Bratteli diagrams where random orders are imperfect

Janssen, J.; Quas, A. and Yassawi, R. (2017). Bratteli diagrams where random orders are imperfect. Proceedings of the American Mathematical Society, 145(2) pp. 721–735.

DOI: https://doi.org/10.1090/proc/13284

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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