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Communications in Mathematical Physics

, Volume 277, Issue 2, pp 305–321 | Cite as

Improved Estimates for Correlations in Billiards

  • N. ChernovEmail author
  • H. -K. Zhang
Article

Abstract

We consider several classes of chaotic billiards with slow (polynomial) mixing rates, which include Bunimovich’s stadium and dispersing billiards with cusps. In recent papers by Markarian and the present authors, estimates on the decay of correlations were obtained that were sub-optimal (they contained a redundant logarithmic factor). We sharpen those estimates by removing that factor.

Keywords

Unstable Manifold Conditional Measure Markov Partition Homogeneous Section Consecutive Collision 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bálint P. and Gouëzel S. (2006). Limit theorems in the stadium billiard. Commun. Math. Phys. 263: 461–512 zbMATHCrossRefGoogle Scholar
  2. 2.
    Benettin G. and Strelcyn J.-M. (1978). Numerical experiments on the free motion of a point in a plane convex region: stochastic transition and entropy. Phys. Rev. A 17: 773–785 CrossRefADSGoogle Scholar
  3. 3.
    Bunimovich L.A. (1974). On billiards close to dispersing. Math. USSR. Sb. 23: 45–67 zbMATHCrossRefGoogle Scholar
  4. 4.
    Bunimovich L.A. (1974). The ergodic properties of certain billiards. Funk. Anal. Prilozh. 8: 73–74 CrossRefGoogle Scholar
  5. 5.
    Bunimovich L.A. (1979). On ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65: 295–312 zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Bunimovich L.A., Sinai Ya.G. and Chernov N.I. (1990). Markov partitions for two-dimensional billiards. Russ. Math. Surv. 45(3): 105–152 zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bunimovich L.A., Sinai Ya.G. and Chernov N.I. (1991). Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46(4): 47–106 CrossRefMathSciNetGoogle Scholar
  8. 8.
    Chernov N. (1991). A new proof of Sinai’s formula for entropy of hyperbolic billiards. Its application to Lorentz gas and stadium. Funct. Anal. Appl. 25: 204–219 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chernov N. (1999). Decay of correlations and dispersing billiards. J. Stat. Phys. 94: 513–556 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chernov N.I. and Haskell C. (1996). Nonuniformly hyperbolic K-systems are Bernoulli. Ergod. Th. Dynam. Sys. 16: 19–44 zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Chernov N. and Zhang H.-K. (2005). Billiards with polynomial mixing rates. Nonlinearity 18: 1527–1553 zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Chernov, N., Markarian, R.: Chaotic Billiards, Mathematical Surveys and Monographs, 127, Providence, RI: Amer. Math. Soc., 2006Google Scholar
  13. 13.
    Chernov N. and Markarian R. (2007). Dispersing billiards with cusps: slow decay of correlations. Commun. Math. Phys. 270: 727–758 zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Dembo A. and Zeitouni O. (1998). Large Deviations, Techniques and Applications. Springer, NY zbMATHGoogle Scholar
  15. 15.
    Machta J. (1983). Power law decay of correlations in a billiard problem. J. Stat. Phys. 32: 555–564 CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Machta J. and Reinhold B. (1986). Decay of correlations in the regular Lorentz gas. J. Stat. Phys. 42: 949–959 CrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Markarian R. (2004). Billiards with polynomial decay of correlations. Ergod. Th. Dynam. Syst. 24: 177–197 zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Ornstein D. and Weiss B. (1998). On the Bernoulli nature of systems with some hyperbolic structure. Ergod. Th. Dynam. Sys. 18: 441–456 zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Reháček J. (1995). On the ergodicity of dispersing billiards. Rand. Comput. Dynam. 3: 35–55 zbMATHGoogle Scholar
  20. 20.
    Sinai Ya.G. (1970). Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv. 25: 137–189 MathSciNetzbMATHGoogle Scholar
  21. 21.
    Szász, D., Varjú, T.: Limit laws and recurrence for the planar Lorentz process with infinite horizon. Manuscript, available at http://www.ma.utexas.edu, archive 06-274, 2006
  22. 22.
    Vivaldi F., Casati G. and Guarneri I. (1983). Origin of long-time tails in strongly chaotic systems. Phys. Rev. Let. 51: 727–730 CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Young L.-S. (1998). Statistical properties of systems with some hyperbolicity including certain billiards. Ann. Math. 147: 585–650 zbMATHCrossRefGoogle Scholar
  24. 24.
    Young L.-S. (1999). Recurrence times and rates of mixing. Israel J. Math. 110: 153–188 zbMATHMathSciNetGoogle Scholar
  25. 25.
    Zaslavsky G. (1981). Stochastisity in quantum systems. Phys. Rep. 80: 157–250 CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Zheng W.-M. (1997). Symbolic dynamics of the stadium billiard. Phys. Rev. E. 56: 1556–1560 CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA
  2. 2.Department of Mathematics and PhysicsNorth China Electric Power UniversityBaodingChina

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