TY - JOUR

T1 - Recognizability for sequences of morphisms

AU - Berthé, Valérie

AU - Steiner, Wolfgang

AU - Thuswaldner, Jörg

AU - Yassawi, Reem

N1 - Publisher Copyright: © Cambridge University Press, 2018.

PY - 2019/11/1

Y1 - 2019/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85073019716&partnerID=8YFLogxK

U2 - 10.1017/etds.2017.144

DO - 10.1017/etds.2017.144

M3 - Article

SN - 1469-4417

VL - 39.2019

SP - 2896

EP - 2931

JO - Ergodic theory and dynamical systems

JF - Ergodic theory and dynamical systems

IS - 11

ER -