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

TL;DR

This paper provides a new combinatorial characterization of minimal dendric shifts using finite graphs, enabling the decision of dendricity in shift spaces generated by morphic words.

Contribution

It introduces an $ ext{S}$-adic characterization of minimal dendric shifts via finite graphs, expanding understanding of their structure and properties.

Findings

Provides an $ ext{S}$-adic characterization using two finite graphs.

Enables deciding dendricity for shift spaces generated by morphic words.

Generalizes known families like Sturmian and Arnoux-Rauzy shifts.

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 give an $\mathcal{S}$-adic characterization of this family by means of two finite graphs. As an application, we are able to decide whether a shift space generated by a uniformly recurrent morphic word is (eventually) dendric.