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Large Deviations for Non-Uniformly Expanding Maps

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Abstract

We obtain large deviation bounds for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states.

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References

  1. J. Alves. Statistical analysis of non-uniformly expanding dynamical systems. Publicaçes Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, 2003. 24° Colóquio Brasileiro de Matematica. [24th Brazilian Mathematics Colloquium].

  2. J. F. Alves. SRB measures for non-hyperbolic systems with multidimensional expansion. Ann. Sci. école Norm. Sup. 33:1–32 (2000).

    MATH  MathSciNet  Google Scholar 

  3. J. F. Alves and V. Araujo. Random perturbations of nonuniformly expanding maps. Astérisque 286:25–62 (2003).

    MATH  MathSciNet  Google Scholar 

  4. J. F. Alves, V. Araiijo, and B. Saussol. On the uniform hyperbolicity of nonuniformly hyperbolic systems. Proc. Am. Math. Soc. 131(4):1303–1309 (2003).

    Article  MATH  Google Scholar 

  5. J. F. Alves and V. Araújo. Hyperbolic times: frequency versus integrability. Ergodic Theory Dyn. Syst. 24:1–18 (2004).

    Article  Google Scholar 

  6. J. F. Alves, C. Bonatti, and M. Viana. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2):351–398 (2000).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. J. F. Alves, S. Luzzatto, and V. Pinheiro. Markov structures and decay of correlations for non-uniformly expanding dynamical systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(6):817–839 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  8. V. Araujo and M. J. Pacifico. Physical measures for infinite-modal maps. Preprint IMPA Serie A, 328 (2004).

  9. V. Araújo and A. Tahzibi. Stochastic stability at the boundary of expanding maps. Nonlinearity 18:939–959 (2005).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. A. Arbieto and C. Matheus. Fast decay of correlations of equilibrium states of open classes of non-uniformly expanding maps and potentials. http://arxiv.org/abs/math.DS/0603629 (2006).

  11. A. Arbieto, C. Matheus, S. Senti, and M. Viana. Maximal entropy measures for viana maps. Discrete and Continuous Dynamical Systems, to appear (2006).

  12. M. Benedicks and L. Carleson. On iterations of 1 − ax2 on (−1, 1). Annals Math. 122:1–25 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Benedicks and L. Carleson. The dynamics of the Hénon map. Annals Math. 133:73–169 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Benedicks and L.-S. Young. Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Ergod. Th. & Dynam. Sys. 12:13–37 (1992).

    MATH  MathSciNet  Google Scholar 

  15. M. Benedicks and L.-S. Young. SBR-measures for certain Hénon maps. Invent. Math. 112:541–576 (1993).

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  17. C. Bonatti, L. J. Díaz, and M. Viana. Dynamics beyond uniform hyperbolicity, volume 102 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective, Mathematical Physics, III.

    Google Scholar 

  18. C. Bonatti and M. Viana. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115:157–193 (2000).

    MATH  MathSciNet  Google Scholar 

  19. R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms, volume 470 of Lect. Notes in Math. Springer Verlag (1975).

  20. R. Bowen and D. Ruelle. The ergodic theory of Axiom A flows. Invent. Math. 29:181–202 (1975).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. L. Breiman. The individual ergodic theorem of information theory. Ann. Math. Statist. 28:809–811 (1957).

    MATH  MathSciNet  Google Scholar 

  22. M. Brin and A. Katok. On local entropy. In Geometric dynamics (Rio de. Janeiro, 1981), volume 1007 of Lecture Notes in Math., pp. 30–38. Springer, Berlin (1983).

    Google Scholar 

  23. J. Buzzi and O. Sari. Uniqueness of equilibrium measures for countable markov shifts and multidimensional piecewise expanding maps. Ergodic Theory Dynam. Systems 23(5):1383–1400 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  24. J. Buzzi, O. Sester, and M. Tsujii. Weakly expanding skew-products of quadratic maps. Ergodic Theory Dynam. Systems 23(5):1401–1414 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  25. W. de Melo and S. van Strien. One-dimensional dynamics. Springer Verlag (1993).

  26. M. Denker. The central limit theorem for dynamical systems. In Dynamical Systems and Ergodic Theory, volume 23. Banach Center Publ., Warsaw (1989).

    Google Scholar 

  27. M. Denker and M. Kesseböhmer. Thermodynamic formalism, large deviation, and multifractals. In Stochastic climate models (Chorin, 1999), volume 49 of Progr. Probab., pp. 159–169. Birkhäuser, Base (2001).

    Google Scholar 

  28. R. S. Ellis. Entropy, large deviations, and statistical mechanics. Reprint of the 1985 original. Classics in Mathematics. Springer-Verlag Berlin (2006).

    Google Scholar 

  29. S. Gouezel. Decay of correlations for nonuniforly expanding systems. Bulletin de la Soc. Math. France 134: 1–31 (2006).

    Google Scholar 

  30. V. Guilleinin and A. Pollack. Differential Topology. Prentice Hall, New Jersey (1974).

    Google Scholar 

  31. M. Hirsch. Differential Topology. Springer-Verlag, New-York (1976).

    MATH  Google Scholar 

  32. M. Jakobson. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 81:39–88 (1981).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  33. A. Katok and J. M. Strelcyn. Invariant manifolds, entropy and billiards. Smooth maps with singularities, volume 1222 of Lect. Notes in Math. Springer Verlag (1986).

  34. Y. Kifer. Large deviations in dynamical systems and stochastic processes. Trans. Am. Math. Soc. 321(2):505–524 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  35. A. Lasota and J. Yorke. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186:481–488 (1973).

    Article  MathSciNet  Google Scholar 

  36. P.-D. Liu. Pesin's Entropy Formula for endomorphisms. Nagoya Math. J. 150:197–209 (1998).

    MATH  MathSciNet  Google Scholar 

  37. S. Luzzatto and M. Viana. Positive Lyapunov exponents for Lorenz-like families with criticalities. Asterisque 261:201–237, 2000. Geometrie complexe et systemes dynamiques (Orsay, 1995).

    MATH  MathSciNet  Google Scholar 

  38. R. Mañé. Ergodic theory and differentiable dynamics. Springer Verlag, New York (1987).

    MATH  Google Scholar 

  39. B. McMillan. The basic theorems of information theory. Ann. Math. Stat. 24:196–219 (1953).

    MATH  MathSciNet  Google Scholar 

  40. R.J. Metzger. Sinai-Ruelle-Bowen measures for contracting Lorenz maps and flows. Ann. Inst. H. Poincaré Anal. Non Lineaire 17(2):247–276 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  41. L. Mora and M. Viana. Abundance of strange attractors. Acta Math. 171(1):1–71 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  42. J. Munkres. Topology, 2nd edition, Prentice-Hall, New Deli (2004).

    Google Scholar 

  43. K. Oliveira and M. Viana. Existence and uniqueness of maximizing measures for robust classes of local diffeomorphisms. Discrete Continuous Dynam. Syst. 15(1): 225–236 (2006).

    Google Scholar 

  44. V. I. Oseledets. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19:197–231 (1968).

    MATH  Google Scholar 

  45. Y. Pesin and Y. Sinai. Gibbs measures for partially hyperbolic attractors. Ergod. Th. & Dynam. Sys. 2:417–438 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  46. V. Pinheiro. SRB measures for weakly expanding maps. Nonlinearity 19(5):1185–1200 (2006).

    Google Scholar 

  47. V. Pliss. On a conjecture due to Smale. Diff. Uravnenija 8:262–268 (1972).

    Google Scholar 

  48. A. Rovella. The dynamics of perturbations of the contracting Lorenz attractor. Bull. Braz. Math. Soc. 24(2):233–259 (1993).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  50. D. Ruelle. An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9:83–87 (1978).

    MATH  MathSciNet  Google Scholar 

  51. D. Ruelle. The thermodynamical formalism for expanding maps. Comm. Math. Phys. 125:239–262 (1989).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  52. D. Ruelle. Thermodynamic formalism. The mathematical structures of equilibrium statistical mechanics. Cambridge Mathematical Library, 2nd edition, Cambridge University Press, Cambridge (2004).

    Google Scholar 

  53. M. Rychlik. Bounded variation and invariant measures. Studia Math. 76:69–80 (1983).

    MATH  MathSciNet  Google Scholar 

  54. C. E. Shannon. A mathematical theory of communication. Bell System Tech. J. 27:379–423, 623–656 (1948).

    MathSciNet  MATH  Google Scholar 

  55. Y. Sinai. Gibbs measures in ergodic theory. Russian Math. Surveys 27:21–69 (1972).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. R. Sturman and J. Stark. Semi-uniform ergodic theorems and applications to forced systems. Nonlinearity 13:113–143 (2000).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  57. M. Viana. Multidimensional nonhyperbolic attractors. Inst. Hautes éatudes Sci. Publ. Math. 85:63–96 (1997).

    MATH  MathSciNet  Google Scholar 

  58. M. Viana. Stochastic dynamics of deterministic systems. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1997. 21° Colóquio Brasileiro de Matematica. [21th Brazilian Mathematics Colloquium].

  59. P. Walters. Antroduction to ergodic theory. Springer Verlag (1982).

  60. L.-S. Young. Some large deviation results for dynamical systems. Trans. Am. Math. Soc. 318(2):525–543 (1990).

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, 21.945-970, Rio de Janeiro, RJ, Brazil

    V. Araújo & M. J. Pacifico

  2. Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    V. Araújo

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  1. V. Araújo
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  2. M. J. Pacifico
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Correspondence to V. Araújo.

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2000 Mathematics Subject Classification: 37D25, 37A50, 37B40, 37C40

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Araújo, V., Pacifico, M.J. Large Deviations for Non-Uniformly Expanding Maps. J Stat Phys 125, 411–453 (2006). https://doi.org/10.1007/s10955-006-9183-y

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