Homeomorphic measures on stationary Bratteli diagrams

Abstract

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. 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 $\mu$ from S with respect to a homeomorphism. The properties of these measures related to the clopen values set $S(\mu )$ are studied. It is shown that for every measure $\mu$ in S there exists a subgroup G of $\mathbb{R}$ such that $S(\mu )$ is the intersection of G with [0,1], i.e. $S(\mu )$ is group-like. A criterion of goodness is proved for such measures. This result is used to classify the measures from S up to a homeomorphism. It is proved that for every good measure $\mu$ in S there exist countably many measures $\{{\mu }_i{\}}_{i\in \mathbb{N}}$ from S such that $\mu$ and ${\mu }_i$ are homeomorphic measures but the tail equivalence relations on corresponding Bratteli diagrams are not orbit equivalent.

36 pages, references added, typos fixed

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