Extreme Value Laws for dynamical systems with countable extremal sets

TL;DR

This paper studies the extremal behavior of dynamical systems with observables evaluated along orbits, focusing on cases where the maximal set is countable and may have accumulation points, extending existing extremal laws.

Contribution

It generalizes extremal laws to dynamical systems with countable maximal sets, including accumulation points, and develops new conditions for distributional limits and clustering analysis.

Findings

Established existence of distributional limits for systems with countable extremal sets.

Analyzed the extremal index and clustering behavior in these systems.

Extended previous extremal laws to more complex maximal set configurations.

Abstract

We consider stationary stochastic processes arising from dynamical systems by evaluating a given observable along the orbits of the system. We focus on the extremal behaviour of the process, which is related to the entrance in certain regions of the phase space, which correspond to neighbourhoods of the maximal set $\mathcal M$, i.e. the set of points where the observable is maximised. The main novelty here is the fact that we consider that the set $\mathcal M$ may have a countable number of points, which are associated by belonging to the orbit of a certain point, and may have accumulation points. In order to prove the existence of distributional limits and study the intensity of clustering, given by the Extremal Index, we generalise the conditions previously introduced in \cite{FFT12,FFT15}.