RAIRO - Theoretical Informatics and Applications

Research Article

A note on univoque self-Sturmian numbers

Jean-Paul Allouche

CNRS, LRI, UMR 8623, Université Paris Sud, Bâtiment 490, 91405 Orsay Cedex, France; allouche@lri.fr

Abstract

We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic Sturmian sequences. As a corollary to our study we obtain that a real number β in (1,2) is univoque and self-Sturmian if and only if the β-expansion of 1 is of the form 1v, where v is a characteristic Sturmian sequence beginning itself in 1.

(Received January 12 2007)

(Accepted November 22 2007)

(Online publication January 4 2008)

Key Words:

  • Sturmian sequences;
  • univoque numbers;
  • self-Sturmian numbers.

Mathematics Subject Classification:

  • 11A63;
  • 68R15
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