Extreme values for Misiurewicz quadratic maps
by
J. Milhazes Freitas
Centro Matemática Universidade Porto
Coauthors: A. C. Moreira Freitas (CMUP and FEP)

We consider the quadratic family of maps given by f_a(x)=1-ax² with x ∈ [-1, 1], where a is a Misiurewicz parameter. On this set of parameters, there is an f_a-invariant measure, m_a, that is absolutely continuous with respect to Lebesgue.

For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic process X₁, X₂, ..., where X_n+1=f_a(X_n), for every positive integer n, and X₁ is a real valued random variable with d.f. given by G_a(x)=m_a((-∞, x]). Using the techniques developed by Benedicks and Carleson, we show that the limiting distribution of M_n=max{X₁, ..., X_n} is the same as that which would apply if the sequence X₁, X₂, ... was i.i.d. This result allows us to obtain that the asymptotic distribution of M_n is of Type III (Weibull), with parameter 1/2.

Date received: July 7, 2006