Laws of rare events for deterministic and random dynamical systems

TL;DR

This paper develops a comprehensive theory of extreme value laws and rare event behavior in deterministic and randomly perturbed dynamical systems, focusing on the extremal index and convergence of rare events.

Contribution

It introduces a unified framework for analyzing rare events and extreme value laws in both deterministic and stochastic dynamical systems, including the impact of perturbations.

Findings

Dichotomy in extremal index depending on periodicity of points

Convergence results for Rare Events Point Processes

Role of decay of correlations in extreme value analysis

Abstract

The object of this paper is twofold. From one side we study the dichotomy, in terms of the Extremal Index of the possible Extreme Value Laws, when the rare events are centred around periodic or non periodic points. Then we build a general theory of Extreme Value Laws for randomly perturbed dynamical systems. We also address, in both situations, the convergence of Rare Events Point Processes. Decay of correlations against $L^1$ observables will play a central role in our investigations.