Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis

Abstract

Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase--space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical r\^ole of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce the concept of extremely rare event, quantified by non--standard Minkowski constants and we study its consequences to extreme value statistics. Limit laws are derived from formal calculations and are verified by numerical experiments.