Point-Shift Foliation of a Point Process
Abstract
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
⊥
, 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.