Extreme Value Distributions for some classes of Non-Uniformly Partially Hyperbolic Dynamical Systems

TL;DR

This paper establishes conditions under which certain non-uniformly hyperbolic dynamical systems exhibit extreme value distributions of the first type, expanding understanding of their statistical behavior.

Contribution

It provides verifiable criteria linking decay of correlations and return times to the emergence of Type I extreme value distributions in complex dynamical systems.

Findings

Type I extreme value distribution proven for S^1 extensions of expanding maps

Conditions established for non-uniformly expanding maps modeled by Young Towers

Distribution shown for skew product extensions with neutral points

Abstract

In this note, we obtain verifiable sufficient conditions for the extreme value distribution for a certain class of skew product extensions of non-uniformly hyperbolic base maps. We show that these conditions, formulated in terms of the decay of correlations on the product system and the measure of rapidly returning points on the base, lead to a distribution for the maximum of $\Phi(p) = -\log(d(p, p_0))$ that is of the first type. In particular, we establish the Type I distribution for $S^1$ extensions of piecewise $C^2$ uniformly expanding maps of the interval, non-uniformly expanding maps of the interval modeled by a Young Tower, and a skew product extension of a uniformly expanding map with a curve of neutral points.