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arXiv · 2111.13134

Automaticity of uniformly recurrent substitutive sequences

Elżbieta Krawczyk·Clemens Müllner
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Abstract

We provide a complete characterisation of automaticity of uniformly recurrent substitutive sequences in terms of the incidence matrix of the return substitution of the underlying purely substitutive sequence. This resolves a recent question posed by Allouche, Dekking and Queff\'elec in the uniformly recurrent case. We show that the same criterion characterizes automaticity of minimal substitutive systems. Furthermore, we construct a minimal substitutive system whose maximal equicontinuous factor is the 2-adic odometer, and for which the corresponding factor map is everywhere uncountable-to-one. We conjecture that a minimal substitutive system is k-automatic if and only if it is an everywhere finite-to-one extension of a k-adic odometer.

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Elżbieta Krawczyk, Clemens Müllner. 2021-11-25. Automaticity of uniformly recurrent substitutive sequences. https://arxiv.org/abs/2111.13134

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