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.

Any amicable pair ϕ, ψ of Sturmian morphisms enables a construction of a ternary morphism η which preserves the set of infinite words coding 3-interval exchange. We determine the number of amicable pairs with the same incidence matrix in SL±(2,ℕ) and we study incidence matrices associated with the corresponding ternary morphisms η.