Zero-dimensional isomorphic dynamical models

By an emph{assignment} we mean a mapping from a Choquet simplex $K$ to probability measure-preserving systems, obeying some natural restrictions. We prove that if $Phi$ is an aperiodic assignment on a Choquet simplex $K$ such that the set of extreme points $mathsf{ex}K$ is a countable union $bigcup_n E_n$, where each set $E_n$ is compact, zero-dimensional, and the restriction of $Phi$ to the Bauer simplex $K_n$ spanned by $E_n$ can be `embedded in some topological dynamical system, then $Phi$ can be `realized in a zero-dimensional system.