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Bratteli diagrams where random orders are imperfect

Authors: J. Janssen, A. Quas and R. Yassawi
Journal: Proc. Amer. Math. Soc. 145 (2017), 721-735
MSC (2010): Primary 37B10; Secondary 37A20
Published electronically: October 31, 2016
MathSciNet review: 3577873
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Abstract: For the simple Bratteli diagrams $ B$ where there is a single edge connecting any two vertices in consecutive levels, we show that a random order has uncountably many infinite paths if and only if the growth rate of the level-$ n$ vertex sets is super-linear. This gives us the dichotomy: a random order on a slowly growing Bratteli diagram admits a homeomorphism, while a random order on a quickly growing Bratteli diagram does not. We also show that for a large family of infinite rank Bratteli diagrams $ B$, a random order on $ B$ does not admit a continuous Vershik map.

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Additional Information

J. Janssen
Affiliation: Department of Mathematics and Statistics, Dalhousie University, 6316 Coburg Road, P.O. Box 15000, Halifax, Nova Scotia, Canada

A. Quas
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada

R. Yassawi
Affiliation: Department of Mathematics, Trent University, 1600 West Bank Drive, Peterborough, ON, Canada

Keywords: Bratteli diagrams, Vershik maps
Received by editor(s): June 22, 2015
Received by editor(s) in revised form: April 13, 2016
Published electronically: October 31, 2016
Additional Notes: The first two authors were partially supported by NSERC Discovery Grants
Communicated by: Nimish Shah
Article copyright: © Copyright 2016 American Mathematical Society

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