Duality between Eigenfunctions and Eigendistributions of Ruelle and Koopman operators via an integral kernel

TL;DR

This paper explores the duality between eigenfunctions and eigendistributions of Ruelle and Koopman operators in symbolic dynamics, using integral kernel techniques to establish a conjugate relationship for isolated eigenvalues.

Contribution

It demonstrates a duality between eigendistributions and eigenvectors of Ruelle operators with conjugate potentials, and extends this relationship to Koopman operators, employing the involution kernel method.

Findings

Eigendistributions are dual to eigenvectors of conjugate Ruelle operators.

Eigenfunctions and eigendistributions of Koopman operators satisfy a similar duality.

The involution kernel is key to establishing these dualities.

Abstract

We consider the classical dynamics given by a one sided shift on the Bernoulli space of $d$ symbols. We study, on the space of H\"older functions, the eigendistributions of the Ruelle operator with a given potential. Our main theorem shows that for any isolated eigenvalue, the eigendistributions of such Ruelle operator are dual to eigenvectors of a Ruelle operator with a conjugate potential. We also show that the eigenfunctions and eigendistributions of the Koopman operator satisfy a similar relationship. To show such results we employ an integral kernel technique, where the kernel used is the involution kernel.