Universidad Complutense de Madrid
E-Prints Complutense

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

Impacto

Downloads

Downloads per month over past year

Castrillón López, Marco and Dominguez,, Manuel and Noll, Thomas (2011) An extension of Christoffel duality to a subset of Sturm numbers and their characteristic words. Theoretical Computer Science, 412 (27). pp. 2942-2954. ISSN 0304-3975

[img] PDF
Restringido a Repository staff only

453kB

Official URL: http://www.sciencedirect.com/science/article/pii/S0304397510007681



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)*.


Item Type:Article
Uncontrolled Keywords:Sturmian words; Christoffel duality; Well-formed scales; Characteristic words; Harmonic words
Subjects:Sciences > Mathematics > Geometry
ID Code:14872
Deposited On:19 Apr 2012 08:23
Last Modified:12 Dec 2018 15:13

Origin of downloads

Repository Staff Only: item control page