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A compatible point-shift F maps, in a translation invariant way, each point of a stationary point process Φ to some point of Φ. It is fully determined by its associated point-map, f, which gives the image of the origin by F. It was proved by J. Mecke that if F is bijective, then the Palm probability of Φ is left invariant by the translation of -f . The initial question motivating this paper is the following generalization of this invariance result: in the nonbijective case, what probability measures on the set of counting measures are left invariant by the translation of -f ? The point-map-probabilities of Φ are defined from the action of the semigroup of point-map translations on the space... 

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

We analyze a class of signal-to-interference-and-noise-ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. Tn contrast to previous studies on STNR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets...