1. CJM 2013 (vol 66 pp. 57)
|Perfect Orderings on Finite Rank Bratteli Diagrams|
Given a Bratteli diagram $B$, we study the set $\mathcal O_B$ of all possible orderings on $B$ and its subset $\mathcal P_B$ consisting of perfect orderings that produce Bratteli-Vershik topological dynamical systems (Vershik maps). We give necessary and sufficient conditions for the ordering $\omega$ 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 $\mathcal O_B$ with product measure $\mu$ and prove that there is some $1 \leq j\leq k$ such that $\mu$-almost 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 $\mu$-almost all orderings are imperfect.
Keywords:Bratteli diagrams, Vershik maps