Finite Rank Bratteli Diagrams: Structure of Invariant Measures

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.