Perfect orderings on Bratteli diagrams

Sergey Bezuglyi; Jan Kwiatkowski; Reem Yassawi

doi:10.48550/arxiv.1204.1621

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Perfect orderings on Bratteli diagrams

Sergey Bezuglyi, Jan Kwiatkowski and Reem Yassawi

ArXiv.org

Cornell University

04/07/2012

DOI: 10.48550/arxiv.1204.1621

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Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

Abstract

Given a Bratteli diagram B, we study the set O(B) of all possible orderings w on a Bratteli diagram B and its subset P(B) consisting of `perfect' orderings that produce Bratteli-Vershik dynamical systems (Vershik maps). We give necessary and sufficient conditions for w 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 affects significantly the values of the entries of the incidence matrices and the structure of the diagram B. Endowing the set O(B) with product measure, we prove that there is some j such that almost all orderings on B have j maximal and minimal paths, and that if j is strictly greater than the number of minimal components that B has, then almost all orderings are imperfect.

Mathematics - Dynamical Systems

Details

Title: Subtitle: Perfect orderings on Bratteli diagrams
Creators: Sergey Bezuglyi
Jan Kwiatkowski
Reem Yassawi
Resource Type: Preprint
Publication Details: ArXiv.org
DOI: 10.48550/arxiv.1204.1621
ISSN: 2331-8422
Publisher: Cornell University; Ithaca, New York
Language: English
Date posted: 04/07/2012
Academic Unit: Mathematics
Record Identifier: 9984936490702771

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