Osaka Journal of Mathematics

Parametrization for a class of Rauzy Fractals

J. Bastos and T. Rodrigues de Souza

Full-text: Open access


In this paper, we study a class of Rauzy fractals ${\mathcal R}_a$ given by the polynomial $x^3- ax^2+x-1$ where $a \geq 2$ is an integer. In particular, we give explicitly an automaton that generates the boundary of ${\mathcal R}_a$ and using an unusual numeration system we prove that ${\mathcal R}_a$ is homeomorphic to a topological disk.

Article information

Osaka J. Math., Volume 56, Number 1 (2019), 101-122.

First available in Project Euclid: 16 January 2019

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx]


Bastos, J.; de Souza, T. Rodrigues. Parametrization for a class of Rauzy Fractals. Osaka J. Math. 56 (2019), no. 1, 101--122.

Export citation


  • S. Akiyama: Cubic Pisot units with finite beta expansions; in Algebraic Number Theory and Diophantine Analysis, ed. F. Halter-Koch and R.F. Tichy, de Gruyter, Berlin, 2000, 11–26.
  • S. Akiyama: Self affine tiling and Pisot numeration system; in Number theory and its Applications (Kyoto, 1997), Kluwer Acad. Publ., Dordrecht, 1999, 7–17.
  • S. Akiyama: On the boundary of self affine tilings generated by Pisot numbers, J. Math. Soc. Japan, 54 (2002), 283–308.
  • S. Akiyama: Pisot number system and its dual tiling; in Physics and Theoretical Computer Science (Cargese, 2006), IOS Press, 2007, 133–54.
  • P. Arnoux, V. Berthé and S. Ito: Discrete planes $\mathbb{Z}^2$-actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier, 52 (2002), 305–349.
  • P. Arnoux and S. Ito: Pisot substitutions and Rauzy fractals, Bull Belg. Math. Soc. Simon Stevin, 8 (2001), 181–207.
  • P. Arnoux, M. Furukado, E. Harriss and S. Ito: Algebraic numbers, free group automorphisms and substitutions on the plane, Tran. Amer. Math. Soc. 363 (2011), 4651–4699.
  • G.Barat, V. Berthé, P. Liardet and J. Thuswaldner: Dynamical Directions in Numeration, Ann. Instit. Fourier (Grenoble), 56 (2006), 1987–2092.
  • J. Bastos, A. Messaoudi, D. Smania and T. Rodrigues: A class of cubic Rauzy fractals, Theoret. Comput. Sci. 588 (2015), 114–130.
  • J. Bastos and T. Rodrigues: Parametrization for a class of Rauzy Fractal, arXiv1409.1168v4 [math.DS].
  • V. Berthé and A. Siegel: Tilings associated with beta-numerations and substitutions, Integers 5 (2005), A2, 46pp.
  • V. Berthé, A. Siegel and J. Thuswaldner: Substitutions, Rauzy Fractals and Tilings; in Combinatorics, automata and number theory, Encyclopedia Math. Appl. 135, Cambridge Univ. Press, Cambridge, 2010, 248–323.
  • V. Canterini and A. Siegel: Geometric representation of substitutions of Pisot type, Trans. Amer. Math. Soc. 353 (2001), 5121–5144.
  • N. Chekhova, P. Hubert and A. Messaoudi: Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, J. Theor. Nombres Bordeaux 13 (2001), 371–394.
  • F. Durand and A. Messaoudi: Boundary of the rauzy fractal sets in $\mathbb{R} \times\mathbb{C}$ generated by $p(x)=x^{4}-x^{3}-x^{2}-x-1$, Osaka J. Math. 48 (2009), 471–469.
  • C. Frougny and B. Solomyak: Finite beta-expansions, Ergodic Theory Dynam. Systems 12 (1992), 713–723.
  • B. Loridant, A. Messaoudi, P. Surer and J.Thuswaldner: Tilings induced by a class of cubic Rauzy fractals, Theoret. Comput. Sci. 477 (2013), 6–31.
  • B. Loridant: Topological Properties of a class of cubic Rauzy fractals, Osaka J. Math. 53 (2016), 161–219.
  • A. Messaoudi: Frontiere du fractal de Rauzy et systemes de numération complexe, Acta Arith. 95 (2000), 195–224.
  • A. Messaoudi: Propriétés arithmétiques et dynamiques du fractal de Rauzy, J. Théor. Nombres Bordeaux, 10 (1998), 135–162.
  • W. Parry: On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.
  • B. Praggastis: Numerations systems and Markov partitions from self similar tilings, Trans. Amer. Math. Soc. 351 (1999), 3315–3349.
  • G. Rauzy: Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147–178.
  • A. Rényi: Représentations for real numbers and their ergodic properties, Acta. Math. Acad. Sci. Hungar 8 (1957), 477–493.
  • A. Siegel: Représentation des systémes dynamiques substitutifs non unimodulaires, Ergodic Theory Dynam. Systems 23 (2003), 1247–1273.
  • W. Thurston: Groups, tilings, and finite state automata, AMS Colloquium lectures, 1989.
  • J.M. Thuswaldner and A. Siegel: Topological properties of Rauzy fractals, Mém Soc. Math. Fr. (N.S) 118 (2009), 144pp.