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Transactions of the American Mathematical Society

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Interplay between finite topological rank minimal Cantor systems, $ \mathcal S$-adic subshifts and their complexity


Authors: Sebastián Donoso, Fabien Durand, Alejandro Maass and Samuel Petite
Journal: Trans. Amer. Math. Soc. 374 (2021), 3453-3489
MSC (2020): Primary 37B10, 68R15
DOI: https://doi.org/10.1090/tran/8315
Published electronically: February 23, 2021
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Abstract: Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable $ {\mathcal S}$-adic subshifts. This is done by establishing necessary and sufficient conditions for a minimal subshift to be of finite topological rank. As an application, we show that minimal subshifts with non-superlinear complexity (like many classical zero-entropy examples) have finite topological rank. Conversely, we analyze the complexity of $ {\mathcal S}$-adic subshifts and provide sufficient conditions for a finite topological rank subshift to have a non-superlinear complexity. This includes minimal Cantor systems given by Bratteli-Vershik representations whose tower levels have proportional heights and the so-called left to right $ {\mathcal S}$-adic subshifts. We also show that finite topological rank does not imply non-superlinear complexity. In the particular case of topological rank two subshifts, we prove their complexity is always subquadratic along a subsequence and their automorphism group is trivial.


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Additional Information

Sebastián Donoso
Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile & UMI-CNRS 2807, Beauchef 851, Santiago, Chile
Email: sdonoso@dim.uchile.cl

Fabien Durand
Affiliation: Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens cedex 1, France
Email: fabien.durand@u-picardie.fr

Alejandro Maass
Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile & UMI-CNRS 2807, Beauchef 851, Santiago, Chile
Email: amaass@dim.uchile.cl

Samuel Petite
Affiliation: Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens cedex 1, France
Email: samuel.petite@u-picardie.fr

DOI: https://doi.org/10.1090/tran/8315
Keywords: ${\mathcal{S}}$-adic subshifts, minimal Cantor systems, finite topological rank, recognizability, complexity
Received by editor(s): March 13, 2020
Received by editor(s) in revised form: August 31, 2020
Published electronically: February 23, 2021
Additional Notes: This research was partially supported by grants PIA-ANID AFB 170001 & Fondap 15090007, and grant ANID/MEC/80180045 hosted by University of O’Higgins.
The first and the third authors thank the hospitality of the LAMFA UMR 7352 CNRS-UPJV and the “poste rouge” CNRS program.
Article copyright: © Copyright 2021 American Mathematical Society