Relatively weakly mixing models for dynamical systems

A classical result in ergodic theory says that there always exists a topological model for any factor map π : (X,X , μ, T ) → (Y,Y, ν, S) of ergodic systems. That is, there is some topological factor map π : (X, T ) → (Ŷ , Ŝ) and invariant measures μ, ν such that the diagram (X,X , μ, T ) φ −−−−→ (X, X , μ, T ) π y yπ (Y,Y, ν, S) ψ −−−−→ (Ŷ , Ŷ, ν, Ŝ) is commutative, where φ and ψ are measure theoretical isomorphisms. In this paper, we show that one can require that in above result π is either weakly mixing or finite-to-one. Also we present some related questions in the paper.