On the dimension group of unimodular \({\mathcal {S}}\)-adic subshifts

Abstract

Dimension groups are complete invariants of strong orbit equivalence for minimal Cantor systems. This paper studies a natural family of minimal Cantor systems having a finitely generated dimension group, namely the primitive unimodular proper \({\mathcal {S}}\)-adic subshifts. They are generated by iterating sequences of substitutions. Proper substitutions are such that the images of letters start with a same letter, and similarly end with a same letter. This family includes various classes of subshifts such as Brun subshifts or dendric subshifts, that in turn include Arnoux–Rauzy subshifts and natural coding of interval exchange transformations. We compute their dimension group and investigate the relation between the triviality of the infinitesimal subgroup and rational independence of letter measures. We also introduce the notion of balanced functions and provide a topological characterization of balancedness for primitive unimodular proper S-adic subshifts.

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Acknowledgements

We would like to thank M. I. Cortez and F. Dolce for stimulating discussions. We also thank warmly the referees of this paper for their careful reading and their very useful comments, concerning in particular the formulation of Corollary 4.3.

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Correspondence to F. Durand.

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This work was supported by the Agence Nationale de la Recherche through the project “Codys” (ANR-18-CE40-0007).

The second author was supported by the PhD Grant CONICYT—PFCHA/Doctorado Nacional/2015-21150544.

Communicated by H. Bruin.

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Berthé, V., Cecchi Bernales, P., Durand, F. et al. On the dimension group of unimodular \({\mathcal {S}}\)-adic subshifts. Monatsh Math (2021). https://doi.org/10.1007/s00605-020-01488-3

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Keywords

  • Dimension group
  • S-adic subshift
  • Orbit equivalence
  • Dendric subshift
  • Balance property

Mathematics Subject Classification

  • 37B10
  • 37A20