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INTERPLAY BETWEEN FINITE TOPOLOGICAL RANK MINIMAL CANTOR SYSTEMS, S-ADIC SUBSHIFTS AND THEIR COMPLEXITY
Donoso, Sebastian
Durand, Fabien
Maass, Alejandro
Petite, Samuel
Universidad de Santiago de Chile [Santiago] (USACH)
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA) ; Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS)
Center for Mathematical Modeling (CMM) ; Universidad de Chile = University of Chile [Santiago] (UCHILE)
International audience
ISSN: 0002-9947
EISSN: 1088-6850
Transactions of the American Mathematical Society
American Mathematical Society
hal-03620379
https://u-picardie.hal.science/hal-03620379
https://u-picardie.hal.science/hal-03620379
Transactions of the American Mathematical Society, 2021, 374 (5), pp.3453-3489. ⟨10.1090/tran/8315⟩
DOI: 10.1090/tran/8315
info:eu-repo/semantics/altIdentifier/doi/10.1090/tran/8315
en
[MATH]Mathematics [math]
info:eu-repo/semantics/article
Journal articles
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 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 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 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.
2021