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Estimates for Correlation in Dynamical Systems: From Hölder Continuous Functions to General Observables

Abstract

For many dynamical systems that are popular in applications, estimates are known for the decay of correlation in the case of Hölder continuous functions. In the present article, we suggest an approach that allows us to obtain estimates for correlation in dynamical systems in the case of arbitrary functions. This approach is based on approximation and estimates are obtained with the use of known estimates for Hölder continuous functions. We apply our approach to transitive Anosov diffeomorphisms and derive the central limit theorem for the characteristic functions of certain sets with boundary of zero measure.

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References

  1. 1.

    J. F. Alves, J. M. Freitas, S. Luzzatto, and S. Vaienti, “From rates of mixing to recurrence times via large deviations,” Adv. Math. 228, 1203 (2011).

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    V. Baladi, Positive Transfer Operators and Decay of Correlations (World Scientific, Singapore, 2000).

    Book  MATH  Google Scholar 

  3. 3.

    J. Bergh and J. Löfström, Interpolation Spaces. An Introduction (Springer-Verlag, Berlin–Heidelberg–New York, 1976).

    Book  MATH  Google Scholar 

  4. 4.

    P. Billingsley, Convergence of Probability Measures (Wiley, Chichester, 1999).

    Book  MATH  Google Scholar 

  5. 5.

    R. Bowen, Methods of Symbolic Dynamics. Collected Papers (Mir, Moscow, 1979) [in Russian].

    Google Scholar 

  6. 6.

    X. Bressaud and C. Liverani, “Anosov diffeomorphisms and coupling,” Ergodic Theory Dynam. Systems 22, 129 (2002).

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    V. I. Burenkov, “On the density of infinitely differentiable functions in Sobolev spaces for an arbitrary open set,” Trudy Mat. Inst. Steklov 131, 39 (1974).[Proc. Steklov Inst. Math. 131, 39 (1974).

    MathSciNet  MATH  Google Scholar 

  8. 8.

    J.-R. Chazottes, P. Collet, and B. Schmitt, “Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems,” Nonlinearity 18, 2341 (2005).

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    N. I. Chernov, “Limit theorems and Markov approximations for chaotic dynamical systems,” Probab. Theory Relat. Fields 101, 321 (1995).

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    N. Chernov, “Advanced statistical properties of dispersing billiards,” J. Statist. Phys. 122, 1061 (2006).

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    N. Chernov and R. Markarian, Chaotic Billiards (Izhevsk Inst. Comp. Sci., Izhevsk, 2006). [Chaotic Billiards (Amer.Math. Soc., Providence, RI, 2006)].

    MATH  Google Scholar 

  12. 12.

    A.-H. Fan, “Decay of correlation for expanding toral endomorphisms,” in Dynamical Systems, 29 (World Scientific, Singapore, 1999).

    Google Scholar 

  13. 13.

    M. Holland, “Slowly mixing systems and intermittency maps,” Ergodic Theory Dynam. Systems 25, 133 (2005).

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    A. G. Kachurovskiĭ, “The rate of convergence in ergodic theorems,” UspekhiMat. Nauk 51, no. 4, 73 (1996).[RussianMath. Surveys 51, 653 (1996).

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    A. G. Kachurovskiĭand I. V. Podvigin, “Correlations, large deviations, and rates of convergence in ergodic theorems for characteristic functions,” Dokl. Akad. Nauk 461, 509 (2015).[Dokl.Math. 91, 204 (2015).

    MathSciNet  Google Scholar 

  16. 16.

    A. G. Kachurovskiĭand I. V. Podvigin, “Large deviations and rates of convergence in the Birkhoff ergodic theorem: from Hölder continuity to continuity,” Dokl. Akad. Nauk 466, 12 (2016).[Dokl.Math. 93, 6 (2016).

    Google Scholar 

  17. 17.

    A.G. Kachurovskiĭand I. V. Podvigin, “Estimates of the rate of convergence in the vonNeumann and Birkhoff ergodic theorems,” Trudy Mosk.Mat. Obshch. 77, 1 (2016).[Trans.MoscowMath. Soc. 1, (2016)].

    Google Scholar 

  18. 18.

    A. G. Kachurovskiĭand I. V. Podvigin, “Large deviations of the ergodic averages: from Hölder continuity to continuity almost everywhere,” Mat. Trudy 20, 97 (2017).[Siberian Adv.Math. 28, 23 (2018).

    MATH  Google Scholar 

  19. 19.

    O. Knill, “Singular continuous spectrum and quantitative rates of weak mixing,” Discrete Contin. Dynam. Systems 4, 33 (1998).

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    R. Leplaideur and B. Saussol, “Large deviations for return times in non-rectangle sets for Axiom A diffeomorphisms,” Discrete Contin. Dynam. Systems 22, 327 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    I. Melbourne, “Large and moderate deviations for slowlymixing dynamical systems,” Proc.Amer.Math. Soc. 137, 1735 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    M. Ruziboev, “Decay of correlations for invertible maps with non-Hölder observables,” Dynam. Systems 30, 341 (2015).

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    O. Sarig, “Decay of correlations,” in Handbook of Dynamical Systems, Vol. 1 B, 244 (Elsevier, Amsterdam, 2006).

    Google Scholar 

  24. 24.

    V. V. Sedalishchev, “Interrelation between the convergence rates in von Neumann’s and Birkhoff’s ergodic theorems,” Siberian Math. J. 55, 336 (2014).[Sibirsk.Mat. Zh. 55, 412 (2014)].

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    M. Stenlund, “A strong pair correlation bound implies the CLT for Sinai billiards,” J. Statist. Phys. 140, 154 (2010).

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    L.-S. Young, “Statistical properties of dynamical systems with some hyperbolicity,” Ann. Math. 147, 585 (1998).

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    L.-S. Young, “Recurrence times and rates of mixing,” Israel J. Math. 110, 153 (1999).

    Google Scholar 

  28. 28.

    H.-K. Zhang, “Decay of correlations on non-Hölder observables,” Internat. J. Nonlinear Sci. 10, 359 (2010).

    MATH  Google Scholar 

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Correspondence to I. V. Podvigin.

Additional information

Original Russian Text © I.V. Podvigin, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 2, pp. 90–119.

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Cite this article

Podvigin, I.V. Estimates for Correlation in Dynamical Systems: From Hölder Continuous Functions to General Observables. Sib. Adv. Math. 28, 187–206 (2018). https://doi.org/10.3103/S1055134418030045

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Keywords

  • correlation
  • the best approximation
  • approximation spaces
  • Anosov diffeomorphisms
  • central limit theorem