2004/07/15 by Valérie Berthé, Valerie Berthe, Anne Siegel +2
Computer Science · Mathematics · #11A63 #11J70 #68R15 #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Primary 37B10 #Secondary 11R06 #math.DS #msc:11A63 #msc:11J70 #msc:11R06 #msc:37B10 #msc:68R15 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0407282
arxiv created 2004/07/15 · openalex publication_date 2004/07/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.