Point-shift foliation of a point process

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 is measure-preserving. The foils are not always stationary point processes. Nevertheless, they admit relative intensities with respect to one another.