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Perfect Orderings on Finite Rank Bratteli Diagrams

Published online by Cambridge University Press:  20 November 2018

S. Bezuglyi ,
J. Kwiatkowski  and
R. Yassawi
S. Bezuglyi
Affiliation:
Institute for Low Temperature Physics, Kharkov, Ukraine e-mail: bezuglyi@ilt.kharkov.ua
J. Kwiatkowski
Affiliation:
Department of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn, Poland e-mail: jkwiat@mat.uni.torun.pl
R. Yassawi
Affiliation:
Department of Mathematics, Trent University, PeterboroughON e-mail: ryassawi@trentu.ca
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Abstract

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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 $1\,\le \,j\,\le \,k$ 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

37B1037A20Bratteli diagramsVershik maps
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 1 , February 2014 , pp. 57 - 101
Copyright
Copyright © Canadian Mathematical Society 2014

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