A stretched exponential bound on time correlations for billiard flows

Journal of Statistical Physics

Volumn 127, Issue 1, 2007, Pages 21-50

  Chernov, N ^a  

^a UNIVERSITY OF ALABAMA AT BIRMINGHAM   (United States)

Author keywords

Decay of correlations; Lorentz gas; Sinai billiards

Indexed keywords

EID: 34247215546     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-007-9293-1     Document Type: Article

References (34)

1. R. Adler and B. Weiss, Entropy, a complete metric invariant for automorphisms of the torus. Proc. Nat. Acad. Sci. USA 57:1573-1576 (1967).
2. V. Baladi and B. Vallée, Exponential decay of correlations for surface semi-flows without finite Markov partitions. Proc. Amer. Math. Soc. 133:865-874 (2005).
3. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lect. Notes Math., vol. 470 (Springer-Verlag, Berlin, 1975).
4. R. Bowen, Markov partitions are not smooth. Proc. Amer. Math. Soc. 71:130-132 (1978).
5. L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Markov partitions for two-dimensional billiards. Russ. Math. Surv. 45:105-152 (1990).
6. L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46:47-106 (1991).
7. N. I. Chernov, G. L. Eyink, J. L. Lebowitz and Ya. G. Sinai, Steady-state electrical conduction in the periodic Lorentz gas. Comm. Math. Phys. 154:569-601 (1993).
8. N. Chernov, Limit theorems and Markov approximations for chaotic dynamical systems. Prob. Th. Rel. Fields 101:321-362 (1995).
9. N. Chernov, On statistical properties of chaotic dynamical systems. AMS transl., Ser. 2 171:57-71 (1995).
10. N. Chernov, Markov approximations and decay of correlations for Anosov flows. Ann. Math. 147:269-324 (1998).
11. N. Chernov, Decay of correlations and dispersing billiards. J. Statist. Phys. 94:513-556 (1999).
12. N. Chernov, Advanced statistical properties of dispersing billiards. J. Statist. Phys. 122:1061-1094 (2006).
13. N. Chernov and R. Markarian, Chaotic billiards. Mathematical Suveys and Monographs, AMS (2006).
14. N. Chernov and L.-S. Young, Decay of correlations of Lorentz gases and hard balls. In D. Sz asz, (Eds.), Hard Ball Systems and Lorentz Gas (Springer, Berlin, 2000), pp. 89-120.
15. N. Chernov and D. Dolgopyat, Brownian Brownian motion - I. Memoirs of AMS (to appear); available at http://www.math.uab.edu/chernov/papers/pubs. html
16. D. Dolgopyat, On decay of correlations in Anosov flows. Ann. Math. 147:357-390 (1998).
17. D. Dolgopyat, Prevalence of rapid mixing. (I) Ergod. Th. Dynam. Syst. 18:1097-1114 (1998);
18. (II) Ergod. Th. Dynam. Syst. 20:1045-1059 (2000).
19. G. Gallavotti and D. S. Ornstein, Billiards and bernoulli schemes. Commun. Math. Phys. 38:83-101 (1974).
20. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, 1995).
21. A. Katok and J.-M. Strelcyn, Invariant Manifolds, Entropy and Billiards: Smooth Maps with Singularities, Lect. Notes Math., vol. 1222 (Springer-Verlag, 1986).
22. I. Kubo and H. Murata, Perturbed billiard systems II, Bernoulli properties. Nagoya Math. J. 81:1-25 (1981).
23. C. Liverani, Flows, Random Perturbations and Rate of Mixing. Ergodic Theory Dyn. Syst. 18:1421-1446 (1998).
24. C. Liverani, On contact Anosov flows. Ann. Math. 159:1275-1312 (2004).
25. I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260:131-146 (2005).
26. I. Melbourne, Rapid decay of correlations for nonuniformly hyperbolic flows. Trans. Amer. Math. Soc. (to appear)
27. G. Nicolis, S. Martinez and E. Tirapegui, Finite coarse-graining and Chapman-Kolmogorov equation in conservative dynamical systems. Chaos, Solutions Fractals 1:25-37 (1991).
28. D. Ruelle, Thermodynamic Formalism (Addison-Wesley, Reading, Mass. 1978).
29. Ya. G. Sinai, Markov partitions and C-diffeomorphisms. Funct. Anal. Appl. 2:61-82 (1968).
30. Ya. G. Sinai, Construction of Markov partitions. Funct. Anal. Appl. 2:245-253 (1968).
31. Ya. G. Sinai, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv. 25:137-189 (1970).
32. Ya. G. Sinai, Gibbs measures in ergodic theory. Russ. Math. Surv. 27: 21-69 (1972).
33. L. Stojanov, Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows. Amer. J. Math. 123:715-759 (2001).
34. L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147:585-650 (1998).