A pseudospectral approach to rigorous numerical estimation of Ruelle resonances of transfer operators

TL;DR

This paper introduces a pseudospectral method inspired by Householder's theorem for rigorously estimating Ruelle resonances of transfer operators in chaotic dynamical systems, applicable to various classes of maps.

Contribution

It presents a general, computer-assisted pseudospectral approach for the rigorous estimation of transfer operator resonances, broadening spectral analysis tools in dynamical systems.

Findings

Successfully implemented for analytic uniformly expanding maps of the circle.

Provides regions where resonances must exist, excluding others.

Framework is broadly applicable to spectral problems in dynamical systems.

Abstract

Resonances, isolated eigenvalues of a transfer operator acting on suitably chosen Banach spaces, play a fundamental role in understanding the statistical properties of chaotic dynamical systems. In this paper, we introduce a pseudospectral approach, inspired by Householder's theorem, for the rigorous, computer-assisted estimation of resonances, providing regions where resonances must exist and precluding the presence of resonances elsewhere. The approach is general, and applies to the transfer operators of a wide variety of chaotic systems, including Anosov/ Axiom A diffeomorphisms and piecewise expanding maps. We implement this approach computationally for a class of analytic uniformly expanding maps of the circle. We anticipate that the pseudospectral framework developed here will be broadly applicable to other spectral problems in dynamical systems and beyond.