$\mathcal{\textbf{S}}$ -adic characterization of minimal ternary dendric shifts. (2nd November 2022)

• Record Type: Journal Article
• Authors: GHEERAERT, FRANCE; LEJEUNE, MARIE; LEROY, JULIEN
• Is Part Of: Ergodic theory and dynamical systems. Volume 42:Number 11(2022)
• Page Start: 3393
• Page End: 3432
• Publication Date: 2022-11-02
• Subjects: symbolic dynamics -- substitutions -- 𝒮-adic -- dendric -- interval exchange -- return words; 37B10 -- 68R15; Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2021.84
• Languages: English
• ISSNs: 0143-3857

Abstract

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.