Generic extensions of ergodic actions

Valery V. Ryzhikov

Published 2022-09-19Version 1

The article is devoted to the generic extensions of measure-preserving actions, the lifting of some their invariants. The work is stimulated by the recent results of Austin, Glasner, Thouvenot and Weiss. We prove that the $P$-entropy of the generic extensions with finite $P$-entropy (an invariant of the Kushnirenko entropy type) is infinite. We get in a different way the result of the mentioned authors that the generic extension of an action with zero classical entropy is not isomorphic to this action. It is shown that typical extensions preserve the singularity of the spectrum, some approximation and asymptotic properties of the base. We also note that the possibility to lift via the typical extensions the algebraic property "to be a composition of two involutions" depends on the statistical properties of the base. We consider also typical measurable automorphism families of the probability space (in short, communities) that arise by extending the identity operator. It is shown that for the typical community, under iterations, the automorphisms included in it behave coherently on some time sequences, although on other sequences they exhibit individual behavior. The topic of the article seems to be extensive as well, since for each invariant of the group actions the question arises about its lifting by the generic extensions. Similar problems arise by studying, in the spirit of Schnurr, the space of actions that preserve a fixed subalgebra and their generic relative invariants. Also of interest may be the typical extensions both for actions on spaces with sigma-finite measure and for mixing actions equipped with the complete Alpern-Tikhonov metric.