Quasi-similarity and joining-stable invariants of ergodic actions

Valery V. Ryzhikov, Jean-Paul Thouvenot

Published 2023-03-16Version 1

Answering Vershik's question we show that quasi-similarity does not conserve the entropy, proving quasi-similarity of all Bernoulli actions of the groups with an element of infinite order. We produce joining-stable invariants considering zero $P$-entropy systems and prove an analog of Pinsker's theorem establishing the disjointness of such systems with the actions of comletely positive $P$-entropy. Then we apply Poisson suspensions to get a class of the corresponding examples.