On weakly mixing and doubly ergodic nonsingular actions
We study weak mixing and double ergodicity for nonsingular actions of locally compact Polish abelian groups. We show that if $T$ is a nonsingular action of $G$, then $T$ is weakly mixing if and only if for all cocompact subgroups $A$ of $G$ the action of $T$ restricted to $A$ is weakly mixing. We show that a doubly ergodic nonsingular action is weakly mixing and construct an infinite measure-preserving flow that is weakly mixing but not doubly ergodic. We also construct an infinite measure-preserving flow whose cartesian square is ergodic.