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

Valerie Berthe (1) , Anne Siegel (2)

1 ARITH - Arithmétique informatique
2 SYMBIOSE - Biological systems and models, bioinformatics and sequences

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.

Keywords: self-affine set; expansion in a non-integral base; Pisot number; beta-shift; beta-numeration; purely periodic expansion; self-affine set.

Domains: Dynamical Systems [math.DS]

Valerie Berthe, Anne Siegel. Purely periodic beta-expansions in the Pisot non-unit case. 2004. ⟨hal-00002208⟩