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Point-Shift Foliation of a Point Process

A point-shift $F$ maps each point of a point process $Φ$ to some point of $Φ$. For all translation invariant point-shifts $F$, the $F$-foliation of $Φ$ is a partition of the support of $Φ$ which is the discrete analogue of the stable manifold of $F$ on $Φ$. 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 $Φ$, and which are measure-preserving. The foils are not always stationary point processes. Nevertheless, they admit relative intensities with respect to one another.