Orders that yield homeomorphisms on Bratteli diagrams

Bezuglyi, Sergey and Yassawi, Reem (2017). Orders that yield homeomorphisms on Bratteli diagrams. Dynamical Systems, 32(2) pp. 249–282.

DOI: https://doi.org/10.1080/14689367.2016.1197888

Abstract

We call an order ω on a Bratteli diagram B perfect if its Vershik map is a homeomorphism. In this paper we study the set of orders on a Bratteli diagram and find necessary and sufficient conditions for an order to be perfect, in particular when the order has several extremal paths. This work generalizes previous results obtained for finite rank Bratteli diagrams. We describe an explicit procedure to create perfect orderings on Bratteli diagrams based on the study of certain relations between the entries of the diagram’s incidence matrices and properties of the associated graphs, with the latter relations characterizing diagrams which support perfect orderings. Also, we apply our theory to give a new combinatorial proof of the fact that the dimension group of a diagram supporting perfect orderings with k maximal paths has a copy of ℤ^k-1 contained in its infinitesimal subgroup. Under certain conditions, we show that a similar result holds if the diagram supports countably many maximal paths. Our results are illustrated by numerous examples.

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