Conjugates of characteristic Sturmian words generated by morphisms

Abstract

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 $\alpha =[0;1+d_1,\overline{d_2,...,d_n}]$ with $d_n\ge d_1\ge 1$. 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 $\alpha =[0;2,\bar{r}]$ ($r\ge 1$), 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 Levé and S\ee bold on conjugates of the infinite Fibonacci word.

11 pages

Cited by in corpus (3)

• Powers in a class of A-strict standard episturmian words
• Occurrences of palindromes in characteristic Sturmian words
• Impartial games whose rulesets produce given continued fractions