2010/08/04 by Sergey Bezuglyi, S. Bezuglyi, Bezuglyi, S. +3
Mathematics · #Mathematical Dynamics and Fractals #advanced mathematical theories #math.DS #msc:28C15 #msc:28D05 #msc:37A05 #msc:37B05
paper · pdf · doi:10.48550/arxiv.1008.0850
36 pages, references added, typos fixed
arxiv created 2010/09/28 · arxiv updated 2010/09/30
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 μ from S with respect to a homeomorphism. The properties of these measures related to the clopen values set S(μ) are studied. It is shown that for every measure μ in S there exists a subgroup G of \mathbb R such that S(μ) is the intersection of G with [0,1], i.e. S(μ) 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 μ in S there exist countably many measures \μi\i∈ \mathbb N from S such that μ and μi are homeomorphic measures but the tail equivalence relations on corresponding Bratteli diagrams are not orbit equivalent.