Finite Rank Bratteli Diagrams: Structure of Invariant Measures

TL;DR

This paper studies invariant measures on finite rank Bratteli diagrams, showing they derive from simple subdiagrams, and explores conditions for unique ergodicity and mixing properties of associated Vershik maps.

Contribution

It characterizes ergodic invariant measures as extensions from simple subdiagrams and introduces conditions for unique ergodicity and non-mixing behavior of Vershik maps.

Findings

Invariant measures are extensions from simple subdiagrams.

Exact finite rank implies non-strong mixing of Vershik map.

Consecutive ordering prevents strong mixing on finite rank diagrams.

Abstract

We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or "regular" infinite) is obtained by an extension from a simple subdiagram. We further investigate quantitative properties of these measures, which are mainly determined by the asymptotic behavior of products of incidence matrices. A number of sufficient conditions for unique ergodicity are obtained. One of these is a condition of exact finite rank, which parallels a similar notion in measurable dynamics. Several examples illustrate the broad range of possible behavior of finite type diagrams and invariant measures on them. We then prove that the Vershik map on the path space of an exact finite rank diagram cannot be strongly mixing, independent of the ordering. On the other hand, for the so-called "consecutive" ordering, the Vershik map is not strongly mixing on all finite rank diagrams.