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.MSC: 37B05, 37A25, 37A20.

Topological Rank Does Not Increase by Natural Extension of Cantor Minimals

“…Thus, we get the following corollary. In [BKMS13, Proposition 2.14], Bezuglyi et al gave a proof to the 'folklore' result that a Bratteli-Vershik system with rank K has at most K ergodic measures (see also [BKMS13,Theorem 3.3(a)] and [S16a, Theorem 6.2]). Therefore, we get the following corollary.…”

On characters of inductive limits of symmetric groups

“…(a) χ is regular, 3 i.e. χ(1) = 1 and χ(g) = 0 for all g = 1; or (b) there exist unique parameters α 1 , .…”

“…either of type II 1 or I n with n < ∞, see the details of this classifications in [37,Chapter 5]. 3 Throughout the paper, by regular characters we mean the characters corresponding to regular representations. Theorem 1.2 shows that the description of characters for each given full group is reduced to the description of ergodic measures on the corresponding Bratteli diagram.…”

“…Theorem 1.2 shows that the description of characters for each given full group is reduced to the description of ergodic measures on the corresponding Bratteli diagram. We mention the following papers [2,3,10,32,33] devoted to the study of ergodic measures on Bratteli diagrams. Theorem 1.2 classifies characters for simple full groups and, as a result, all II 1 -factor representations of G up to quasi-equivalence.…”

“…We mention that the proof in [16] is based on the asymptotic theory of characters. For more examples of Bratteli diagrams and, hence, of full groups with different number of ergodic measures, we refer the reader to the papers [3,10].…”