Poisson suspensions and Sushis
Résumé
In this paper, we prove that ergodic point processes with moments of all orders, driven by particular infinite measure preserving transformations,
have to be a superposition of shifted Poisson processes.
This rigidity result has a lot of implications in terms of joining and disjointness for the corresponding Poisson suspension.
In particular, we prove that its ergodic self-joinings are Poisson joinings, which provides an analog, in the Poissonian context, of the GAG property for Gaussian dynamical systems.
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