Monatshefte für Mathematik

, Volume 187, Issue 2, pp 275–291 | Cite as

Multiple tilings associated to d-Bonacci beta-expansions

  • Tomáš HejdaEmail author


Let \(\beta \in (1,2)\) be a Pisot unit and consider the symmetric \(\beta \)-expansions. We give a necessary and sufficient condition for the associated Rauzy fractals to form a tiling of the contractive hyperplane. For \(\beta \) a d-Bonacci number, i.e., Pisot root of \(x^d-x^{d-1}-\dots -x-1\) we show that the Rauzy fractals form a multiple tiling with covering degree \(d-1\).


Beta-expansions Rauzy fractals Tiling Multiple tiling 

Mathematics Subject Classification

11A63 52C23 (11R06 37B10) 



We acknowledge support by Czech Science Foundation (GAČR) grant 17-04703Y and ANR/FWF project “FAN – Fractals and Numeration” (ANR-12-IS01-0002, FWF grant I1136). We are also grateful for Sage and TikZ software which were used to prepare the Figs. [13, 15].


  1. 1.
    Akiyama, S.: On the boundary of self affine tilings generated by Pisot numbers. J. Math. Soc. Jpn. 54(2), 283–308 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akiyama, S., Scheicher, K.: Symmetric shift radix systems and finite expansions. Math. Pannon. 18(1), 101–124 (2007)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Barge, M.: Pure discrete spectrum for a class of one-dimensional substitution tiling systems. Discrete Contin. Dyn. Syst. 36(3), 1159–1173 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barge, M.: The Pisot conjecture for $\beta $-substitutions. Ergodic Theory Dyn. Syst. 38(2), 444–472 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brauer, A.: On algebraic equations with all but one root in the interior of the unit circle. Math. Nachr. 4, 250–257 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ito, S., Rao, H.: Atomic surfaces, tilings and coincidence. I. Irreducible case. Isr. J. Math. 153, 129–155 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kalle, C., Steiner, W.: Beta-expansions, natural extensions and multiple tilings associated with Pisot units. Trans. Am. Math. Soc. 364(5), 2281–2318 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Li, T.-Y., Yorke, J.A.: Ergodic transformations from an interval into itself. Trans. Am. Math. Soc. 235, 183–192 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Neukirch, J.: Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, Translated from the 1992 German original and with a note by Norbert Schappacher, with a foreword by G. Harder (1999)Google Scholar
  10. 10.
    Rauzy, G.: Nombres algébriques et substitutions. Bull. Soc. Math. France 110(2), 147–178 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rao, H., Wen, Z.-Y., Yang, Y.M.: Dual systems of algebraic iterated function systems. Adv. Math. 253, 63–85 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Siegel A., Thuswaldner, J.M.: Topological properties of Rauzy fractals. Mém. Soc. Math. Fr. (N.S.) , no. 118, 140 (2009)Google Scholar
  13. 13.
    The Sage Group, Sage: Open source mathematical software (version 6.4) (2014). [2015-03-01]
  14. 14.
    Thurston, W.P.: Groups, tilings and finite state automata, AMS Colloquium lectures (1989)Google Scholar
  15. 15.
    Tantau, T. et al.:, TikZ & PGF (version 3.0.0) (2014). 01 March 2015

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IRIF, CNRS UMR 8243, Université Paris Diderot – Paris 7ParisFrance
  2. 2.Department of Mathematics FCEUniversity of Chemistry and Technology, PraguePragueCzechia
  3. 3.Department of Algebra FMFCharles UniversityPragueCzechia

Personalised recommendations