Fixed points of Sturmian morphisms and their derivated words

TL;DR

This paper studies fixed points of Sturmian morphisms, providing an algorithm to list morphisms fixing derivated words of Sturmian words and establishing bounds on the list length.

Contribution

It introduces an algorithm to identify morphisms fixing derivated words of Sturmian fixed points and determines an upper bound on the list size.

Findings

Algorithm for listing morphisms fixing derivated words

Sharp upper bound on list length

Application to Sturmian words fixed by primitive morphisms

Abstract

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