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# Perfect Orderings on Finite Rank Bratteli Diagrams

Published:2013-12-04
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
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## 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 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
 MSC Classifications: 37B10 - Symbolic dynamics [See also 37Cxx, 37Dxx] 37A20 - Orbit equivalence, cocycles, ergodic equivalence relations

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