Skip to main content Accessibility help

Extreme values for Benedicks–Carleson quadratic maps


We consider the quadratic family of maps given by fa(x)=1−ax2 with x∈[−1,1], where a is a Benedicks–Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes X0,X1,… , given by Xn=fan, for every integer n≥0, where each random variable Xn is distributed according to the unique absolutely continuous, invariant probability of fa. Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of Mn=max {X0,…,Xn−1} is the same as that which would apply if the sequence X0,X1,… was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of Mn is of type III (Weibull).

Hide All
[1]Balakrishnan, V., Nicolis, C. and Nicolis, G.. Extreme value distributions in chaotic dynamics. J. Stat. Phys. 80(1–2) (1995), 307336.
[2]Benedicks, M. and Carleson, L.. On iterations of 1−ax 2 on (−1,1). Ann. Math. 122 (1985), 125.
[3]Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. Math. 133 (1991), 73169.
[4]Benedicks, M. and Young, L. S.. Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Ergod. Th. & Dynam. Sys. 12 (1992), 1327.
[5]Collet, P. and Eckmann, J. P.. Positive Lyapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dynam. Sys. 3 (1983), 1346.
[6]Collet, P.. Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Th. & Dynam. Sys. 21(2) (2001), 401420.
[7]Freitas, J. M.. Continuity of SRB measure and entropy for Benedicks–Carleson quadratic maps. Nonlinearity 18 (2005), 831854.
[8]Freitas, J. M.. Statistical stability for chaotic dynamical systems. PhD Thesis. Universidade do Porto, 2006,
[9]Freitas, A. C. M. and Freitas, J. M.. On the link between dependence and independence in extreme value theory for dynamical systems. Statist. Probab. Lett. to appear.
[10]Haiman, G.. Extreme values of the tent map process. Statist. Probab. Lett. 65(4) (2003), 451456.
[11]Jakobson, M.. Absolutely continuous invariant measures for one parameter families of one-dimensional maps. Comm. Math. Phys. 81 (1981), 3988.
[12]Keller, G.. Exponential weak Bernoulli mixing for Collet–Eckmann maps. Israel J. Math. 86 (1994), 301310.
[13]Keller, G. and Nowicki, T.. Spectral theory, zeta functions and the distribution of periodic points for Collet–Eckmann maps. Comm. Math. Phys. 149 (1992), 3169.
[14]Kingman, J. F. C. and Taylor, S. J.. Introduction to Measure and Probability. Cambridge University Press, Cambridge, 1966.
[15]Lindgren, G., Leadbetter, M. R. and Rootzén, H.. Extremes and Related Properties of Stationary Sequences and Processes (Springer Series in Statistics, XII). Springer, New York, 1983.
[16]de Melo, W. and van Strien, S.. One-Dimensional Dynamics. Springer, Berlin, 1993.
[17]Moreira, F. J.. Chaotic dynamics of quadratic maps, Informes de Matemática, IMPA, Série A, 092/93, 1993,
[18]Nowicki, T.. Symmetric S-unimodal mappings and positive Liapunov exponents. Ergod. Th. & Dynam. Sys. 5 (1985), 611616.
[19]Rychlik, M.. Another proof of Jakobson’s theorem and related results. Ergod. Th. & Dynam. Sys. 8(1) (1988), 93109.
[20]Young, L. S.. Decay of correlations for certain quadratic maps. Comm. Math. Phys. 146 (1992), 123138.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed