Fixed points of Sturmian morphisms and their derivated words

Abstract 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 ψ lists the morphisms fixing the derivated words of the Sturmian word u = ψ ( u ) . We provide a sharp upper bound on length of the list.