Perfect orderings on Bratteli diagrams

TL;DR

This paper characterizes perfect orderings on Bratteli diagrams that produce Vershik maps, describes classes of diagrams without such maps, and analyzes the typical structure of orderings under a natural measure.

Contribution

It provides necessary and sufficient conditions for perfect orderings, describes non-simple diagrams without Vershik maps, and studies the measure-theoretic properties of orderings.

Findings

Almost all orderings have a fixed number of maximal and minimal paths.

Existence of perfect orderings depends on incidence matrix entries.

Most orderings are imperfect if the number of maximal paths exceeds minimal components.

Abstract

Given a Bratteli diagram B, we study the set O(B) of all possible orderings w on a Bratteli diagram B and its subset P(B) consisting of `perfect' orderings that produce Bratteli-Vershik dynamical systems (Vershik maps). We give necessary and sufficient conditions for w to be perfect. On the other hand, a wide class of non-simple Bratteli diagrams that do not admit Vershik maps is explicitly described. In the case of finite rank Bratteli diagrams, we show that the existence of perfect orderings with a prescribed number of extreme paths affects significantly the values of the entries of the incidence matrices and the structure of the diagram B. Endowing the set O(B) with product measure, we prove that there is some j such that almost all orderings on B have j maximal and minimal paths, and that if j is strictly greater than the number of minimal components that B has, then almost all orderings are imperfect.