(i) says $ \{ \uu'\} \subset \Der(\uu)$ and Item (ii) says $ \Der(\uu)  \subset \Der(\uu') \cup \{ \uu'\}.$ Proposition \[prop:der\_of\_preimage\_fi\_b\] says $\Der(\uu')  \subset \Der(\uu) $. Proofs of these propositions use the following simple property of the injective morphism $\varphi_{b}$.

\[lem:unique\_preimage\_under\_fi\_b\] Let $\uu = \varphi_{b}(\uu')$ be a Sturmian word. If $p0 \in \mathcal{L}(\uu)$ and $0$ is a prefix of $p$, then there exists a unique factor $p'\in \mathcal{L}(\uu')$ such that $p0= \varphi_b(p')0$.

\[prop:der\_of\_images\_fi\_b\] Let $\uu$ and $\uu'$ be Sturmian words such that $\uu = \varphi_{b}(\uu')$ and let $w$ be a prefix of $\uu$.

(i) If $|w|=1$, then $d_{\uu}(w) = \uu'$ (up to a permutation of letters). \[it:der\_of\_images\_fi\_b\_1\]

(ii) If $|w|>1$, then there exists a prefix $w'$ of $\uu'$ such that $|w'|<|w|$ and $d_{\uu}(w) = d_{\uu'}(w')$ (up to a permutation of letters). \[it:der\_of\_images\_fi\_b\_2\] Moreover, if $w$ is right special, $w'$ is right special as well.

Since $\varphi_b(0) =0$ and $\varphi_b(1)=01$, the word $\uu = \varphi_{b}(\uu')$ has a prefix $0$ and the letter $1$ is in $\uu$ separated by blocks $0^k$ with $k\geq 1$. Therefore, the two return words in $\uu$ to the word $w=0$ are $r_0=0$ and $r_1=01$. We may write $\uu = r_{s_0} r_{s_1}  r_{s_2}  \cdots$, where $r_{s_j} \in \{r_0,r_1\}$ and 