Conjugates of characteristic Sturmian words generated by morphisms

TL;DR

This paper investigates conjugates of characteristic Sturmian words generated by morphisms, providing a decomposition method for certain cases and extending previous results on Fibonacci words.

Contribution

It introduces a decomposition of conjugates of characteristic Sturmian words for specific irrational slopes, extending known results on Fibonacci words to more general cases.

Findings

Decomposition of conjugates into generalized adjoining singular words

Extension of Fibonacci word conjugate results to broader Sturmian words

Analysis of morphism conjugates for specific continued fraction expansions

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 $\alpha = [0;1+d_1,\bar{d_2,...,d_n}]$ 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.