Quasi-Similarity, Entropy and Disjointness of Ergodic Actions

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.