Regular PI metric flows are equicontinuous

Abstract:

Let $(X,T)$ be a metrizable minimal flow. We show that a homomorphism $X \stackrel {\pi }{\to } Y$, which is regular, and PI can be decomposed as $X \stackrel {\sigma }{\to } Z \stackrel {\rho }{\to } Y$, $\pi = \rho \circ \sigma$, where $\rho$ is proximal and $\sigma$ is a compact group extension. In particular, assuming further that $T$ is abelian and taking $Y$ to be the trivial one point flow, we find that a metric regular PI flow is a compact group rotation.