My recollection is that in one of the last chapters of the book by Denker-Grillenberger-Sigmund, there is a theorem which says something like: given $n$ ergodic measure-preserving transformations with entropy strictly less than $\log d$, there is a minimal subshift of the full shift on $d$ symbols which has precisely these invariant measures. Going to the Springer website, I think it's chapter 31, but unfortunately I can't get through the paywall, so I could be quite wrong...