Zero-dimensional isomorphic dynamical models

By an assignment we mean a mapping from a Choquet simplex $K$ to probability measure-preserving systems obeying some natural restrictions. We prove that if $\unicode[STIX]{x1D6F7}$ 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 $\unicode[STIX]{x1D6F7}$ to the Bauer simplex $K_{n}$ spanned by $E_{n}$ can be ‘embedded’ in some topological dynamical system, then $\unicode[STIX]{x1D6F7}$ can be ‘realized’ in a zero-dimensional system.