Finitary isomorphisms of Poisson point processes

TL;DR

This paper presents an elementary, finitary construction of an isomorphism between Poisson point processes, advancing the understanding of measure-preserving actions in the context of amenable groups.

Contribution

It provides the first explicit, elementary construction of a finitary isomorphism between Poisson point processes, building on prior theoretical results.

Findings

Finitary isomorphism constructed explicitly

Simplifies previous abstract proofs

Enhances understanding of Poisson process symmetries

Abstract

As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss (J. Anal. Math. 48:1-141,1987) proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary.