  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 $d\geq
1$ and every Toeplitz flow $(X,T)$, there exists a Toeplitz ${\mathbb
Z}^d$-subshift which is topologically orbit equivalent to $(X,T)$.
