On weakly mixing and doubly ergodic nonsingular actions
Volume 103 / 2005
Colloquium Mathematicum 103 (2005), 247-264 MSC: Primary 37A40. DOI: 10.4064/cm103-2-10
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.