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Point-shift foliation of a point process. (English) Zbl 1454.60061
Summary: 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.
Reviewer: Reviewer (Berlin)

##### MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 60G10 Stationary stochastic processes 60G57 Random measures
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