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

  • Authors:

  • Manuel Domínguez;Thomas Noll;Marco Castrillón

  • Affiliations:

  • Universidad Complutense de Madrid, Departamento de Geometría y Topología, Spain;ESMUC (Escola Superior de Musica de Catalunya, Barcelona), Departament de Teoria i Composició, Spain;Universidad Complutense de Madrid, Departamento de Geometría y Topología, Spain

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2011

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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 slopeg"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 0g"N^*^* has exactly one fixed point, @f"M"N@?[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