A square root map on Sturmian words

Abstract

We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope $\alpha$, there exists exactly six minimal squares in its language (a minimal square does not have a square as a proper prefix). A Sturmian word $s$ of slope $\alpha$ can be written as a product of these six minimal squares: $s=X_1^2{X}_2^2{X}_3^2\cdots$. The square root of $s$ is defined to be the word $\sqrt{s}=X_1{X}_2{X}_3\cdots$. The main result of this paper is that that $\sqrt{s}$ is also a Sturmian word of slope $\alpha$. Further, we characterize the Sturmian fixed points of the square root map, and we describe how to find the intercept of $\sqrt{s}$ and an occurrence of any prefix of $\sqrt{s}$ in $s$. Related to the square root map, we characterize the solutions of the word equation $X_1^2{X}_2^2\cdots X_n^2=(X_1{X}_2\cdots X_n{)}^2$ in the language of Sturmian words of slope $\alpha$ where the words $X_i^2$ are minimal squares of slope $\alpha$. We also study the square root map in a more general setting. We explicitly construct an infinite set of non-Sturmian fixed points of the square root map. We show that the subshifts $\Omega$ generated by these words have a curious property: for all $w\in \Omega$ either $\sqrt{w}\in \Omega$ or $\sqrt{w}$ is periodic. In particular, the square root map can map an aperiodic word to a periodic word.

Extended version. 40 pages, 5 figures, 2 tables