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 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:The square root of s is defined to be the wordThe main result of this paper is that 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 X 2 1 X 2 2 · · · X 2 n = (X 1 X 2 · · · X n ) 2 in the language of Sturmian words of slope α where the words X 2 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.

More on the Dynamics of the Symbolic Square Root Map

“…All known constructions of fixed points rely on this method. For example, the two Sturmian words 01c α and 10c α of slope α and intercept 1 − α both have arbitrarily long squares u 2 as prefixes, where u = L( s ) for a reversed standard word s [5,Proposition 27]. In the next subsection, we see that the dynamical system studied in this paper is also fundamentally linked to fixed points obtained from solutions of (3).…”

“…The distance to 1 − α is measured in the interval I 0 or I 1 depending on which of these intervals the point ρ belongs to. In [5], we proved the following result relating the intercepts of a Sturmian word and its square root. 3 is true also if α is rational provided that the continued fraction expansion of α has enough partial quotients.…”

“…It is natural to ask if there are non-Sturmian optimal squareful words whose languages the square root map preserves. In [5], we proved by an explicit construction that such words indeed exist. The construction is as follows.…”

“…3 is true also if α is rational provided that the continued fraction expansion of α has enough partial quotients. More precisely, the proof of Theorem 2.3 in [5] considers only certain properties of the denominators of the (semi)convergents q 2,1 and q 3,1 of α. Thus if the continued fraction expansion of α has at least three partial quotients, then Theorem 2.3 holds.…”

“…In what follows, we often consider infinite products of γ k and γ k , and we wish to argue independently of the index k. Hence we make a convention that γ and γ respectively stand for γ k and γ k for some k ≥ 0. The words γ and γ are primitive; see [5,Lemma 39]. For simplification, we abuse notation and write S for γ 0 and L for γ 0 .…”