Recognizability of morphisms

We investigate several questions related to the notion of recognizable morphism. The main result is a new proof of Mosse's theorem and actually of a generalization to a more general class of morphisms due to Berthe et al [Recognizability for sequences of morphisms. Ergod. Th. & Dynam. Sys. 39(11) (2019), 2896-2931]. We actually prove the result of Berthe et al for the most general class of morphisms, including ones with erasable letters. Our result is derived from a result concerning elementary morphisms for which we also provide a new proof. We also prove some new results which allow us to formulate the property of recognizability in terms of finite automata. We use this characterization to show that for an injective morphism, the property of being recognizable on the full shift for aperiodic points is decidable.