Extreme values for Benedicks-Carleson quadratic maps

• Type: Article in International Scientific Journal
• Date: 2008

Authors

Ana Cristina M Moreira Freitas (Author) FEP; Jorge Milhazes Freitas (Author) FCUP

Journal

Title: Ergodic Theory and Dynamical Systems

Vol. 28 No. 4

Pages: 1117-1133

ISSN: 0143-3857

Publisher: Cambridge University Press

Indexing

ISI Web of Knowledge - 23 Citations

ISI Web of Science

Scopus - 24 Citations

Scientific classification

FOS: Natural sciences > Mathematics

CORDIS: Physical sciences > Mathematics > Chaos theory

Other information

Authenticus ID: P-003-X76

DOI: 10.1017/s0143385707000624

Abstract (EN): We consider the quadratic family of maps given by f(a)(x) = 1 - ax(2) with x epsilon [- 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 X(0), X(1), ..., given by X(n) = f(a)(n), for every integer n >= 0, where each random variable Xn is distributed according to the unique absolutely continuous, invariant probability of f(a). Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of M(n) = max {X(0), ..., X(n) - 1} is the same as that which would apply if the sequence X(0), X(1), ... was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of M(n) is of type III (Weibull).

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 17