Deterministic dynamics on stationary point process in Rd are built upon compatible point-shifts: translation invariant mappings from each point of the process to another. When a point-shift is applied multiple times to a point-process it creates a sequence of distributions, namely, the distributions of point process given there is a point of the nth iteration of the point-shift at the origin. We will introduce the notion of marked stochastic point-shifts. Marked Stochastic point-shifts use not only the realization of the point-process to decide how to map each point of the process, but also extra information coming from enlarging the probability space through marks. Examples are provided and the discussion is motivated by an application in the field of theoretical Computer Science.
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