Lifting mixing properties by Rokhlin cocycles

TL;DR

This paper investigates how mixing properties of a base automorphism can be transferred to skew product systems using Rokhlin cocycles, revealing conditions under which mixing properties are preserved or fail.

Contribution

It provides new criteria for lifting mixing properties via Rokhlin cocycles and characterizes when these properties are not preserved in skew products.

Findings

Mixing properties can be lifted from base automorphisms to skew products under certain conditions.

Non-preservation of mixing implies the existence of coboundary or quasi-coboundary Rokhlin cocycles on factors.

The study links ergodic and mixing behavior of skew products to properties of associated cocycles.

Abstract

We study the problem of lifting various mixing properties from a base automorphism $T\in {\rm Aut}\xbm$ to skew products of the form $\tfs$, where $\va:X\to G$ is a cocycle with values in a locally compact Abelian group $G$, $\cs=(S_g)_{g\in G}$ is a measurable representation of $G$ in ${\rm Aut}\ycn$ and $\tfs$ acts on the product space $(X\times Y,\cb\ot\cc,\mu\ot\nu)$ by $$\tfs(x,y)=(Tx,S_{\va(x)}(y)).$$ It is also shown that whenever $T$ is ergodic (mildly mixing, mixing) but $\tfs$ is not ergodic (is not mildly mixing, not mixing), then on a non-trivial factor $\ca\subset\cc$ of $\cs$ the corresponding Rokhlin cocycle $x\mapsto S_{\va(x)}|_{\ca}$ is a coboundary (a quasi-coboundary).