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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).

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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