Homeomorphic measures on stationary Bratteli diagrams

We study the set S of ergodic probability Borel measures on stationary non-simple Bratteli diagrams which are invariant with respect to the tail equivalence relation R. Equivalently, the set S is formed by ergodic probability measures invariant with respect to aperiodic substitution dynamical systems. The paper is devoted to the classification of measures μ from S with respect to a homeomorphism. The properties of the clopen values set S(μ) are studied. It is shown that for every measure μ∈S there exists a subgroup G⊂R such that S(μ)=G∩[0,1]. A criterion of goodness is proved for such measures. Based on this result, the measures from S are classified up to a homeomorphism. We prove that for every good measure μ∈S there exist countably many measures {μi}_i∈N⊂S such that the measures μ and μi are homeomorphic but the tail equivalence relations on the corresponding Bratteli diagrams are not orbit equivalent.