Homeomorphic measures on stationary Bratteli diagrams

TL;DR

This paper classifies ergodic measures on stationary Bratteli diagrams up to homeomorphism, revealing their structure as group-like sets and distinguishing measures by their topological properties.

Contribution

It introduces a criterion for measure goodness and demonstrates the existence of measures that are homeomorphic but not orbit equivalent, advancing the understanding of measure classification.

Findings

Measures in S have group-like clopen value sets.

A criterion for measure goodness is established.

Existence of measures that are homeomorphic but not orbit equivalent.

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.