Rare events, exponential hitting times and extremal indices via spectral perturbation

TL;DR

This paper explores the connection between spectral perturbation of transfer operators and the statistical properties of rare events, such as exponential hitting times and extremal indices, in chaotic dynamical systems.

Contribution

It introduces an eigenvalue perturbation approach linking transfer operators to extreme value theory and applies it to various piecewise expanding and hyperbolic systems.

Findings

Eigenvalue perturbation formulas relate to exponential hitting time distributions.

The theory applies to several classes of piecewise expanding systems.

Potential extension to piecewise hyperbolic systems is discussed.

Abstract

We discuss how an eigenvalue perturbation formula for transfer operators of dynamical systems is related to exponential hitting time distributions and extreme value theory for processes generated by chaotic dynamical systems. We also list a number of piecewise expanding systems to which this general theory applies and discuss the prospects to apply this theory to some classes of piecewise hyperbolic systems.