Point-Shift Foliation of a Point Process

TL;DR

This paper introduces the concept of point-shift foliations in point processes, classifying their behavior and establishing properties of the resulting partitions, with implications for understanding the structure of point processes.

Contribution

It provides a novel classification framework for point-shifts via foliations and constructs measure-preserving shifts related to these foliations, advancing the theoretical understanding of point process dynamics.

Findings

Existence of a measure-preserving point-shift with orbits as F-foils.

Foliations lead to a classification of point-shift behaviors.

Foils may not always be stationary but have relative intensities.

Abstract

A point-shift $F$ maps each point of a point process $\Phi$ to some point of $\Phi$. For all translation invariant point-shifts $F$, the $F$-foliation of $\Phi$ is a partition of the support of $\Phi$ which is the discrete analogue of the stable manifold of $F$ on $\Phi$. It is first shown that foliations lead to a classification of the behavior of point-shifts on point processes. Both qualitative and quantitative properties of foliations are then established. It is shown that for all point-shifts $F$, there exists a point-shift $F_\bot$, the orbits of which are the $F$-foils of $\Phi$, and which are measure-preserving. The foils are not always stationary point processes. Nevertheless, they admit relative intensities with respect to one another.