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Extreme values for Benedicks–Carleson quadratic maps

Published online by Cambridge University Press:  01 August 2008

ANA CRISTINA MOREIRA FREITAS  and
JORGE MILHAZES FREITAS
ANA CRISTINA MOREIRA FREITAS
Affiliation:
Centro de Matemática & Faculdade de Economia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-464 Porto, Portugal (email: amoreira@fep.up.pt)
JORGE MILHAZES FREITAS
Affiliation:
Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal (email: jmfreita@fc.up.pt)
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Abstract

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).

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 4 , August 2008 , pp. 1117 - 1133
Copyright
Copyright © 2008 Cambridge University Press

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References

[1]Balakrishnan, V., Nicolis, C. and Nicolis, G.. Extreme value distributions in chaotic dynamics. J. Stat. Phys. 80(1–2) (1995), 307–336.CrossRefGoogle Scholar
[2]Benedicks, M. and Carleson, L.. On iterations of 1−ax 2 on (−1,1). Ann. Math. 122 (1985), 1–25.CrossRefGoogle Scholar
[3]Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. Math. 133 (1991), 73–169.CrossRefGoogle Scholar
[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), 13–27.CrossRefGoogle Scholar
[5]Collet, P. and Eckmann, J. P.. Positive Lyapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dynam. Sys. 3 (1983), 13–46.CrossRefGoogle Scholar
[6]Collet, P.. Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Th. & Dynam. Sys. 21(2) (2001), 401–420.CrossRefGoogle Scholar
[7]Freitas, J. M.. Continuity of SRB measure and entropy for Benedicks–Carleson quadratic maps. Nonlinearity 18 (2005), 831–854.CrossRefGoogle Scholar
[8]Freitas, J. M.. Statistical stability for chaotic dynamical systems. PhD Thesis. Universidade do Porto, 2006, http://www.fc.up.pt/pessoas/jmfreita/homeweb/publications.htm.Google Scholar
[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.Google Scholar
[10]Haiman, G.. Extreme values of the tent map process. Statist. Probab. Lett. 65(4) (2003), 451–456.CrossRefGoogle Scholar
[11]Jakobson, M.. Absolutely continuous invariant measures for one parameter families of one-dimensional maps. Comm. Math. Phys. 81 (1981), 39–88.CrossRefGoogle Scholar
[12]Keller, G.. Exponential weak Bernoulli mixing for Collet–Eckmann maps. Israel J. Math. 86 (1994), 301–310.CrossRefGoogle Scholar
[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), 31–69.CrossRefGoogle Scholar
[14]Kingman, J. F. C. and Taylor, S. J.. Introduction to Measure and Probability. Cambridge University Press, Cambridge, 1966.CrossRefGoogle Scholar
[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.Google Scholar
[16]de Melo, W. and van Strien, S.. One-Dimensional Dynamics. Springer, Berlin, 1993.CrossRefGoogle Scholar
[17]Moreira, F. J.. Chaotic dynamics of quadratic maps, Informes de Matemática, IMPA, Série A, 092/93, 1993, http://www.fc.up.pt/cmup/fsmoreir/downloads/BC.pdf.Google Scholar
[18]Nowicki, T.. Symmetric S-unimodal mappings and positive Liapunov exponents. Ergod. Th. & Dynam. Sys. 5 (1985), 611–616.CrossRefGoogle Scholar
[19]Rychlik, M.. Another proof of Jakobson’s theorem and related results. Ergod. Th. & Dynam. Sys. 8(1) (1988), 93–109.CrossRefGoogle Scholar
[20]Young, L. S.. Decay of correlations for certain quadratic maps. Comm. Math. Phys. 146 (1992), 123–138.CrossRefGoogle Scholar