Abstract view
Perfect Orderings on Finite Rank Bratteli Diagrams


Published:20131204
Printed: Feb 2014
S. Bezuglyi,
Institute for Low Temperature Physics, Kharkov, Ukraine
J. Kwiatkowski,
The University of Computer Science and Economics, Olsztyn, Poland
R. Yassawi,
Department of Mathematics, Trent University, Peterborough, ON
Abstract
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
BratteliVershik 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 nonsimple 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.