Extreme values for Benedicks-Carleson quadratic maps

TL;DR

This paper investigates the extreme value distribution of quadratic maps with Benedicks-Carleson parameters, showing that the maximum of the process follows a Weibull distribution similar to an i.i.d. sequence.

Contribution

It proves that for these chaotic quadratic maps, the maximum distribution converges to Weibull, matching the i.i.d. case, using Benedicks and Carleson techniques.

Findings

Maximum distribution is Weibull (Type III)

Distribution matches that of an i.i.d. sequence

Results apply to chaotic quadratic maps with Benedicks-Carleson parameters

Abstract

We consider the quadratic family of maps given by $f_{a}(x)=1-a x^2$ with $x\in [-1,1]$, 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 $n\geq0$, where each random variable $X_n$ 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).