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 Phi is an aperiodic assignment on a Choquet simplex K such that the set of extreme points exK is a countable union boolean OR(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.