Ergodic properties that lift to compact group extensions

Abstract:

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.