Geodesic
Quasi-similarity, entropy and disjointness of ergodic actions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 58 (2024) no. 1, pp. 117-124

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We answer a question posed by Vershik regarding connections between quasi-similarity of dynamical systems and Kolmogorov entropy. We prove that all Bernoulli actions of a given countably infinite group are quasi-similar to each other. The existence of non-Bernoulli actions in the same quasi-similarity class is an open problem. A notion opposite to quasi-similarity is that of disjointness (or independence) of actions. Pinsker proved that a deterministic action is independent from an action with completely positive entropy. Using joinings, we obtain the following generalization of Pinsker's theorem: an action with zero $P$-entropy (an invariant defined by Kirillov and Kushnirenko) and an action with completely positive $P$-entropy are disjoint.
Keywords: disjointness of measure-preserving actions, quasi-similarity, entropy invariants
Mots-clés : Poisson suspensions. 
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     author = {Valerii Ryzhikov and Jean-Paul Thouvenot},
     title = {Quasi-similarity, entropy and disjointness of ergodic actions},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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     year = {2024},
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Valerii Ryzhikov; Jean-Paul Thouvenot. Quasi-similarity, entropy and disjointness of ergodic actions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 58 (2024) no. 1, pp. 117-124. http://geodesic.mathdoc.fr/item/FAA_2024_58_1_a7/