Isomorphisms of Poisson systems over locally compact groups

TL;DR

This paper demonstrates that all Poisson systems over certain non-compact, non-amenable locally compact groups are finitarily isomorphic, extending classical results and providing new examples of isomorphisms in non-amenable group actions.

Contribution

It proves all Poisson systems over a broad class of non-amenable, non-compact groups are finitarily isomorphic, and generalizes splitting results to these groups.

Findings

All Poisson systems over the considered groups are finitarily isomorphic.

Poisson systems and their products are finitarily isomorphic.

Poisson systems can be decomposed into sums of systems with smaller intensities.

Abstract

A Poisson system is a Poisson point process and a group action, together forming a measure-preserving dynamical system. Ornstein and Weiss proved Poisson systems over many amenable groups were isomorphic in their 1987 paper. We consider Poisson systems over non-discrete, non-compact, locally compact Polish groups, and we prove by construction all Poisson systems over such a group are finitarily isomorphic, producing examples of isomorphisms for non-amenable group actions. As a corollary, we prove Poisson systems and products of Poisson systems are finitarily isomorphic. For a Poisson system over a group belonging to a slightly more restrictive class than above, we further prove it splits into two Poisson systems whose intensities sum to the intensity of the original, generalizing the same result for Poisson systems over Euclidean space proved by Holroyd, Lyons, and Soo.