Edita Pelantová
- Štěpán Starosta
title: Fixed points of Sturmian morphisms and their derivated words
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Introduction
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Sturmian words are probably the most studied object in combinatorics on words. They are aperiodic words over a binary alphabet having the least factor complexity possible. Many properties, characterizations and generalizations are known, see for instance [@BeSe_Lothaire; @Be_survey_corr; @BaPeSta2].

One of their characterizations is in terms of return words to their factors. Let $\uu = u_0u_1u_2 \cdots$ be a binary infinite word with $u_i \in \{0,1\}$. Let $w = u_iu_{i+1} \cdots u_{i+n-1}$ be its factor. The integer $i$ is called an *occurrence* of the factor $w$. A return word to a factor $w$ is a word $u_iu_{i+1} \cdots u_{j-1}$ with $i$ and $j$ being two consecutive occurrences of $w$ such that $i < j$. In [@Vu], Vuillon showed that an infinite word $\uu$ is Sturmian if and only if each nonempty factor $w$ has exactly two distinct return words. A straightforward consequence of this characterization is that if $w$ is a prefix of $\uu$, we may write $$\uu = r_{s_0}r_{s_1}r_{s_2}r_{s_3}\cdots$$ with $s_i \in \{0,1\}$ and $r_0$ and $r_1$ being the two return words to $w$. The coding of these return words, 