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arXiv: math.DS/0407282

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Purely periodic beta-expansions in the Pisot non-unit case

Valerie Berthe 1, Anne Siegel 2

(2004-07-15)

It is well known that real numbers with a purely periodic decimal expansion are the rationals having, when reduced, a denominator coprime with 10. The aim of this paper is to extend this result to beta-expansions with a Pisot base beta which is not necessarily a unit: we characterize real numbers having a purely periodic expansion in such a base; this characterization is given in terms of an explicit set, called generalized Rauzy fractal, which is shown to be a graph-directed self-affine compact subset of non-zero measure which belongs to the direct product of Euclidean and p-adic spaces.

1:	Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM)
	CNRS : UMR5506 – Université Montpellier II - Sciences et Techniques du Languedoc
2:	SYMBIOSE (INRIA - IRISA)
	CNRS : UMR6074 – INRIA – INSA Rennes – Université de Rennes 1

Subject

:

Mathematics/Dynamical Systems

Keyword(s): expansion in a non-integral base – Pisot number – beta-shift – beta-numeration – purely periodic expansion – self-affine set.

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From: Valerie Berthe <>

Submitted on: Wednesday, 14 July 2004 20:28:22

Updated on: Tuesday, 13 March 2007 13:46:26