奇异测度的极值分布
Extreme Value distribution for singular measures
摘要：
本文对具有奇异测度的离散动力系统的极值分布进行了分析和数值研究。使用Faranda等人[2011]中描述的块最大值方法，我们在数值上表明，这些映射的极值分布可以与广义极值族相关联，其中参数随信息维度缩放。对几个低维映射进行了数值分析。对于中间三分之一的康托尔集和Sierpinskij三角形，使用迭代函数系统，实验参数显示出与理论值非常好的协议。对于像Lozi和Hènon映射这样的奇怪吸引子，观察到向广义极值分布的收敛速度较慢。即使在大的统计量的存在下，所观察到的收敛速度是缓慢的，如果与具有绝对连续的不变测度的地图。然而，在计算的不确定度范围内，结果与理论估计吻合得很好。
英文摘要：
In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems that have a singular measure. Using the block maxima approach described in Faranda et al. [2011] we show that, numerically, the Extreme Value distribution for these maps can be associated to the Generalised Extreme Value family where the parameters scale with the information dimension. The numerical analysis are performed on a few low dimensional maps. For the middle third Cantor set and the Sierpinskij triangle obtained using Iterated Function Systems, experimental parameters show a very good agreement with the theoretical values. For strange attractors like Lozi and Hènon maps a slower convergence to the Generalised Extreme Value distribution is observed. Even in presence of large statistics the observed convergence is slower if compared with the maps which have an absolute continuous invariant measure. Nevertheless and within the uncertainty computed range, the results are in good agreement with the theoretical estimates.
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