar
[11]Denker, M.. The central limit theorem for dynamical systems. Dyn. Syst. Ergod. Th., Banach Center Publ. 23, Warsaw: PWN-Polish Sci. Publ., 1989.Google Scholar
[12]Gallavotti, G. and Ornstein, D. S.. Billiards and Bernoulli schemes. Commun. Math. Phys. 38 (1974), 83–101.CrossRefGoogle Scholar
[13]Katznelson, Y.. Ergodic automorphisms of Tn are Bernoulli shifts. Israel J. Math. 10 (1971), 186–195.CrossRefGoogle Scholar
[14]Kubo, I. and Murata, H.. Perturbed billiard systems II, Bernoulli properties. Nagoya Math. J. 81 (1981), 1–25.CrossRefGoogle Scholar
[15]Katok, A. and Strelcyn, J.-M.. Invariant Manifolds, Entropy and Billiards: Smooth Maps with Singularities, Springer Lecture Notes in Mathematics 1222. Springer-Verlag, New York, 1986.Google Scholar
[16]Ledrappier, F.. Some properties of absolutely continuous invariant measures on an interval. Ergod. Th. & Dynam. Sys. 1 (1981), 77–94.CrossRefGoogle Scholar
[17]Lind, D. A.. Ergodic automorphisms of the infinite torus are Bernoulli. Israel J. Math. 17, (1974), 162–168.CrossRefGoogle Scholar
[18]Markarian, R.. Billiards with Pesin region of measure one. Commun. Math. Phys. 118 (1988), 87–97.CrossRefGoogle Scholar
[19]Ornstein, D. S.. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 (1970), 337–352.CrossRefGoogle Scholar
[20]Ornstein, D. S.. Bernoulli shifts with infinite entropy are isomorphic. Adv. Math. 5 (1971), 339–483.CrossRefGoogle Scholar
[21]Ornstein, D. S.. Imbedding Bernoulli shifts in flows. Contributions to Ergodic Theory and Probability, Springer Lecture Notes in Mathematics 160, pp. 178–218, Springer-Verlag, New York, 1970.CrossRefGoogle Scholar
[22]Ornstein, D. S. and Weiss, B.. Geodesic flows are Bernoullian. Israel J. Math. 14 (1973), 184–198.CrossRefGoogle Scholar
[23]Ornstein, D. S. and Weiss, B.. Statistical properties of chaotic systems. Bull. Amer. Math. Soc. 24 (1991), 11–116.CrossRefGoogle Scholar
[24]Oseledec, V.. The multiplicative ergodic theorem, the Lyapunov characteristic numbers of dynamical systems. Trans. Most: Math. Soc. 19 (1968), 197–231.Google Scholar
[25]Pesin, Ya. B.. Geodesic flows on closed Riemannian manifolds without focal points. Math. USSR Izv. 11 (1977), 1195–1228.CrossRefGoogle Scholar
[26]Pesin, Ya. B.. Characteristic Lyapunov exponents and smooth ergodic theory. Runs. Math. Surv. 32:4 (1977), 55–114.CrossRefGoogle Scholar
[27]Pesin, Ya. B. and Sinai, Ya. G.. Hyperbolicity and stochasticity of dynamical systems. Math. Phys. Rev. 2, 53–115, Soviet Sci. Reviews, Harwood Acad. Pub., 1981.Google Scholar
[28]Ratner, M.. Anosov flows with Gibbs measures are also Bernoullian. lsrael J. Math. 17 (1974), 380–391.CrossRefGoogle Scholar
[29]Ratner, M.. Bernoulli flows over maps of the interval. Israel J. Math. 31 (1978), 298–314.CrossRefGoogle Scholar
[30]Sinai, Ya. G.. Dynamical systems with elastic reflections. Ergodic properties of scattering billiards. Russ. Math. Surv. 25 (1970), 137–189.CrossRefGoogle Scholar
[31]Sinai, Ya. G. and Chernov, N. I.. Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls. Russ. Math. Surv. 42 (1987), 181–207.CrossRefGoogle Scholar
[32]Smorodinsky, M.. β-automorphisms are Bernoulli shifts. Acta Math. Acad. Sci. Hungar. 24 (1973), 273–278.CrossRefGoogle Scholar
[33]Szasz, D.. On the K-property of some planar hyperbolic billiards. Commun. Math. Phys. 145 (1992), 595–604.CrossRefGoogle Scholar
[34]Szasz, D.. Ergodicity of classical billiard balls. Phys. A 194 (1993), 86–92.CrossRefGoogle Scholar
[35]Wojtkowski, M.. Principles for the design of billiards with nonvanishing Lyapunov exponents. Commun. Math. Phys. 105 (1986), 391–414.CrossRefGoogle Scholar