Some improvements of the S-adic conjecture

In [S. Ferenczi, Rank and symbolic complexity, Ergodic Theory Dynam. Systems 16 (1996) 663–682], S. Ferenczi proved that the language of any uniformly recurrent sequence with an at most linear complexity is S-adic. In this paper we adapt his proof in a more structured way and improve this result stating that any such sequence is itself S-adic. We also give some properties on the constructed morphisms.