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Recognizability for sequences of morphisms

Published online by Cambridge University Press:  24 January 2018

VALÉRIE BERTHÉ ,
WOLFGANG STEINER ,
JÖRG M. THUSWALDNER  and
REEM YASSAWI
VALÉRIE BERTHÉ
Affiliation:
IRIF, CNRS UMR 8243, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France email berthe@irif.fr, steiner@irif.fr
WOLFGANG STEINER
Affiliation:
IRIF, CNRS UMR 8243, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France email berthe@irif.fr, steiner@irif.fr
JÖRG M. THUSWALDNER
Affiliation:
Chair of Mathematics and Statistics, University of Leoben, A-8700 Leoben, Austria email joerg.thuswaldner@unileoben.ac.at
REEM YASSAWI
Affiliation:
ICJ, CNRS UMR 5208, Université Claude Bernard Lyon 1, F-69622 Villeurbanne Cedex, France email ryassawi@gmail.com
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Abstract

We investigate different notions of recognizability for a free monoid morphism $\unicode[STIX]{x1D70E}:{\mathcal{A}}^{\ast }\rightarrow {\mathcal{B}}^{\ast }$. Full recognizability occurs when each (aperiodic) point in ${\mathcal{B}}^{\mathbb{Z}}$ admits at most one tiling with words $\unicode[STIX]{x1D70E}(a)$, $a\in {\mathcal{A}}$. This is stronger than the classical notion of recognizability of a substitution $\unicode[STIX]{x1D70E}:{\mathcal{A}}^{\ast }\rightarrow {\mathcal{A}}^{\ast }$, where the tiling must be compatible with the language of the substitution. We show that if $|{\mathcal{A}}|=2$, or if $\unicode[STIX]{x1D70E}$’s incidence matrix has rank $|{\mathcal{A}}|$, or if $\unicode[STIX]{x1D70E}$ is permutative, then $\unicode[STIX]{x1D70E}$ is fully recognizable. Next we investigate the classical notion of recognizability and improve earlier results of Mossé [Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci.99(2) (1992), 327–334] and Bezuglyi et al [Aperiodic substitution systems and their Bratteli diagrams. Ergod. Th. & Dynam. Sys.29(1) (2009), 37–72], by showing that any substitution is recognizable for aperiodic points in its substitutive shift. Finally we define recognizability and also eventual recognizability for sequences of morphisms which define an $S$-adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable. As an application, for a recognizable sequence of morphisms, we obtain an almost everywhere bijective correspondence between the $S$-adic shift it generates, and the measurable Bratteli–Vershik dynamical system that it defines.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 11 , November 2019 , pp. 2896 - 2931
Copyright
© Cambridge University Press, 2018 

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