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On topological rank of factors of Cantor minimal systems

Part of: Connections with other structures, applications Topological dynamics

Published online by Cambridge University Press:  08 June 2021

NASSER GOLESTANI  and
MARYAM HOSSEINI
NASSER GOLESTANI
Affiliation:
Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P. O. Box 14115-134, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran (e-mail: n.golestani@modares.ac.ir)
MARYAM HOSSEINI*
Affiliation:
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran (e-mail: n.golestani@modares.ac.ir)
*
e-mail: maryhoseini@ipm.ir
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Abstract

A Cantor minimal system is of finite topological rank if it has a Bratteli–Vershik representation whose number of vertices per level is uniformly bounded. We prove that if the topological rank of a minimal dynamical system on a Cantor set is finite, then all its minimal Cantor factors have finite topological rank as well. This gives an affirmative answer to a question posed by Donoso, Durand, Maass, and Petite in full generality. As a consequence, we obtain the dichotomy of Downarowicz and Maass for Cantor factors of finite-rank Cantor minimal systems: they are either odometers or subshifts.

Keywords

Cantor minimal systemtopological ranktopological factorordered Bratteli diagramordered premorphism

MSC classification

Primary: 54H20: Topological dynamics
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 37B10: Symbolic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 9 , September 2022 , pp. 2866 - 2889
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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