E\varphi_bE .$$

Our aim is to study derivated words of fixed points of Sturmian morphisms. If $\uu$ is a fixed point of $\psi$, it is also a fixed point of $\psi^2$. Due to , the square $\psi^2$ always belongs to $\mathcal{M}$. To illustrate why this is true, assume, e.g., that $\psi \in St = \langle E, \varphi_a, \varphi_b \rangle$ equals $\psi = \varphi_a E \varphi_b \varphi_a$. Using  and the fact that $E^2$ is the identity morphism, we have $$\psi = \varphi_a E \varphi_b E E \varphi_a E E = \varphi_a  \varphi_\beta \varphi_\alpha E$$ and hence $$\psi^2 = \varphi_a \varphi_\beta \varphi_\alpha E \varphi_a \varphi_\beta \varphi_\alpha E = \varphi_a \varphi_\beta \varphi_\alpha E \varphi_a E E\varphi_\beta E E\varphi_\alpha E = \varphi_a \varphi_\beta \varphi_\alpha \varphi_\alpha \varphi_b \varphi_a \in \mathcal{M}.$$ Therefore we may restrict ourselves to fixed points of morphisms from the special Sturmian monoid $\mathcal{M}$. Note that this would not be true if we consider only the morphisms from $\langle \varphi_a, \varphi_b \rangle$, see also .

\[fibonacci1\] The Fibonacci word is the fixed point of the morphism $\tau: 0\to 01, 1\to 0$. The morphism $\tau$ is Sturmian, but $\tau  \notin \mathcal{M}$. We see that $\tau =  \varphi_b\circ E$ and by the relations we have 