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On Decision Problems for Substitutions in Symbolic Dynamics

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Reachability Problems (RP 2020)

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Abstract

In this survey, we discuss decidability issues for symbolic dynamical systems generated by substitutions. Symbolic dynamical systems are discrete dynamical systems made of infinite sequences of symbols, with the shift acting on them. Substitutions are simple rules that replace letters by string of letters and allow the generation of infinite words. We focus here on symbolic dynamical systems that are generated by infinite compositions of substitutions, allowing to go beyond the case of the iteration of a single substitution. This is the so-called S-adic framework. Motivated by decidability and ergodic questions, we focus on questions dealing with the convergence of products of nonnegative matrices and associated Lyapunov exponents.

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

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Notes

  1. 1.

    Here \(\mu \)-almost everywhere refers to directive sequences of substitutions chosen in D with respect to the measure \(\mu \).

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Authors and Affiliations

  1. Université de Paris, IRIF, CNRS, 75013, Paris, France

    Valérie Berthé

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  1. Valérie Berthé
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Correspondence to Valérie Berthé .

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  1. Université de Paris, Paris, France

    Sylvain Schmitz

  2. University of Liverpool, Liverpool, UK

    Igor Potapov

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Berthé, V. (2020). On Decision Problems for Substitutions in Symbolic Dynamics. In: Schmitz, S., Potapov, I. (eds) Reachability Problems. RP 2020. Lecture Notes in Computer Science(), vol 12448. Springer, Cham. https://doi.org/10.1007/978-3-030-61739-4_1

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  • DOI: https://doi.org/10.1007/978-3-030-61739-4_1

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