On Sturmian graphs

Authors:
Chiara Epifanio;Filippo Mignosi;Jeffrey Shallit;Ilaria Venturini
Affiliations:
Dipartimento di Matematica e Applicazioni, Universití di Palermo, Italy;Dipartimento di Informatica, Universití dell'Aquila, Italy;School of Computer Science, University of Waterloo, Ont., Canada;TSI, ENST, Paris, France
Venue:
Discrete Applied Mathematics
Year:
2007

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Abstract

In this paper we define Sturmian graphs and we prove that all of them have a certain ''counting'' property. We show deep connections between this counting property and two conjectures, by Moser and by Zaremba, on the continued fraction expansion of real numbers. These graphs turn out to be the underlying graphs of compact directed acyclic word graphs of central Sturmian words. In order to prove this result, we give a characterization of the maximal repeats of central Sturmian words. We show also that, in analogy with the case of Sturmian words, these graphs converge to infinite ones.