Conjugates of characteristic Sturmian words generated by morphisms

This article is concerned with characteristic Sturmian words of slope $\alpha$ and $1-\alpha$ (denoted by $c_\alpha$ and $c_{1-\alpha}$ respectively), where $\alpha \in (0,1)$ is an irrational number such that $\alpha = [0;1+d_1,\bar{d_2,...,d_n}]$ with $d_n \geq d_1 \geq 1$. It is known that both $c_\alpha$ and $c_{1-\alpha}$ are fixed points of non-trivial (standard) morphisms $\sigma$ and $\hat{\sigma}$, respectively, if and only if $\alpha$ has a continued fraction expansion as above. Accordingly, such words $c_\alpha$ and $c_{1-\alpha}$ are generated by the respective morphisms $\sigma$ and $\hat{\sigma}$. For the particular case when $\alpha = [0;2,\bar{r}]$ ($r\geq1$), we give a decomposition of each conjugate of $c_\alpha$ (and hence $c_{1-\alpha}$) into generalized adjoining singular words, by considering conjugates of powers of the standard morphism $\sigma$ by which it is generated. This extends a recent result of Lev\'{e} and S\ee bold on conjugates of the infinite Fibonacci word.