Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis *Dedicated to the memory of Joseph Ford, on the twentieth anniversary of his departure. (19th August 2016)

• Record Type: Journal Article
• Authors: Mantica, Giorgio; Perotti, Luca
• Is Part Of: Journal of physics. Volume 49:Number 37(2016)
• Publication Date: 2016-08-19
• Subjects: extreme value laws -- Minkowski dimension and content -- Minkowski question-mark function -- non-standard Minkowski behavior -- fractal landscapes -- extremely rare events -- Möbius iterated function systems; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8113/49/37/374001
• Languages: English
• ISSNs: 1751-8113

Abstract

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 rôle 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.