Quasi-similarity, entropy and disjointness of ergodic actions

Valerii Ryzhikov ; Jean-Paul Thouvenot

Funkcionalʹnyj analiz i ego priloženiâ, Tome 58 (2024) no. 1, pp. 117-124.

Résumé

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.

Détail

Keywords: disjointness of measure-preserving actions, quasi-similarity, entropy invariants
Mots-clés : Poisson suspensions.

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