Tilings for Pisot beta numeration

被引：5
作者：
Minervino, Milton [1 ]
Steiner, Wolfgang [2 ]
机构：
[1] Univ Leoben, Dept Math & Informat Technol, A-8700 Leoben, Austria
[2] Univ Paris 07, CNRS UMR 7089, LIAFA, F-75205 Paris 13, France
来源：
INDAGATIONES MATHEMATICAE-NEW SERIES | 2014年 / 25卷 / 04期
基金：
奥地利科学基金会;
关键词：
Beta-expansion; Pisot number; Tiling; Rauzy fractal; ATOMIC SURFACES; SYSTEMS; SUBSTITUTIONS; EXPANSIONS; NUMBERS; COINCIDENCE; BOUNDARY; SHIFTS;
D O I：
10.1016/j.indag.2014.04.008
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
For a (non-unit) Pisot number beta, several collections of tiles are associated with beta-numeration. This includes an aperiodic and a periodic one made of Rauzy fractals, a periodic one induced by the natural extension of the beta-transformation and a Euclidean one made of integral beta-tiles. We show that all these collections (except possibly the periodic translation of the central tile) are tilings if one of them is a tiling or, equivalently, the weak finiteness property (W) holds. We also obtain new results on rational numbers with purely periodic beta-expansions; in particular, we calculate gamma(beta) for all quadratic beta with beta(2) = alpha beta + b, gcd(a, b) = 1. (C) 2014 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
引用
收藏
页码：745 / 773
页数：29
相关论文
共 36 条
[21]  
Hama M., 1997, Commentarii mathematici Universitatis Sancti Pauli, V46, P103
[22]  
HARDER G, 1999, FUNDAMENTAL PRINCIPL, V322
[23]  
Hollander M., 1996, THESIS U WASHINGTON
[24]   Purely periodic β-expansions with Pisot unit base [J].
Ito, S ;
Rao, H .
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2005, 133 (04) :953-964
[25]   Atomic surfaces, tilings and coincidence I. Irreducible case [J].
Ito, Shunji ;
Rao, H. .
ISRAEL JOURNAL OF MATHEMATICS, 2006, 153 (1) :129-155
[26]   BETA-EXPANSIONS, NATURAL EXTENSIONS AND MULTIPLE TILINGS ASSOCIATED WITH PISOT UNITS [J].
Kalle, Charlene ;
Steiner, Wolfgang .
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 2012, 364 (05) :2281-2318
[27]  
Minervino M., 2014, ANN I FOURI IN PRESS
[28]  
Parry W., 1960, Acta Math. Acad. Sci. Hung, V11, P401, DOI [10.1007/BF02020954, DOI 10.1007/BF02020954]
[29]   Numeration systems and Markov partitions from self similar tilings [J].
Praggastis, B .
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 1999, 351 (08) :3315-3349
[30]  
RAUZY G, 1982, B SOC MATH FR, V110, P147
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