Generalized radix representations and dynamical systems. IV |
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Authors: | Shigeki Akiyama |
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Institution: | a Department of Mathematics, Faculty of Science Niigata University, Ikarashi 2-8050, Niigata 950-2181, Japan b Haus-Endt-Strasse 88, D-40593 Düsseldorf, Germany c Department of Computer Science, University of Debrecen, Number Theory Research Group, Hungarian Academy of Sciences and University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary d Chair of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria |
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Abstract: | For r = (r1,…, rd) ∈ ?d the mapping τr:?d →?d given byτr(a1,…,ad) = (a2, …, ad,−⌊r1a1+…+ rdad⌋)where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ?d there exists an integer k > 0 with τrk(a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F). |
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Keywords: | 11A63 11K06 11R06 |
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