A Square Root Map on Sturmian Words

We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope α, 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 α can be written as a product of these six minimal squares: s = X2 1X 2 2X 2 3 · · · . The square root of s is defined to be the word √ s = X1X2X3 · · · . The main result of this paper is that √ s is also a Sturmian word of slope α. Further, we characterize the Sturmian fixed points of the square root map, and we describe how to find the intercept of √ s and an occurrence of any prefix of √ s in s. Related to the square root map, we characterize the solutions of the word equation X2 1X 2 2 · · ·X2 n = (X1X2 · · ·Xn) in the language of Sturmian words of slope α where the words X2 i are minimal squares of slope α. 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 Ω generated by these words have a curious property: for all w ∈ Ω either √w ∈ Ω or √w is periodic. In particular, the square root map can map an aperiodic word to a periodic word.