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

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 such that the upper Sturmian word of slope 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 is an interesting and somewhat natural choice and we provide geometrical and music-theoretical motivations for its definition.In general, the second-order companion does not coincide with the original g. We show that, given a rational number , the map has exactly one fixed point, , 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 . Its root is the singleton {2}, which represents the odd mirror number . The characteristic word of slope remains fixed under a standard morphism which can be computed from the semi-period of . Finally, we prove that the characteristic word 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 , 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 .