A class of multipliers for $$\mathcal{W}^ \bot $$

Abstract

Let\(\mathcal{W}^ \bot \) denote the class of ergodic probability preserving transformations which are disjoint from every weakly mixing system. Let\(\mathcal{M}(\mathcal{W}^ \bot )\) be the class of multipliers for\(\mathcal{W}^ \bot \), i.e. ergodic transformations whose all ergodic joinings with any element of\(\mathcal{W}^ \bot \) are also in\(\mathcal{W}^ \bot \). Fix an ergodic rotationT, a mildly mixing actionS of a locally compact second countable groupG and an ergodic cocycle ϕ forT with values inG. The main result of the paper is a sufficient (and also necessary by [LeP] whenG is countable Abelian andS is Bernoullian) condition for the skew product build fromT, ϕ andS to be an element of\(\mathcal{M}(\mathcal{W}^ \bot )\). Moreover, the self-joinings of such extensions ofT are described with an application to study semisimple extensions of rotations.