Extreme Value distribution for singular measures

TL;DR

This paper investigates the behavior of Extreme Value distributions in discrete dynamical systems with singular measures, showing their relation to the Generalized Extreme Value family and how parameters scale with the information dimension.

Contribution

It provides an analytical and numerical analysis linking Extreme Value distributions to singular measures in dynamical systems, extending the understanding of their statistical properties.

Findings

Extreme Value distribution parameters scale with the information dimension.

Good agreement between numerical results and theoretical predictions for fractal sets.

Slower convergence observed in strange attractors like Lozi and Hénon maps.

Abstract

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 H\`enon 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.