Generic extensions of ergodic actions

TL;DR

This paper investigates the properties of generic measure-preserving extensions, revealing their infinite P-entropy, non-isomorphism to original actions, and various spectral and mixing characteristics, highlighting complex dynamic behaviors.

Contribution

It provides new insights into the ergodic properties of generic extensions, including entropy, spectral, and mixing features, and explores the behavior of generic cocycles and automorphisms.

Findings

Generic extensions with finite P-entropy have infinite P-entropy.

A generic extension of a deterministic action is almost surely not isomorphic to it.

Generic cocycles are recurrent and typical extensions preserve spectral singularity.

Abstract

The article considers generic extensions of measure-preserving actions. We prove that the P-entropy of the generic extensions with finite P-entropy is infinite. This is exploited to obtain the result by Austin, Glasner, Thouvenot, and Weiss that the generic extension of an deterministic action is not isomorphic to it. We show also that generic cocycles are recurrent; as well as typical extensions preserve the singularity of the spectrum, partial rigidity, mildly mixing, and mixing. At the same time, the lifting of some algebraic properties under the generic extension may depend on the statistical properties of the base. The typical measurable families of automorphisms are considered as well. The dynamic behavior of such families is a bit unusual. It is characterized by a combination of the dynamic conformism with the dynamic individualism of the representatives of the generic family.