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Poisson suspensions and entropy for infinite transformations. (English) Zbl 1196.37015
This paper considers the relationship between Krengel, Parry, and Poisson entropies for \(\sigma\)-finite measure preserving transformations. In particular, it is shown that for type \(II_{\infty}\) transformations, the Parry entropy is a lower bound for the Poisson entropy. Another result shows that for a quasi-finite measure-preserving transformation of type \(II_{\infty}\), the three entropies are equal. Also discussed are Pinsker factors associated to these entropies. The paper finishes with some relationships between Poisson entropy and joinings, Poisson factors and disjointness, Poisson suspensions and distality, and open questions relating the three types of entropy.

MSC:
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37A05 Dynamical aspects of measure-preserving transformations
28D20 Entropy and other invariants
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