Generalized balances in Sturmian words

  • Authors:

  • Isabelle Fagnot;Laurent Vuillon

  • Affiliations:

  • Institut Gaspard Monge, Cité Descartes, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-vallée Cedex 2, France;LIAFA Université Paris 7, 2, place Jussieu, Case 7014, 75251 Paris Cedex 05, France

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2002

Quantified Score

Hi-index 0.05

Visualization

Abstract

One of the numerous characterizations of Sturmian words is based on the notion of balance. An infinite word X on the {0, 1} alphabet is balanced if, given two factors of X, w and w', having the same length, the difference between the number of 0's in w (denoted by |w|0) and the number of 0's in w' is at most 1, i.e. ||w|0 - |w'|0| ≤ 1. It is well known that an aperiodic word is Sturmian if and only if it is balanced.In this paper, the balance notion is generalized by considering the number of occurrences of a word u in w (denoted by |w|u,) and w'. The following is obtained. Theorem. Let x be a Sturmian word. Let u, w and w' be three factors of x. Then, |w| = |w'| ⇒ ||w|u - |w'|u| ≤ |u|.Another balance property, called equilibrium, is also given. This notion permits us to give a new characterization of Sturmian words. The main techniques used in the proofs are word graphs and return words.