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 $\sqrt{s} = X_1 X_2 X_3 \cdots$. We prove that $\sqrt{s}$ is also a Sturmian word of slope $\alpha$. Moreover, 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 $X_1^2 X_2^2 \cdots X_m^2 = (X_1 X_2 \cdots X_m)^2$ in the language of Sturmian words of slope $\alpha$ where the words $X_i^2$ are minimal squares of slope $\alpha$.