Extreme Values for Benedicks-carleson Quadratic Maps

We consider the quadratic family of maps given by fa(x) = 1− ax 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 = f n a , 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).