Minimal models for noninvertible and not uniquely ergodic systems

Let (Y, S) be a (not necessarilly invertible) topological dynamical system on a zero-dimensional metric space Y without periodic points. Then there exists a minimal system (X, T) with the same siniplex of invariant measures as (Y, S). More precisely, there exists a Borel isomorphisin between full sets in Y and X such that the adjoint map on measures is a homeomorphism between the corresponding sets of invariant measures in the weak* topology. As an application we construct a minimal system carrying isomorphic copies of all nonatomic invariant measures.