i.e. $\mathcal{R}_{\uu}(w) = \{r_0, r_1, \ldots, r_{k-1}\}$. The word $\uu$ can be written as unique concatenation of the return words $\uu = r_{s_0}r_{s_1}r_{s_2}\cdots$. The *derivated* word of $\uu$ with respect to the prefix $w$ is the infinite word $$d_{\uu}(w) =  s_0s_1s_2 \cdots$$ over the alphabet of cardinality $ \#\mathcal{R}_{\uu}(w)=k $. In his original definition, Durand [@Durand98] fixed the alphabet of the derivated word to the set $\{0,1, \ldots, k-1\}$. Moreover, Durand’s definition requires that for $i< j$ the first occurrence of $r_i$ in $\uu$ is less than the first occurrence of $r_j$ in $\uu$. In particular, a derivated word always starts with the letter $0$. In the article [@AraBru05], where derivated words of standard Sturmian words are studied, the authors required that the starting letters of the original word and its derivated word coincide. For our purposes, we do not need to fix the alphabet of derivated words: two derivated words which differ only by a permutation of letters are identified one with another.

In the sequel, we work only with infinite words which are *uniformly recurrent*, i.e. each prefix $w$ of $\uu$ occurs in $\uu$ infinitely many times and the set $\mathcal{R}_{\uu}(w)$ is finite. Our aim is to describe 