Fixed points of Sturmian morphisms and their derivated words

Abstract

Any infinite uniformly recurrent word $\mathbf{u}$ can be written as concatenation of a finite number of return words to a chosen prefix $w$ of $\mathbf{u}$. Ordering of the return words to $w$ in this concatenation is coded by derivated word $d_{\mathbf{u}}(w)$. In 1998, Durand proved that a fixed point $\mathbf{u}$ of a primitive morphism has only finitely many derivated words $d_{\mathbf{u}}(w)$ and each derivated word $d_{\mathbf{u}}(w)$ 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 $\mathbf{u}=\psi (\mathbf{u})$. We provide a sharp upper bound on length of the list.

16 pages