Conjugates of characteristic Sturmian words generated by morphisms

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

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