Subdiagrams and invariant measures on Bratteli diagrams

TL;DR

This paper investigates invariant measures on Bratteli diagrams, providing criteria for measure extension from subdiagrams, conditions for positive measure, and classifying ergodic measures for finite rank diagrams, including rank two cases.

Contribution

It offers new criteria for extending measures from subdiagrams and classifies ergodic measures on finite rank Bratteli diagrams, especially for rank two.

Findings

Criteria for measure extension from subdiagrams.

Conditions for positive measure of subdiagram path spaces.

Complete classification of ergodic measures for rank two diagrams.

Abstract

We study ergodic finite and infinite measures defined on the path space $X_B$ of a Bratteli diagram $B$ which are invariant with respect to the tail equivalence relation on $X_B$. Our interest is focused on measures supported by vertex and edge subdiagrams of $B$. We give several criteria when a finite invariant measure defined on the path space of a subdiagram of $B$ extends to a finite invariant measure on $B$. Given a finite ergodic measure on a Bratteli diagram $B$ and a subdiagram $B'$ of $B$, we find the necessary and sufficient conditions under which the measure of the path space $X_{B'}$ of $B'$ is positive. For a class of Bratteli diagrams of finite rank, we determine when they have maximal possible number of ergodic invariant measures. The case of diagrams of rank two is completely studied. We include also an example which explicitly illustrates the proved results.