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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Purely periodic $\beta$-expansions with Pisot unit base


Authors: Shunji Ito and Hui Rao
Journal: Proc. Amer. Math. Soc. 133 (2005), 953-964
MSC (2000): Primary 11R06; Secondary 37B50
Published electronically: November 19, 2004
MathSciNet review: 2117194
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Abstract: Let $\beta>1$ be a Pisot unit. A family of sets $\{X_i\}_{1\leq i\leq q}$ defined by a $\beta$-numeration system has been extensively studied as an atomic surface or Rauzy fractal. For the purpose of constructing a Markov partition, a domain $\hat X=\bigcup_{i=1}^q \hat X_i$ constructed by an atomic surface has appeared in several papers. In this paper we show that the domain $\hat X$ completely characterizes the set of purely periodic $\beta$-expansions.


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Additional Information

Shunji Ito
Affiliation: Department of Information and Systems Engineering, Kanazawa University, Kanazawa, Japan
Email: ito@t.kanazawa-u.ac.jp

Hui Rao
Affiliation: Department of Mathematics, Tsinghua University, Beijing, People’s Republic of China
Email: hrao@math.tsinghua.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07794-9
PII: S 0002-9939(04)07794-9
Keywords: Pisot number, $\beta$-expansion, atomic surface
Received by editor(s): May 28, 2003
Published electronically: November 19, 2004
Additional Notes: The second author was supported by the Japanese Science Promotion Society (JSPS)
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2004 American Mathematical Society