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Slow Rates of Mixing for Dynamical Systems with Hyperbolic Structures

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Abstract

We consider invertible discrete-time dynamical systems having a hyperbolic product structure in some region of the phase space with infinitely many branches and variable return time. We show that the decay of correlations of the SRB measure associated to that hyperbolic structure is related to the tail of the recurrence times. We also give sufficient conditions for the validity of the Central Limit Theorem. This extends previous results by Young in (Ann. Math. 147: 585–650, [1998]; Israel J. Math. 110: 153–188, [1999]).

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References

  1. Alves, J.F., Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140, 351–398 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Benedicks, M., Young, L.-S.: Markov extensions and decay of correlations for certain Hénon maps. Astérisque 261, 13–56 (2000)

    MathSciNet  Google Scholar 

  3. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, vol. 470. Springer, New York (1975)

    MATH  Google Scholar 

  4. Gouëzel, S.: Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. France 134(1), 1–31 (2006)

    MATH  MathSciNet  Google Scholar 

  5. Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180, 119–140 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  6. Pianigiani, G.: First return maps and invariant measure. Israel J. Math. 35, 32–48 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ruelle, D.: A measure associated with Axiom A attractors. Am. J. Math. 98, 619–654 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  8. Sinai, Ya.: Gibbs measure in ergodic theory. Russ. Math. Surv. 27, 21–69 (1972)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Viana, M.: Multidimensional nonhyperbolic attractors. Publ. Math. IHES 85, 69–96 (1997)

    Google Scholar 

  10. Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147, 585–650 (1998)

    Article  MATH  Google Scholar 

  11. Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática Pura, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    José F. Alves

  2. Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110, Salvador, Brazil

    Vilton Pinheiro

Authors
  1. José F. Alves
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  2. Vilton Pinheiro
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Correspondence to José F. Alves.

Additional information

Work carried out at the Federal University of Bahia, University of Porto and IMPA. J.F.A. was partially supported by FCT through CMUP and POCI/MAT/61237/2004. V.P. was partially supported by PADCT/CNPq and POCI/MAT/61237/2004.

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Alves, J.F., Pinheiro, V. Slow Rates of Mixing for Dynamical Systems with Hyperbolic Structures. J Stat Phys 131, 505–534 (2008). https://doi.org/10.1007/s10955-008-9482-6

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  • DOI: https://doi.org/10.1007/s10955-008-9482-6

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