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.

### 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 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.
##### Scheda breve Scheda completa Scheda completa (DC)
2016
http://iopscience.iop.org/article/10.1088/1751-8113/49/37/374001/pdf
extreme value laws; extremely rare events; fractal landscapes; Minkowski dimension and content; Minkowski question-mark function; Mobius iterated function systems; non-standard Minkowski behavior; Mathematical Physics; Physics and Astronomy (all); Statistical and Nonlinear Physics; Modeling and Simulation; Statistics and Probability
Mantica, GIORGIO DOMENICO PIO; Perotti, LUCA CAMILLO
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11383/2059454`
##### Attenzione

L'Ateneo sottopone a validazione solo i file PDF allegati

• ND
• 6
• 6