Perfect orderings on finite rank Bratteli diagrams

Given a Bratteli diagram B, we study the set OB of all possible orderings on B and its subset PB consisting of perfect orderings that produce Bratteli–Vershik topological dynamical systems (Vershik maps). We give necessary and sufficient conditions for the ordering ω 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 constrains significantly the values of the entries of the incidence matrices and the structure of the diagram B. Our proofs are based on the new notions of skeletons and associated graphs, defined and studied in the paper. For a Bratteli diagram B of rank k, we endow the set OB with product measure µ and prove that there is some 1 ≤ j ≤ k such that µalmost all orderings on B have j maximal and j minimal paths. If j is strictly greater than the number of minimal components that B has, then µ-almost all orderings are imperfect