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

TL;DR

This paper characterizes minimal ternary dendric shifts using an $ ext{S}$-adic framework, extending understanding of their combinatorial structure and providing a new graph-based description.

Contribution

It provides an $ ext{S}$-adic characterization of minimal ternary dendric shifts, involving a directed graph with 2 vertices, advancing the combinatorial understanding of this family.

Findings

Characterization of minimal ternary dendric shifts via $ ext{S}$-adic representations.

Introduction of a directed graph with 2 vertices for this characterization.

Extension of known results to the ternary case.

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 2 vertices.