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Communications in Mathematical Physics

, Volume 359, Issue 3, pp 1123–1138 | Cite as

Equidistribution for Nonuniformly Expanding Dynamical Systems, and Application to the Almost Sure Invariance Principle

  • Alexey Korepanov
Article

Abstract

Let \({T : M \to M}\) be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let \({v : M \to \mathbb{R}^d}\) be an observable and \({v_n = \sum_{k=0}^{n-1} v \circ T^k}\) denote the Birkhoff sums. Given a probability measure \({\mu}\) on M, we consider v n as a discrete time random process on the probability space \({(M, \mu)}\). In smooth ergodic theory there are various natural choices of \({\mu}\), such as the Lebesgue measure, or the absolutely continuous T-invariant measure. They give rise to different random processes. We investigate relation between such processes. We show that in a large class of measures, it is possible to couple (redefine on a new probability space) every two processes so that they are almost surely close to each other, with explicit estimates of “closeness”. The purpose of this work is to close a gap in the proof of the almost sure invariance principle for nonuniformly hyperbolic transformations by Melbourne and Nicol.

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References

  1. 1.
    Berkes I., Philipp W.: Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7, 29–54 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, vol. 470. Springer, Berlin (1975)Google Scholar
  3. 3.
    Chernov N.: Advanced statistical properties of dispersing billiards. J. Stat. Phys. 122, 1061–1094 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chernov, N., Markarian, R.: Chaotic Billiards, volume 127 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2006)Google Scholar
  5. 5.
    Cuny C., Merlevède F.: Strong invariance principles with rate for “reverse” martingales and applications. J. Theor. Probab. 28, 137–183 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Denker M., Philipp W.: Approximation by Brownian motion for Gibbs measures and flows under a function. Ergod. Theory Dyn. Syst 4, 541–552 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gibbs A.L., Su F.E.: On choosing and bounding probability metrics. Int. Stat. Rev. 70, 419–435 (2002)CrossRefzbMATHGoogle Scholar
  8. 8.
    Gouëzel S.: Almost sure invariance principle for dynamical systems by spectral methods. Ann. Probab. 38, 1639–1671 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gouëzel S.: Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139, 29–65 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hu H.: Decay of correlations for piecewise smooth maps with indifferent fixed points. Ergod. Theory Dyn. Syst. 24, 495–524 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Korepanov A.: Linear response for intermittent maps with summable and nonsummable decay of correlations. Nonlinearity 29, 1735–1754 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Korepanov A., Kosloff Z., Melbourne I.: Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems. Stud. Math. 238, 59–89 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Korepanov, A., Kosloff, Z., Melbourne, I.: Explicit coupling argument for nonuniformly hyperbolic transformations. Proc. R. Soc. Edinb. Sect. A (2016) (to appear) Google Scholar
  14. 14.
    Korepanov, A., Kosloff, Z., Melbourne, I.: Martingal e-Coboundary Decomposition for Families of Dynamical Systems. Ann. Inst. H. Poincaré Anal. Non Linéaire (2016) (to appear) Google Scholar
  15. 15.
    Liverani C., Saussol B., Vaienti S.: A probabilistic approach to intermittency. Ergod. Theory Dyn. Syst. 19, 671–685 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Melbourne I., Nicol M.: Almost sure invariance principle for nonuniformly hyperbolic systems. Commun. Math. Phys. 260, 131–146 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Melbourne I., Nicol M.: A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann. Probab. 37, 478–505 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Philipp W., Stout W.: Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables, vol. 161. American Mathematical Society, Providence (1975)zbMATHGoogle Scholar
  19. 19.
    Sarig O.: Subexponential decay of correlations. Invent. Math. 150, 629–653 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shortt R.M.: Universally measurable spaces: an invariance theorem and diverse characterizations. Fund. Math. 121, 169–176 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sinai Y. G.: Gibbs measures in ergodic theory. Russ. Math. Surv. 27, 21–70 (1972)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Young L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147, 585–650 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Young L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zweimüller R.: Measure preserving transformations similar to Markov shifts. Israel J. Math. 173, 421–443 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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