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$\mathcal{\textbf{S}}$-adic characterization of minimal ternary dendric shifts

Part of: Topological dynamics Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  02 September 2021

FRANCE GHEERAERT ,
MARIE LEJEUNE  and
JULIEN LEROY Open the ORCID record for JULIEN LEROY [Opens in a new window]
FRANCE GHEERAERT
Affiliation:
Department of Mathematics, University of LiĂšge, LiĂšge, 4000, Belgium (e-mail: france.gheeraert@uliege.be, m.lejeune@uliege.be)
MARIE LEJEUNE
Affiliation:
Department of Mathematics, University of LiĂšge, LiĂšge, 4000, Belgium (e-mail: france.gheeraert@uliege.be, m.lejeune@uliege.be)
JULIEN LEROY*
Affiliation:
Department of Mathematics, University of LiĂšge, LiĂšge, 4000, Belgium (e-mail: france.gheeraert@uliege.be, m.lejeune@uliege.be)
*
e-mail: j.leroy@uliege.be
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Abstract

Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux–Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive $\mathcal {S}$ -adic representation where the morphisms in $\mathcal {S}$ are positive tame automorphisms of the free group generated by the alphabet. In this paper, we investigate those $\mathcal {S}$ -adic representations, heading towards an $\mathcal {S}$ -adic characterization of this family. We obtain such a characterization in the ternary case, involving a directed graph with two vertices.

Keywords

symbolic dynamicssubstitutions𝒼-adicdendricinterval exchangereturn words

MSC classification

Primary: 37B10: Symbolic dynamics 68R15: Combinatorics on words
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 11 , November 2022 , pp. 3393 - 3432
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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