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Communications in Mathematical Physics

, Volume 260, Issue 1, pp 131–146 | Cite as

Almost Sure Invariance Principle for Nonuniformly Hyperbolic Systems

  • Ian Melbourne
  • Matthew Nicol
Article

Abstract

We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon.

Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.

Keywords

Neural Network Discrete Time Limit Theorem Statistical Limit General Classis 
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.
    Aaronson, J.: An Introduction to Infinite Ergodic Theory. Math. Surveys and Monographs 50, Providence, RI: Amer. Math. Soc., 1997Google Scholar
  2. 2.
    Aaronson, J., Denker, M.: Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps. Stoch. Dyn. 1, 193–237 (2001)CrossRefGoogle Scholar
  3. 3.
    Alves, J., Luzzatto, S., Pinheiro, V.: Markov structures and decay of correlations for non-uniformly expanding dynamical systems. Ann. Inst. H. Poincaré, Anal. Non Linéaire, to appearGoogle Scholar
  4. 4.
    Baladi, V.: Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics 16, Singapore: World Scientific, 2000Google Scholar
  5. 5.
    Baladi, V.: Decay of correlations. In: Katok, A. (ed.) et al., Smooth Ergodicity Theory and its Applications Proc. Symp. Pure Math. 69, Providence, RI: Amer. Math. Soc., 2001, pp. 297–325Google Scholar
  6. 6.
    Benedicks, M., Young, L.-S.: Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Ergod. Th. & Dynam. Sys. 12, 13–37 (1992)Google Scholar
  7. 7.
    Benedicks, M., Young, L.-S.: Sinai-Bowen-Ruelle measures for certain Hénon maps. Invent. Math. 112, 541–576 (1993)CrossRefGoogle Scholar
  8. 8.
    Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math. 470, Berlin: Springer, 1975Google Scholar
  9. 9.
    Bruin, H., Holland, M., Nicol, M.: Livsic regularity for Markov systems. Ergod. Th. and Dyn. Syst. To appearGoogle Scholar
  10. 10.
    Bruin, H., Luzzatto, S., van Strien, S.: Decay of correlations in one-dimensional dynamics. Ann. Sci. École Norm. Sup. 36, 621–646 (2003)Google Scholar
  11. 11.
    Bunimovich, L.A., Sinai, Y.G., Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Uspekhi Mat. Nauk 46, 43–92 (1991)Google Scholar
  12. 12.
    Castro, A.: Backward inducing and exponential decay of correlations for partially hyperbolic attractors with mostly contracting direction. Ph. D. Thesis, IMPA (1998)Google Scholar
  13. 13.
    Chernov, N.: Statistical properties of piecewise smooth hyperbolic systems in high dimensions. Discrete Contin. Dynam. Systems 5, 425–448 (1999)Google Scholar
  14. 14.
    Chernov, N.: Decay of correlations and dispersing billiards. J. Stat. Phys. 94, 513–556 (1999)CrossRefGoogle Scholar
  15. 15.
    Chernov, N., Young, L.S.: Decay of correlations for Lorentz gases and hard balls. In: Hard ball systems and the Lorentz gas. Encyclopaedia Math. Sci. 101, Berlin: Springer, 2000, pp. 89–120Google Scholar
  16. 16.
    Conze, J.-P., Le Borgne, S.: Méthode de martingales et flow géodésique sur une surface de courbure constante négative. Ergod. Th. & Dyn. Sys. 21, 421–441 (2001)Google Scholar
  17. 17.
    Denker, M., Philipp, W.: Approximation by Brownian motion for Gibbs measures and flows under a function. Ergod. Th. & Dyn. Sys. 4, 541–552 (1984)Google Scholar
  18. 18.
    Dolgopyat, D.: On dynamics of mostly contracting diffeomorphisms. Commun. Math. Phys. 213, 181–201 (2000)CrossRefGoogle Scholar
  19. 19.
    Field, M.J., Melbourne, I., Török, A.: Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions. Ergod. Th. & Dyn. Sys. 23, 87–110 (2003)Google Scholar
  20. 20.
    Gordin, M.I.: The central limit theorem for stationary processes. Soviet Math. Dokl. 10, 1174–1176 (1969)Google Scholar
  21. 21.
    Gouëzel, S., Statistical properties of a skew product with a curve of neutral points. Preprint, 2004Google Scholar
  22. 22.
    Gouëzel, S.: Vitesse de décorrélation et théorèmes limites pour les applications non uniformément dilatantes. Ph. D. Thesis, Ecole Normale Supérieure, 2004Google Scholar
  23. 23.
    Hennion, H.: Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc. 118, 627–634 (1993)Google Scholar
  24. 24.
    Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180, 119–140 (1982)CrossRefGoogle Scholar
  25. 25.
    Keller, G.: Un théorème de la limite centrale pour une classe de transformations monotones per morceaux. C. R. Acad. Sci. Paris 291, 155–158 (1980)Google Scholar
  26. 26.
    Liverani, C.: Central limit theorem for deterministic systems. In: International Conference on Dynamical Systems (F. Ledrappier, J. Lewowicz and S. Newhouse, eds.), Pitman Research Notes in Math. 362, Harlow: Longman Group Ltd, 1996, pp. 56–75Google Scholar
  27. 27.
    Liverani, C., Saussol, B., Vaienti, S.: A probabilistic approach to intermittency. Ergod Th. and Dyn. Sys. 19, 671–685 (1999)CrossRefGoogle Scholar
  28. 28.
    Melbourne, I., Nicol, M.: Statistical properties of endomorphisms and compact group extensions. J. London Math. Soc. 70, 427–446 (2004)CrossRefGoogle Scholar
  29. 29.
    Melbourne, I., Török, A.: Central limit theorems and invariance principles for time-one maps of hyperbolic flows. Commun. Math. Phys. 229, 57–71 (2002)CrossRefGoogle Scholar
  30. 30.
    Melbourne, I., Török, A.: Statistical limit theorems for suspension flows. Israel J. Math. 194, 191–210 (2004)MathSciNetGoogle Scholar
  31. 31.
    Parry, W., Pollicott, M.: Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics. Astérique 187–188, Montrouge: Société Mathématique de France, 1990Google Scholar
  32. 32.
    Philipp, W., Stout, W.F.: Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables. Mem. of the Amer. Math. Soc. 161, Providence, RI: Amer. Math. Soc., 1975Google Scholar
  33. 33.
    Pollicott, M., Sharp, R.: Invariance principles for interval maps with an indifferent fixed point. Commun. Math. Phys. 229, 337–346 (2002)CrossRefGoogle Scholar
  34. 34.
    Ratner, M.: The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Israel J. Math. 16, 181–197 (1973)Google Scholar
  35. 35.
    Ruelle, D.: Thermodynamic Formalism. Encyclopedia of Math. and its Applications 5, Reading, Massachusetts: Addison Wesley, 1978Google Scholar
  36. 36.
    Sinai, Y.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat. Nauk 25, 141–192 (1970)Google Scholar
  37. 37.
    Sinai, Y.G.: Gibbs measures in ergodic theory. Russ. Math. Surv. 27, 21–70 (1972)Google Scholar
  38. 38.
    Viana, M.: Stochastic dynamics of deterministic systems. Col. Bras. de Matemática, 1997Google Scholar
  39. 39.
    Walkden, C.P.: Invariance principles for iterated maps that contract on average. Preprint, 2003Google Scholar
  40. 40.
    Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147, 585–650 (1998)Google Scholar
  41. 41.
    Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ian Melbourne
    • 1
  • Matthew Nicol
    • 2
  1. 1.Department of Maths. and Stats.University of SurreyGuildfordUK
  2. 2.Department of Math.University of HoustonHoustonUSA

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