Sampling local properties of attractors via Extreme Value Theory

TL;DR

This paper develops formulas to analyze the local geometric properties of attractors in dynamical systems using Extreme Value Theory, applicable to both chaotic and contractive systems under noise perturbations.

Contribution

It introduces a method to compute asymptotic distribution coefficients for maxima of observables, revealing local properties of stationary measures in perturbed dynamical systems.

Findings

Chaotic and contractive systems exhibit similar behavior under noise.

The method provides a way to quantify local geometrical properties.

Applicable to systems with additive and observational noise.

Abstract

We provide formulas to compute the coefficients entering the affine scaling needed to get a non-degenerate function for the asymptotic distribution of the maxima of some kind of observable computed along the orbit of a randomly perturbed dynamical system. This will give information on the local geometrical properties of the stationary measure. We will consider systems perturbed with additive noise and with observational noise. Moreover we will apply our techniques to chaotic systems and to contractive systems, showing that both share the same qualitative behavior when perturbed.