Invariant measures and orbit equivalence for generalized Toeplitz subshifts

We show that for every metrizable Choquet simplex $K$ and for every group $G$, which is infinite, countable, amenable and residually finite, there exists a Toeplitz $G$-subshift whose set of shift-invariant probability measures is affine homeomorphic to $K$. Furthermore, we get that for every integer $d\geq 1$andeveryToeplitzflow$(X,T)$,thereexistsaToeplitz${\mathbb Z}^d$-subshiftwhichistopologicallyorbitequivalentto$(X,T)$.