Sturmian words, Lyndon words and trees

Abstract

We prove some new combinatorial properties of the set PER of all words w having two periods p and q which are coprimes and such that w = p + q − 2 [4, 3]. We show that aPERb∪{ a, b} = St ∩ Lynd, where St is the set of the finite factors of all infinite Sturmian words and Lynd is the set of the Lyndon words on the alphabet { a, b}. It is also shown that aPERb∪{ a, b} = CP, where CP is the set of Christoffel primitive words. Such words can be defined in terms of the ‘slope’ of the words and of their prefixes [1]. From this result one can derive in a different way, by using a theorem of Borel and Laubie, that the elements of the set aPERb are Lyndon words. We prove the following correspondence between the ratio p q of the periods p, q, p⩽ q of w ϵ PER ∩ a{a,b} ∗ and the slope ρ = (¦w¦ b + 1) (¦w¦ a + 1) of the corresponding Christoffel primitive word awb: If p q has the development in continued fractions [0, h 1,…, h n − 1 , h n + 1], then ρ has the development in continued fractions [0, h n ,…, h 2, h 1 + 1]. This and other related results can be also derived by means of a theorem which relates the developments in continued fractions of the Stern-Brocot and the Raney numbers of a node in a complete binary tree. However, one needs some further results. More precisely we label the binary tree with standard pairs (standard tree), Christoffel pairs (Christoffel tree) and the elements of PER (Farey tree). The main theorem is the following: If the node W is labeled by the standard pair ( u, v), by the Christoffel pair ( x, y) and by w ϵ PER, then uv = wab, xy = awb. The Stern-Brocot number SB( W) is equal to the slope of the standard word uv and of the Christoffel word xy while the Raney number Ra( W) is equal to the ratio of the minimal period of wa and the minimal period of wb. Some further auxiliary results are also derived.