can be obtained from $w$ by applying the rewriting rules .

Note that the presentation of a generalization of the Sturmian monoid, the so-called *episturmian monoid*, is also known, see [@Ri03]. The next lemma summarizes several simple and well-known properties of Sturmian morphisms we exploit in the sequel.

\[lem:properties\_of\_sturm\_morph\] Let $w \in \{a,b,\alpha,\beta\}^+$.

(i) The morphism $\varphi_w$ is primitive if and only if $w$ contains at least one Greek letter $\alpha$ or $\beta$ and at least one Latin letter $a$ or $b$.

(ii) If $\varphi_w$ is primitive, then each of its fixed points is aperiodic and uniformly recurrent.

(iii) If $\varphi_w$ is primitive, then it has two fixed points if and only if $w$ belongs to $\{a,\alpha\}^* $.

For $w \in  \{a,b,\alpha,\beta\}^*$ the rules  preserve positions in $w$ occupied by Latin letters $ \{a,b\}$ and positions occupied by Greek letters $ \{\alpha,\beta\}$. We define that $a<b$ and $\alpha < \beta$ which allows the following definition.

Let $w \in  \{a,b,\alpha,\beta\}^*$. The lexicographically greatest word in $  \{a,b,\alpha,\beta\}^*$ which can be obtained from $w$ by application of rewriting rules is denoted $N(w)$. If $\psi = \varphi_w$, then the word $N(w)$ is the *normalized name* of the morphism $\psi$ and it is also 