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

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