Journal of the Mathematical Society of Japan

Boundary parametrization of self-affine tiles

Shigeki AKIYAMA and Benoît LORIDANT

Full-text: Open access


A standard way to parametrize the boundary of a connected fractal tile T is proposed. The parametrization is Hölder continuous from R/Z to ∂T and fixed points of ∂T have algebraic preimages. A class of planar tiles is studied in detail as sample cases and a relation with the recurrent set method by Dekking is discussed. When the tile T is a topological disk, this parametrization is a bi-Hölder homeomorphism.

Article information

J. Math. Soc. Japan, Volume 63, Number 2 (2011), 525-579.

First available in Project Euclid: 25 April 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A80: Fractals [See also 37Fxx] 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20] 68Q70: Algebraic theory of languages and automata [See also 18B20, 20M35]
Secondary: 05B45: Tessellation and tiling problems [See also 52C20, 52C22] 28A78: Hausdorff and packing measures 37F20: Combinatorics and topology 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] 54D05: Connected and locally connected spaces (general aspects)

self-affine tile graph directed set Hausdorff measure Büchi automata


AKIYAMA, Shigeki; LORIDANT, Benoît. Boundary parametrization of self-affine tiles. J. Math. Soc. Japan 63 (2011), no. 2, 525--579. doi:10.2969/jmsj/06320525.

Export citation


  • S. Akiyama and B. Loridant, Boundary parametrization of planar self-affine tiles with collinear digit set, Sci. China Math., 53 (2010), 2173–2194.
  • S. Akiyama and J. M. Thuswaldner, Topological properties of two-dimensional number systems, J. Théor. Nombres Bordeaux, 12 (2000), 69–79.
  • S. Akiyama and J. M. Thuswaldner, A survey on topological properties of tiles related to number systems, Geom. Dedicata, 109 (2004), 89–105.
  • S. Akiyama and J. M. Thuswaldner, The topological structure of fractal tilings generated by quadratic number systems, Comput. Math. Appl., 49 (2005), 1439–1485.
  • P. Arnoux, Sh. Ito and Y. Sano, Higher dimensional extensions of substitutions and their dual maps, J. Anal. Math., 83 (2001), 183–206.
  • C. Bandt and M. Mesing, Self-affine fractals of finite type, Banach Center Publication, 2009.
  • H. Brunotte, Characterization of CNS trinomials, Acta Sci. Math. (Szeged), 68 (2002), 673–679.
  • F. M. Dekking, Replicating superfigures and endomorphisms of free groups, J. Combin. Theory Ser. A, 32 (1982), 315–320.
  • F. M. Dekking, Recurrent sets, Adv. in Math., 44 (1982), 78–104.
  • F. M. Dekking and P. van der Wal, The boundary of the attractor of a recurrent iterated function system, Fractals, 10 (2002), 77–89.
  • J. M. Dumont and A. Thomas, Systèmes de numération et fonctions fractales relatifs aux substitutions, Theoret. Comput. Sci., 65 (1989), 153–169.
  • P. Duvall, J. Keesling and A. Vince, The Hausdorff dimension of the boundary of a self-similar tile, J. London Math. Soc. (2), 61 (2000), 748–760.
  • K. J. Falconer, Techniques in Fractal Geometry, John Wiley and Sons, Chichester, New York, Weinheim, Brisbane, Singapore, Toronto, 1997.
  • W. J. Gilbert, Radix representations of quadratic fields, J. Math. Anal. Appl., 83 (1981), 264–274.
  • W. J. Gilbert, Complex bases and fractal similarity, Ann. Sci. Math. Québec, 11 (1987), 65–77.
  • K. Gröchenig and A. Haas, Self-similar lattice tilings, J. Fourier Anal. Appl., 1 (1994), 131–170.
  • K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of $R\sp n$, IEEE Trans. Inform. Theory, 38 (1992), 556–568.
  • B. Grünbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman and Company, New York, 1987.
  • M. Hata, On the structure of self-similar sets, Japan J. Appl. Math., 2 (1985), 381–414.
  • X. G. He and K. S. Lau, On a generalized dimension of self-affine fractals, Math. Nachr., 281 (2008), 1142–1158.
  • J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713–747.
  • K. H. Indlekofer, I. Katai and P. Racsko, Number systems and fractal geometry, Probability theory and applications, Math. Appl., Kluwer Acad. Publ., Dordrecht, 80 (1992), 319–334.
  • Sh. Ito, On the fractal curves induced from the complex radix expansion, Tokyo J. Math., 12 (1989), 299–320.
  • I. Kátai, Number systems and fractal geometry, University of Pécs, 1995.
  • I. Kátai and I. Kőrnyei, On number systems in algebraic number fields, Publ. Math. Debrecen, 41 (1992), 289–294.
  • I. Kátai and B. Kovács, Kanonische Zahlensysteme in der Theorie der Quadratischen Zahlen, Acta Sci. Math. (Szeged), 42 (1980), 99–107.
  • K. Kuratowski, Topology, II, Academic Press, New York, 1968.
  • J. Lagarias and Y. Wang, Self-affine tiles in $\bm{R}^n$, Adv. Math., 121 (1996), 21–49.
  • J. Lagarias and Y. Wang, Integral self-affine tiles in $\bm{R}^ n$, I, Standard and nonstandard digit sets, J. London Math. Soc. (2), 54 (1996), 161–179.
  • J. Lagarias and Y. Wang, Haar bases for $L\sp 2(R\sp n)$ and algebraic number theory, J. Number Theory, 57 (1996), 181–197.
  • J. Lagarias and Y. Wang, Integral self-affine tiles in $R\sp n$, II, Lattice tilings, J. Fourier Anal. Appl., 3 (1997), 83–102.
  • B. Loridant and J. M. Thuswaldner, Interior components of a tile associated to a quadratic canonical number system, Topology Appl., 155 (2008), 667–695.
  • J. Luo, S. Akiyama and J. M. Thuswaldner, On the boundary connectedness of connected tiles, Math. Proc. Cambridge Philos. Soc., 137 (2004), 397–410.
  • J. Luo and Y. M. Yang, On single matrix graph directed iterated function systems, J. Math. Anal. Appl., 372 (2010), 8–18.
  • R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811–829.
  • 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. Müller, J. M. Thuswaldner and R. F. Tichy, Fractal properties of number systems, Periodica Math. Hungar., 42 (2001), 51–68.
  • S. Z. Ngai and T. M. Tang, A technique in the topology of connected self-similar tiles, Fractals, 12 (2004), 389–403.
  • S. M. Ngai and N. Nguyen, The Heighway dragon revisited, Discrete Comput. Geom., 29 (2003), 603–623.
  • D. Perrin and J.-E. Pin, Infinite Words – Automata, Semigroups, Logic and Games, Pure and Applied Mathematics, 141, Elsevier, 2004.
  • K. Scheicher and J. M. Thuswaldner, Canonical number systems, counting automata and fractals, Math. Proc. Cambridge Philos. Soc., 133 (2002), 163–182.
  • K. Scheicher and J. M. Thuswaldner, Neighbors of self-affine tiles in lattice tilings, in Fractals in Graz 2001, (eds. P. Grabner and W. Woess), Birkhäuser Verlag, 2002, pp.,241–262.
  • A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122 (1994), 111–115.
  • B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems, 17 (1997), 695–738.
  • H. J. Song, Replicating fractiles derived from digit systems in lattices, unpublished.
  • R. S. Strichartz and Y. Wang, Geometry of self-affine tiles I, Indiana Univ. Math. J., 48 (1999), 1–23.
  • T. M. Tang, Arcwise connectedness of the boundary of connected self-similar sets, Acta Math. Hungar., 109 (2005), 295–303.
  • W. Thomas, Automata on infinite objects, Handbook of theoretical computer science, Vol.,B, Elsevier, Amsterdam, 1990, pp.,133–191.
  • A. Vince, Self-replicating tiles and their boundary, Discrete Comput. Geom., 21 (1999), 463–476.
  • A. Vince, Digit tiling of euclidean space, in Directions in Mathematical Quasicrystals, Amer. Math. Soc., Providence, RI, 2000, pp.,329–370.
  • G. T. Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, 28, Amer. Math. Soc., New York, 1942.