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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
DOI: https://doi.org/10.1090/S0002-9939-04-07794-9
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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  • 1. R. Adler and B. Weiss, Similarities of automorphisms of the torus, Memoirs of the American Mathematical Society, 98, 1970. MR 0257315 (41:1966)
  • 2. S. Akiyama, Pisot numbers and greedy algorithm, Number Theory, Diophantine, Computational and Algebraic Aspects, Edited by K. Gyory, A. Petho and V. T. Sos, de Gruyter 1998, pp 9-21. MR 1628829 (99d:11007)
  • 3. S. Akiyama, Self-affine tilings and Pisot numeration system, Number Theory and Its Applications, Edited by K. Gyory and S. Kanemitsu, Kluwer, 1999, pp 7-17. MR 1738803 (2001b:11094)
  • 4. S. Akiyama, On the boundary of self-affine tilings generated by Pisot numbers, J. Math. Soc. Japan 54:2 (2002), 283-308. MR 1883519 (2002k:11132)
  • 5. S. Akiyama, H. Rao and W. Steiner, A certain finiteness property of Pisot number system, J. Number Theory 107 (2004), no. 1, 135-160. MR 2059954
  • 6. P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. 8 (2001), 181-207. MR 1838930 (2002j:37018)
  • 7. V. Canterini and A. Sigel, Geometric representation of primitive substitutions of Pisot type, Trans. Amer. Math. Soc. 353 (2001), 5121-5144. MR 1852097 (2002f:37023)
  • 8. H. Ei, S. Ito and H. Rao, Atomic surfaces, tilings and coincidence II: Reducible case. preprint 2002.
  • 9. C. Frougny and B. Solomyak, Finite beta-expansions, Ergodic. Th. & Dynam. Sys. 12 (1992), 713-723. MR 1200339 (94a:11123)
  • 10. Y. Hara and S. Ito, On real quadratic fields and periodic expansions, Tokyo J. Math. 12 (1989), 357-370. MR 1030499 (90m:11021)
  • 11. S. Ito and H. Rao, Atomic surfaces, tilings and coincidence I: Irreducible case. preprint 2001.
  • 12. S. Ito and Y. Sano, On periodic $\beta$-expansions of Pisot numbers and Rauzy fractals, Osaka J. Math. 38 (2001), 349-368. MR 1833625 (2002d:11124)
  • 13. S. Ito and Y. Takahashi, Markov subshifts and the realization of $\beta$-expansions, J. Math. Soc. Japan 26 (1974), 33-55. MR 0346134 (49:10860)
  • 14. W. Parry, On the $\beta$-expansion of real numbers, Acta Math. Acad. Sci. Hung. 11 (1960), 401-416. MR 0142719 (26:288)
  • 15. B. Praggastis, Markov partition for hyperbolic toral automorphism, Ph.D. Thesis, Univ. of Washington, 1992.
  • 16. G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147-178. MR 0667748 (84h:10074)
  • 17. K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), 269-278. MR 0576976 (82c:12003)
  • 18. A. Siegel, Représentations géométriques, combinatoire et arithmétique des systèmes substitutifs de type Pisot, Thèsis de Doctorat, Université de la Méditérranée, 2000.
  • 19. V. Sirvent and Y. Wang, Self-affine tiling via substitution dynamical systems and Rauzy fractals. Pacific J. Math. 206 (2002), no. 2, 465-485. MR 1926787 (2003g:37026)
  • 20. W. Thurston, Groups, tilings, and finite state automata, AMS Colloquium Lecture Notes, Boulder, 1989.

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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: https://doi.org/10.1090/S0002-9939-04-07794-9
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

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