Probabilistic potential theory and induction of dynamical systems

TL;DR

This paper develops a probabilistic potential theory approach to measure-preserving dynamical systems, introducing a balayage formula that generalizes measure invariance under induced maps and explores invariance properties of certain dynamical invariants.

Contribution

It introduces a probabilistic balayage formula for measure-preserving systems and proves invariance of the Green-Kubo formula and a new degree 3 invariant under induction.

Findings

Balayage formula generalizes measure invariance under induced maps.

Green-Kubo formula remains invariant under induction in some cases.

A new degree 3 invariant is shown to be invariant under induction.

Abstract

In this article, we outline a version of a balayage formula in probabilistic potential theory adapted to measure-preserving dynamical systems. This balayage identity generalizes the property that induced maps preserve the restriction of the original invariant measure. As an application, we prove in some cases the invariance under induction of the Green-Kubo formula, as well as the invariance of a new degree 3 invariant.