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Recognizability of morphisms

Part of: Topological dynamics

Published online by Cambridge University Press:  13 January 2023

MARIE-PIERRE BÉAL Open the ORCID record for MARIE-PIERRE BÉAL [Opens in a new window] ,
DOMINIQUE PERRIN  and
ANTONIO RESTIVO
MARIE-PIERRE BÉAL
Affiliation:
Université Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France (e-mail: marie-pierre.beal@univ-eiffel.fr)
DOMINIQUE PERRIN*
Affiliation:
Université Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France (e-mail: marie-pierre.beal@univ-eiffel.fr)
ANTONIO RESTIVO
Affiliation:
Dipartimento di Matematica e Informatica, Università di Palermo, Italy (e-mail: antonio.restivo@unipa.it)
*
e-mail: dominique.perrin@esiee.fr
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Abstract

We investigate several questions related to the notion of recognizable morphism. The main result is a new proof of Mossé’s theorem and actually of a generalization to a more general class of morphisms due to Berthé et al [Recognizability for sequences of morphisms. Ergod. Th. & Dynam. Sys. 39(11) (2019), 2896–2931]. We actually prove the result of Berthé et al for the most general class of morphisms, including ones with erasable letters. Our result is derived from a result concerning elementary morphisms for which we also provide a new proof. We also prove some new results which allow us to formulate the property of recognizability in terms of finite automata. We use this characterization to show that for an injective morphism, the property of being recognizable on the full shift for aperiodic points is decidable.

Keywords

symbolic dynamicssubstitutionsautomata

MSC classification

Primary: 37B10: Symbolic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 11 , November 2023 , pp. 3578 - 3602
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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