Metric properties of transformations of $G$-spaces

Abstract:

The measure-preserving transformation $T$ acts on a Lebesgue space $(M,\mathcal {B},\mu )$ which is also a $G$-space for a compact separable group $G$. It is proved that if the factor-transformation on the space of $G$-orbits has completely positive entropy and a certain condition regarding the relations between the actions of $G$ and $T$ is satisfied, then $T$ weakly mixing implies $T$ has completely positive entropy.