Exact number of ergodic invariant measures for Bratteli diagrams

TL;DR

This paper characterizes the number of ergodic invariant measures for Bratteli diagrams, providing criteria for unique ergodicity, explicit descriptions for finite rank cases, and conditions for prescribed numbers of measures.

Contribution

It introduces new criteria and explicit descriptions for the number of ergodic measures in Bratteli diagrams, including conditions for unique ergodicity and support structures.

Findings

Criteria for unique ergodicity of Bratteli diagrams

Explicit structure description for diagrams with a finite number of ergodic measures

Conditions for diagrams to have prescribed numbers of ergodic measures

Abstract

For a Bratteli diagram $B$, we study the simplex $\mathcal{M}_1(B)$ of probability measures on the path space of $B$ which are invariant with respect to the tail equivalence relation. Equivalently, $\mathcal{M}_1(B)$ 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 $\mathcal{M}_1(B)$ 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 $\mathcal{M}_1(B)$ is a singleton. For a finite rank $k$ Bratteli diagram $B$ having exactly $l \leq k$ 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.