Ergodic properties that lift to compact group extensions

Source abstract

Let T T and R R be measure preserving, T T weakly mixing, R R ergodic, and let S S be conservative ergodic and nonsingular. Let T ~ \tilde T be a weakly mixing compact abelian group extension of T T . If T × S T \times S is ergodic then T ~ × S \tilde T \times S is ergodic. A corollary is a new proof that if T T is mildly mixing then so is T ~ \tilde T . A similar statement holds for other ergodic multiplier properties. Now let T ~ \tilde T be a weakly mixing type α \alpha compact affine G G extension of T T where α \alpha is an automorphism of G G . If T T and R R are disjoint and α \alpha or R R has entropy zero, then T ~ \tilde T and R R are disjoint. T ~ \tilde T is uniquely ergodic if and only if T T is uniquely ergodic and α \alpha has entropy zero. If T T is mildly mixing and T ~ \tilde T is weakly mixing then T ~ \tilde T is mildly mixing. We also provide a new proof that if T ~ \tilde T is weakly mixing then T ~ \tilde T has the K K -property if T T does.