Real numbers having ultimately periodic representations in abstract numeration systems

Authors:
P. Lecomte;M. Rigo
Affiliations:
Institut de Mathématique. Université de Liège, Grande Traverse 12 (B 37), B-4000 Liège, Belgium;Institut de Mathématique. Université de Liège, Grande Traverse 12 (B 37), B-4000 Liège, Belgium
Venue:
Information and Computation
Year:
2004

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Abstract

Using a genealogically ordered infinite regular language, we know how to represent an interval of R. Numbers having an ultimately periodic representation play a special role in classical numeration systems. The aim of this paper is to characterize the numbers having an ultimately periodic representation in generalized systems built on a regular language. The syntactical properties of these words are also investigated. Finally, we show the equivalence of the classical θ-expansions with our generalized representations in some special case related to a Pisot number θ.