On Infinite Terms Having a Decidable Monadic Theory

We study a transformation on terms consisting of applying an inverse deterministic rational mapping followed by an unfolding. Iterating these transformations from the regular terms gives a hierarchy of families of terms having a decidable monadic theory. In particular, the family at level 2 contains the morphic infinite words investigated by Carton and Thomas. We show that this hierarchy coincides with the hierarchy considered by Knapik, Niwiński and Urzyczyn: the families of terms that are solutions of higher order safe schemes. We also show that this hierarchy coincides with the hierarchy defined by Damm, and recently considered by Courcelle and Knapik: the families of terms obtained by iterating applications of first order substitutions to the set of regular terms. Finally, using second order substitutions yields the same terms.