Fixed points of Sturmian morphisms and their derivated words

Any infinite uniformly recurrent word u can be written as concatenation of a finite number of return words to a chosen prefix w of u. Ordering of the return words to w in this concatenation is coded by derivated word d(u)(w). In 1998, Durand proved that a fixed point u of a primitive morphism has only finitely many derivated words d(u)(w) and each derivated word d(u)(w) is fixed by a primitive morphism as well. In our article we focus on Sturmian words fixed by a primitive morphism. We provide an algorithm which to a given Sturmian morphism psi lists the morphisms fixing the derivated words of the Sturmian word u = psi(u). We provide a sharp upper bound on length of the list. (C) 2018 Published by Elsevier B.V.