Statistical Properties of the Maximum for Non-Uniformly Hyperbolic Dynamics

  • Ana Cristina Moreira FreitasEmail author
  • Jorge Milhazes Freitas
  • Mike Todd
Part of the Springer Proceedings in Mathematics book series (PROM, volume 1)


We study the asymptotic distribution of the partial maximum of observable random variables evaluated along the orbits of some particular dynamical systems. Moreover, we show the link between Extreme Value Theory and Hitting Time Statistics for discrete time non-uniformly hyperbolic dynamical systems. This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa.


Critical Orbit Stationary Stochastic Process Extreme Value Theory Partial Maximum Poisson Limit 
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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MT is supported by FCT grant SFRH/BPD/26521/2006. All three authors are supported by FCT through CMUP.


  1. 1.
    Benedicks, M., Carleson, L.: On iterations of 1 − ax 2 on ( − 1, 1). Ann. Math. 122, 1–25 (1985)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bruin, H., Saussol, B., Troubetzkoy, S., Vaienti, S.: Return time statistics via inducing. Ergod. Theory Dyn. Syst. 23, 991–1013 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bruin, H., Todd, M.: Return time statistics for invariant measures for interval maps with positive Lyapunov exponent. Stoch. Dyn. (to appear)Google Scholar
  4. 4.
    Bruin, S., Vaienti, S.: Return time statistics for unimodal maps. Fund. Math. 176, 77–94 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Collet, P.: Some ergodic properties of maps of the interval. Dynamical Systems (Temuco, 1991/1992), (Travaux en cours, 52), pp. 55–91. Herman, Paris (1996)Google Scholar
  6. 6.
    Collet, P.: Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Theory Dyn. Syst. 21, 401–420 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Freitas, A.C.M., Freitas, J.M.: Extreme values for Benedicks Carleson maps. Ergod. Theory Dyn. Syst. 28, 1117–1133 (2008)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Freitas, A.C.M., Freitas, J.M.: On the link between dependence and independence in Extreme Value Theory for Dynamical Systems. Stat. Probab. Lett. 78, 1088–1093 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Freitas, A.C.M., Freitas, J.M., Todd, M.: Hitting Time Statistics and Extreme Value Theory. arXiv:0804.2887Google Scholar
  10. 10.
    Haydn, N., Lacroix, Y., Vaienti, S.: Hitting and return times in ergodic dynamical systems. Ann. Probab. 33, 2043–2050 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Hirata, M.: Poisson law for Axiom A diffeomorphisms. Ergod. Theory Dyn. Syst. 13, 533–556 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hirata, M., Saussol, B., Vaienti, S.: Statistics of return times: a general framework and new applications. Comm. Math. Phys. 206, 33–55 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Holland, M., Nicol, M., Torok, A.: Extreme value distributions for non-uniformly hyperbolic dynamical systems. Preprint ( ∼ nicol/papers.html)Google Scholar
  14. 14.
    Lindgren, G., Leadbetter, M.R., Rootzén, H.: Extremes and related properties of random sequences and processes. Springer Series in Statistics. Springer, New York, Berlin (1983)zbMATHGoogle Scholar
  15. 15.
    Pitskel, B.: Poisson limit law for Markov chains. Ergod. Theory Dyn. Syst. 11, 501–513 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Rychlik, M.: Bounded variation and invariant measures. Studia Math. 76, 69–80 (1983)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ana Cristina Moreira Freitas
    • 1
    Email author
  • Jorge Milhazes Freitas
    • 2
  • Mike Todd
    • 2
  1. 1.Centro de Matemática & Faculdade de Economia daUniversidade do PortoPortoPortugal
  2. 2.Centro de Matemática daUniversidade do PortoPortoPortugal

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