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Decidability Problems for Self-induced Systems Generated by a Substitution

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Machines, Computations, and Universality (MCU 2015)

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Abstract

In this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties of the substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems.

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Authors and Affiliations

  1. Université Paul Sabatier, Institut de Mathmatique de Toulouse, Toulouse, France

    Timo Jolivet

  2. CNRS, Université de Rennes 1, IRISA-UMR 6074, Rennes, France

    Anne Siegel

  3. Inria Rennes - Bretagne Atlantique, Team Dyliss, Rennes, France

    Anne Siegel

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  1. Timo Jolivet
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  1. Université d'Orléans, Orléans, France

    Jerome Durand-Lose

  2. Eastern Mediterranean University, Famagusta, Cyprus

    Benedek Nagy

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Jolivet, T., Siegel, A. (2015). Decidability Problems for Self-induced Systems Generated by a Substitution. In: Durand-Lose, J., Nagy, B. (eds) Machines, Computations, and Universality. MCU 2015. Lecture Notes in Computer Science(), vol 9288. Springer, Cham. https://doi.org/10.1007/978-3-319-23111-2_1

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