Characteristic Sturmian words are extremal for the Critical Factorization Theorem

  • Authors:

  • Filippo Mignosi;Antonio Restivo

  • Affiliations:

  • Dipartimento di Informatica, Università dellAquila, LAquila, Italy;Dipartimento di Matematica e Informatica, Università di Palermo, Palermo, Italy

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2012

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Abstract

We prove that characteristic Sturmian words are extremal for the Critical Factorization Theorem (CFT) in the following sense. If p"x(n) denotes the local period of an infinite word x at point n, we prove that x is a characteristic Sturmian word if and only if p"x(n) is smaller than or equal to n+1 for all n=1 and it is equal to n+1 for infinitely many integers n. This result is extremal with respect to the CFT since a consequence of the CFT is that, for any infinite recurrent word x, either the function p"x is bounded, and in such a case x is periodic, or p"x(n)=n+1 for infinitely many integers n. As a byproduct of the techniques used in the paper we extend a result of Harju and Nowotka (2002) in [18] stating that any finite Fibonacci word f"n,n=5, has only one critical point. Indeed we determine the exact number of critical points in any finite standard Sturmian word.