Extreme Value distribution for singular measures

نویسنده

  • Valerio Lucarini
چکیده

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. The existence of extreme value laws for dynamical systems preserving an absolutely continuous invariant measure or a singular continuous invariant measure has been recently proven if strong mixing properties or exponential hitting time statistics on balls are satisfied. In our previous work we have shown that there exists an algorithmic way

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تاریخ انتشار 2011