Benedicks-Carleson二次映射的极值
Extreme values for Benedicks-Carleson quadratic maps
摘要：
本文考虑由$f_{a}(x)=1-a x^2$给出的二次映射族，其中$f_{a}(x)=1-a x^2$是Benedicks-Carleson参数，具有$x\in [-1,1]$.对于每一个混沌动力系统，我们研究了平稳随机过程$X_0,X_1,...$的极值分布，由$X_{n}=f_a^n$给出，对于每一个整数$n\geq0$，其中每个随机变量$X_n$是根据$f_a$的唯一绝对连续，不变的概率分布。使用技术开发的Benedicks和Carleson，我们表明，极限分布的$M_n=\max\{X_0,...,X_{n-1}\}$是相同的，将适用于如果序列$X_0,X_1,...$是独立和同分布的。这个结果使我们可以得出结论，$M_n$的渐近分布是III型（威布尔）。
英文摘要：
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).
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