the set $$\Der(\uu) = \{ d_{\uu}(w) \colon w \text{ is a prefix of }  \uu\}.$$ Clearly, if a prefix $w$ is not right special, then there exists a unique letter $x$ such that $wx \in \mathcal{L}(\uu)$. Thus the occurrences of $w$ and $wx$ coincide, $\mathcal{R}_{\uu}(w) = \mathcal{R}_{\uu}(wx)$ and $d_{\uu}(w)$ = $d_{\uu}(wx)$. If $\uu$ is not eventually periodic, then $w$ is a prefix of a right special prefix of $\uu$. Therefore for an aperiodic uniformly recurrent word $\uu$ we have $$\Der(\uu) = \{ d_{\uu}(w) \colon w \text{ is a right special prefix of }  \uu \}.$$

Sturmian words
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Any Sturmian word $\uu$ can be identified with an upper or lower mechanical word. A mechanical word is described by two parameters: slope and intercept. The slope is an irrational number $\gamma \in (0,1)$ and the intercept is a real number $\rho \in [0,1)$. To define the lower mechanical word ${\bf s}(\gamma, \rho) = \left(s_{n} (\gamma, \rho) \right)_{n \in \N}$ we put $I_0 =  [0,1-\gamma)$. The $n^{th}$ letter of ${\bf s}(\gamma, \rho)$ is as follows: $$s_n(\gamma, \rho) =
\begin{cases}
0 & \text{if the number }    \gamma n + \rho \mod 1 \text{ belongs to }  I_0, \\
1 & \text{otherwise.}
\end{cases}$$ The 