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Ergodic properties that lift to compact group extensions. (English) Zbl 0636.28007
Summary: Let T and R be measure preserving, T weakly mixing, R ergodic, and let S be conservative ergodic and nonsingular. Let $\tilde T$ be a weakly mixing compact Abelian group extension of T. If $T\times S$ is ergodic then $\tilde T\times S$ is ergodic. A corollary is a new proof that if T is mildly mixing then so is $\tilde T.$ A similar statement holds for other ergodic multiplier properties. Now let $\tilde T$ be a weakly mixing type $\alpha$ compact affine G extension of T where $\alpha$ is an automorphism of G. If T and R are disjoint and $\alpha$ or R has entropy zero, then $\tilde T$ and R are disjoint. $\tilde T$ is uniquely ergodic if and only if T is uniquely ergodic and $\alpha$ has entropy zero. If T is mildly mixing and $\tilde T$ is weakly mixing then $\tilde T$ is mildly mixing. We also provide a new proof that if $\tilde T$ is weakly mixing then $\tilde T$ has the K-property if T does.

28D05Measure-preserving transformations
28D20Entropy and other measure-theoretic invariants
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