146. Interestingly enough, the term *derivated* sequence is also used in [@HeMo], however, its definition differs from our one (as taken from [@Durand98]): Using again our terminology, their derivated word is a derivated word with respect to a one-letter word in a biinfinite Sturmian word.

Sturmian morphisms {#sec:stu_mor}
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A *morphism* over $\mathcal{A}^*$ is a mapping $\psi : \mathcal{A}^* \mapsto  \mathcal{A}^*$ such that $\psi(vw) = \psi(v)\psi(w)$ for all $v,w \in \mathcal{A}^*$. The domain of the morphism $\psi$ can be naturally extended to $\mathcal{A}^{\mathbb{N}}$ by $$\psi(u_0u_1u_2\cdots ) = \psi(u_0)\psi(u_1) \psi(u_2)\cdots .$$ A morphism $\psi$ is *primitive* if there exists a positive integer $k$ such that the letter $a$ occurs in the word $\psi^k(b)$ for each pair of letters $a,b \in  \mathcal{A}$. A *fixed point* of a morphism $\psi$ is an infinite word $\uu$ such that $\psi(\uu) = \uu$.

A morphism $\psi$ is a *Sturmian morphism* if $\psi(\uu)$ is a Sturmian word for any Sturmian word $\uu$. The set of Sturmian morphisms together with composition forms the so-called *Sturmian monoid* usually denoted [*St*]{}. We work with these four elementary Sturmian morphisms: $$\varphi_a: \begin{cases} 0 \to 0 \\ 1 \to 10 \end{cases} \quad
    \varphi_b: \begin{cases} 0 \to 0 \\ 1 \to 01 \end{cases} 