Sturmian images of non Sturmian words and standard morphisms

We prove that if a Sturmian word is the image by a morphism of a word which is a fixed point of another morphism, then this latter word is mostly a Sturmian word, and the involved morphisms are Sturmian. This gives a characterization of Sturmian words that are generated by HDOL systems. We also characterize the non Sturmian words which can be sent on Sturmian words by morphism, and the involved morphisms. We prove that the same Sturmian images can be obtained by using the standard morphisms of which the above morphisms are the conjugates, and we show how to obtain these Sturmian morphisms from their standard representatives. (C) 2017 Elsevier B.V. All rights reserved.