S-adic characterization of minimal ternary dendric shifts

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 S-adic representation where the morphisms in S are positive tame automorphisms of the free group generated by the alphabet. In this paper, we investigate those S-adic representations, heading towards an S-adic characterization of this family. We obtain such a characterization in the ternary case, involving a directed graph with two vertices.