An extension of Christoffel duality to a subset of Sturm numbers and their characteristic words

Abstract

The paper investigates an extension of Christoffel duality to a certain family of Sturmian words. Given an Christoffel prefix w of length N of an Sturmian word of slope g we associate a N-companion slope g(N)* such that the upper Sturmian word of slope g(N)* has a prefix w* of length N which is the upper Christoffel dual of w. Although this condition is satisfied by infinitely many slopes, we show that the companion slope g(N)* is an interesting and somewhat natural choice and we provide geometrical and music-theoretical motivations for its definition. In general, the second-order companion (g(N)*)(N)* = g(N)** does not coincide with the original g. We show that, given a rational number 0 < M/N < 1, the map g -> g(N)** has exactly one fixed point, phi(M/N) is an element of [0, 1), called odd mirror number. We show that odd mirror numbers are Sturm numbers and their continued fraction expansion is purely periodic with palindromic periods of even length. The semi-periods are of odd length and form a binary tree in bijection to the Farey tree of ratios 0 < M/N < 1. Its root is the singleton {2}, which represents the odd mirror number -1+root 8/2 = [0; (22) over bar]. The characteristic word c(phi M/N) of slope phi(M/N) remains fixed under a standard morphism which can be computed from the semi-period of phi(M/N). Finally, we prove that the characteristic word G(c(phi M/N)) is a harmonic word. As a minor open question we ask for the properties of even mirror numbers. A final conjecture provides a proper word-theoretic meaning to the extended duality for odd mirror number slopes: given a characteristic word c(phi M/N), the succession of those letters which immediately precede the occurrences of the left special factor of length N coincides - up to letter exchange - with the G-image of the dual word c(phi M/N)*.