Osaka Journal of Mathematics

On a class of Rauzy fractals without the finiteness property

Gustavo A. Pavani

Full-text: Open access


We present some topological and arithmetical aspects of a class of Rauzy fractals $\mathcal{R}_{a,b}$ related to the polynomials of the form $P_{a,b}(x)=x^{3}-ax^{2}-bx-1$, where $a$ and $b$ are integers satisfying $-a+1 \leq b \leq -2$. This class has the property that $0$ lies on the boundary of $\mathcal{R}_{a,b}$. We construct explicit finite automata that recognize the boundaries of these fractals. This allows to establish the number of neighbors of $\mathcal{R}_{a,b}$ in the tiling it generates. Furthermore, we prove that if $2a+3b+4 \leq 0$ then $\mathcal{R}_{a,b}$ is not homeomorphic to a topological disk. We also show that the boundary of the set $\mathcal{R}_{3,-2}$ is generated by two infinite iterated function systems.

Article information

Osaka J. Math., Volume 56, Number 3 (2019), 577-599.

First available in Project Euclid: 16 July 2019

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B85: Automata sequences
Secondary: 28A80: Fractals [See also 37Fxx] 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx] 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20]


Pavani, Gustavo A. On a class of Rauzy fractals without the finiteness property. Osaka J. Math. 56 (2019), no. 3, 577--599.

Export citation


  • S. Akiyama: Cubic Pisot units with finite beta expansions; in Algebraic Number Theory and Diophantine Analysis 2000, 11–26.
  • P. Arnoux, V. Berthé, H. Ei and S. Ito: Tilings, quasicrystals, discrete planes, generalized substitutions, and multidimensional continued fractions; in Discrete Math. Theor. Comput. Sci. AA 2001, 59–78.
  • P. Arnoux, V. Berthé and S. Ito: Discrete planes, $\mathbb{Z}^{2}$-actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier (Grenoble) 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, S. Ito and Y. Sano: Higher dimensional extensions of substitutions and their dual maps, J. Anal. Math. 83 (2001), 183–206.
  • P. Arnoux and G. Rauzy: Représentation géométrique des suites de complexité $2n+1$, Bull. Soc. Math. France 119 (1991), 101–117.
  • J. Bastos, A. Messaoudi, T. Rodrigues and D. Smania: A class of cubic Rauzy fractals, Theoret. Comput. Sci. 588 (2015), 114–130.
  • V. Berthé and A. Siegel: Purely periodic beta-expansions in the non-unit case, J. Number Theory 127 (2007), 153–172.
  • V. Canterini: Connectedness of geometric representation of substitutions of Pisot type, Bull. Belg. Math. Soc. Simon Stevin 10 (2003), 77–89.
  • 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. Théor. Nombres Bordeaux 13 (2001), 371–394.
  • C. Frougny: Number representation and finite automata; in Topics in Symbolic Dynamics and Applications, London Math. Soc. Lectures Note Ser. 279, 2000, 207–228.
  • B. Grünbaum and G.C. Shephard: Tilings and Patterns, W.H. Freeman and Company, New York, 1987.
  • C. Holton and L. Zamboni: Geometric realizations of substitutions, Bull. Soc. Math. France 126 (1998), 149–179.
  • P. Hubert and A. Messaoudi: Best simultaneous diophantine approximation of Pisot numbers and Rauzy fractals, Acta Arith. 124 (2006), 1–15.
  • T. Jolivet: Timo Jolivet's research page, available at
  • R. Kenyon and A. Vershik: Arithmetic construction of sofic partitions of hyperbolic toral automorphisms, Ergodic Theory Dynam. Systems 18 (1998), 357–372.
  • J.C. Lagarias, H.A. Porta and K.B. Stolarsky: Asymmetric tent map expansions. I. Eventually periodic points, J. London Math. Soc. 47 (1993), 542–556.
  • B. Loridant: Topological properties of a class of cubic Rauzy fractals, Osaka J. Math. 53 (2016), 161–219.
  • B. Loridant, A. Messaoudi, J. Thuswaldner and P. Surer: Tilings induced by a class of Rauzy fractals, Theoret. Comput. Sci. 477 (2013), 6–31.
  • A. Messaoudi: Propriétés arithmétiques et dynamiques du fractal de Rauzy, J. Théor. Nombres Bordeaux 10 (1998), 135–162.
  • A. Messaoudi: Frontière du fractal de Rauzy et système de numération complexe, Acta Arith. 95 (2000), 195–224.
  • A. Messaoudi: Propriétés arithmétiques et topologiques d'une classe d'ensembles fractales, Acta Arith. 121 (2006), 341–366.
  • W. Parry: On the beta-expansion of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.
  • G.A. Pavani: Fractais de Rauzy, autômatos e fraçoes contínuas (Portuguese). Rauzy Fractals, Automata, and Continued Fractions, PhD Thesis, Saõ Paulo State University (2015), available at
  • B. Praggastis: Numeration 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: Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.
  • S. Ito and M. Kimura: On Rauzy fractal, Japan J. Indust. Appl. Math. 8 (1991), 461–486.
  • K. Schmidt: On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), 269–278.
  • A. Siegel and J. Thuswaldner: Topological properties of Rauzy fractals, Mém. Soc. Math. Fr. 118 (2009), 144pp.
  • V. Sirvent and Y. Wang: Self-affine tiling via substitution dynamical systems and Rauzy fractals, Pacific J. Math. 206 (2002), 465–485.
  • C.J. Smyth: There are only eleven Special Pisot numbers, Bull. London Math. Soc. 31 (1999), 1–5.
  • W. Thurston: Groups, tilings, and finite state automata, AMS Colloquium Lecture Notes 1 (1989).
  • J.M. Thuswaldner: Unimodular Pisot substitutions and their associated tiles, J. Théor. Nombres Bordeaux 18 (2006), 487–536.