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

Dendric subshifts are defined by combinatorial restrictions of the extensions of the words in its language. This family generalizes well-known families of subshifts such as Sturmian subshifts, Arnoux-Rauzy subshifts and codings of interval exchange transformations. It is known that any minimal dendric subshifts 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 ot this family. We obtain such a characterization in the ternary case, involving a directed graph with 9 vertices.