Invariant measures and orbit equivalence for generalized Toeplitz subshifts

Abstract

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