## Abstract

We investigate different notions of recognizability for a free monoid morphism . Full recognizability occurs when each (aperiodic) point in admits at most one tiling with words , . This is stronger than the classical notion of recognizability of a substitution , where the tiling must be compatible with the language of the substitution. We show that if , or if 's incidence matrix has rank , or if is permutative, then 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 -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 -adic shift it generates, and the measurable Bratteli-Vershik dynamical system that it defines.

Originalsprache | Englisch |
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Seiten (von - bis) | 2896-2931 |

Seitenumfang | 36 |

Fachzeitschrift | Ergodic theory and dynamical systems |

Jahrgang | 39.2019 |

Ausgabenummer | 11 |

Frühes Online-Datum | 24 Jan. 2018 |

DOIs | |

Publikationsstatus | Veröffentlicht - 1 Nov. 2019 |

### Bibliographische Notiz

Funding Information:Acknowledgements. The fourth author thanks IRIF, Université Paris Diderot–Paris 7, for its hospitality and support. This work was supported by the Agence Nationale de la Recherche and the Austrian Science Fund through the projects ‘Fractals and Numeration’ (ANR-12-IS01-0002, FWF I1136), ‘Dyna3S’ (ANR-13-BS02-0003), ‘Discrete Mathematics’ (FWF W1230), and ‘Fractals and Words: Topological, Dynamical, and Combinatorial Aspects’ (FWF P27050). It also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant Agreement No 648132.

Publisher Copyright:

© Cambridge University Press, 2018.