Journal article
Perfect orderings on finite rank Bratteli diagrams
Canadian journal of mathematics, Vol.66(1), pp.57-101
2014
DOI: 10.4153/CJM-2013-041-6
Abstract
Abstract Given a Bratteli diagram B , we study the set 𝒪 B of all possible orderings on B and its subset P B consisting of perfect orderings that produce Bratteli–Vershik topological dynamical systems (Vershik maps). We give necessary and sufficient conditions for the ordering ω 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 𝒪 B with product measure μ and prove that there is some 1 ≤ j ≤ k such that μ -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 μ -almost all orderings are imperfect.
Details
- Title: Subtitle
- Perfect orderings on finite rank Bratteli diagrams
- Creators
- S BezuglyiJ KwiatkowskiR Yassawi
- Resource Type
- Journal article
- Publication Details
- Canadian journal of mathematics, Vol.66(1), pp.57-101
- DOI
- 10.4153/CJM-2013-041-6
- ISSN
- 0008-414X
- eISSN
- 1496-4279
- Language
- English
- Date published
- 2014
- Academic Unit
- Mathematics
- Record Identifier
- 9983985709402771
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