Exact number of ergodic invariant measures for Bratteli diagrams

[摘要] For a Bratteli diagram B, we study the simplex M-1(B) of probability measures on the path space of B which are invariant with respect to the tail equivalence relation. Equivalently, M-1 (B) is formed by probability measures invariant with respect to a homeomorphism of a Cantor set. We study relations between the number of ergodic measures from M-1(B) and the structure and properties of the diagram B. We prove a criterion and find sufficient conditions of unique ergodicity of a Bratteli diagram, in which case the simplex M-1(B) is a singleton. For a finite rank k Bratteli diagram B having exactly l <= k ergodic invariant measures, we explicitly describe the structure of the diagram and find the subdiagrams which support these measures. We find sufficient conditions under which: (i) a Bratteli diagram has a prescribed number (finite or infinite) of ergodic invariant measures, and (ii) the extension of a measure from a uniquely ergodic subdiagram gives a finite ergodic invariant measure. Several examples, including stationary Bratteli diagrams, Pascal-Bratteli diagrams, and Toeplitz flows, are considered. (C) 2019 Elsevier Inc. All rights reserved.

[发布日期] 2019-12-15

[关键词] Brattcli diagram;Ergodic invariant measure;Tail equivalence relation;Unique ergodicity;Finite rank