$Q$-Factorization of suffixes of two-way infinite extensions of irrational characteristic words

Abstract

Let $\alpha$ be an irrational number between 0 and 1 with continued fraction expansion $[0;a_1+1,a_2,a_3,\dots ]$, where $a_n\ge 1$ ($n\ge 1$). Define a sequence of numbers ${\left\{q_n\right\}}_{n\ge -1}$ by $q_{-1}=1$, $q_0=1$, $q_n=a_n{q}_{n-1}+q_{n-2}$ ($n\ge 1$). For each integer $k\ge -1$, we consider the kth-order Q-factorization of each suffix $H$ of a two-way infinite extension $X$ of the characteristic word of $\alpha$ of the form: $H=u_k{u}_{k+1}{u}_{k+2}\cdots$, where the length of the factor $u_i$ is $q_i$ ($i\ge k$). We show that in such a factorization of the simple Sturmian word $H$, either all $u_i$ are singular words, or there exists a nonnegative integer $q$ such that $H$ begins with $q$ singular words, and $u_{k+q},u_{k+q+1},u_{k+q+2},\dots$ are $\alpha$-words. Moreover, the number $q$ and the labels of these $\alpha$-words are uniquely determined by the $Q$-representation of a nonnegative integer obtained from the position of $H$ in $X$. These results lead, naturally, to a new method for generating suffixes of $X$.