A note on univoque self-Sturmian numbers

We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of a unimodal continuous map from the unit interval into itself, but it also characterizes univoque real numbers; the other is an equivalent definition of characteristic Sturmian sequences. As a corollary to our study we obtain that a real number $\beta$ in $(1,2)$ is univoque and self-Sturmian if and only if the $\beta$-expansion of 1 is of the form $1v$, where $v$ is a characteristic Sturmian sequence beginning itself in 1.