Geometric & Functional Analysis GAFA

, Volume 6, Issue 3, pp 471–488 | Cite as

The construction of self-similar tilings

  • R. Kenyon


We give a construction of a self-similar tiling of the plane with any prescribed expansion coefficient λɛℂ (satisfying the necessary algebraic condition of being a complex Perron number).

For any integerm>1 we show that there exists a self-similar tiling with 2π/m-rotational symmetry group and expansion λ if and only if either λ or λe2π∿/m is a complex Perron number for which e2π∿/m is in ℚ[λ], respectivelyQe2πı/m].


Expansion Coefficient Symmetry Group Algebraic Condition Perron Number Prescribe Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AWe]
    R. Adler, B. Weiss, Entropy is a complete metric invariant for automorphisms of the torus, Proc. Nat. Acad. Sci. 57:6 (1967), 1573–1576.Google Scholar
  2. [B]
    C. Bandt, Self-similar sets 5. Integer matrices and fractal tilings of ℝn, Proc. AMS. 112:2 (1991), 549–562.Google Scholar
  3. [Bo]
    R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer Lecture Notes in Math 470 (1975), 78–83.Google Scholar
  4. [D1]
    F.M. Dekking, Recurrent sets, Adv. in Math. 44 (1982), 78–104.Google Scholar
  5. [D2]
    F.M. Dekking, Replicating superfigures and endomorphisms of free groups, J. Combin. Th. Ser. A 32 (1982), 315–320.Google Scholar
  6. [G]
    M. Gardner, Extraordinary nonperiodic tiling that enriches the theory of tiles, Scientific American (January 1977), 116–119.Google Scholar
  7. [Gi]
    W. Gilbert, Radix representations of quadratic fields, Journal of Math. Anal. and Appl. 83 (1981), 264–274.Google Scholar
  8. [Go]
    S. Golomb, Replicating figures in the plane, Math. Gaz. 48 (1964), 403–412.Google Scholar
  9. [GrH]
    K. Gröchenig, A. Haas, Self-similar lattice tilings, J. Fourier Analysis, to appear.Google Scholar
  10. [K1]
    R. Kenyon, Self-similar tilings, Thesis, Princeton Univ., 1990.Google Scholar
  11. [K2]
    R. Kenyon, Inflationary similarity-tilings, Comment. Math. Helv., 69 (1994), 169–198.Google Scholar
  12. [LW]
    J. Lagarias, Y. Wang, Integral self-affine tiles in ℝn. I: Standard and nonstandard digit sets, J. London Math. Soc., to appear.Google Scholar
  13. [Li]
    D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Erg. Th. Dyn. Sys. 4 (1984), 283–300.Google Scholar
  14. [P]
    B. Praggastis, Markov partitions for hyperbolic toral automorphisms, Thesis, Univ. of Washington, Seattle (1992).Google Scholar
  15. [RWo]
    C. Radin, M. Wolff, Space tilings and local isomorphism, Geometriae Dedicata 42 (1992), 355–360.Google Scholar
  16. [S]
    Y. Sinai, Constructions of Markov partitions, Func. Anal. and its Appl. 2:2 (1968), 70–80.Google Scholar
  17. [St]
    R.S. Strichartz, Wavelets and self-affine tilings, Constructive Approximation 9 (1993), 327–346.Google Scholar
  18. [T]
    W.P. Thurston, Groups, tilings, and finite state automata, Lecture notes, AMS colloquium lectures, (1990).Google Scholar
  19. [V]
    A.M. Vershik, Arithmetic isomorphism of hyperbolic toral automorphisms and sofic shifts, Func. Anal. and its Appl. 26:3 (1992), 170–173.Google Scholar
  20. [Vi]
    A. Vince, Replicating tessellations, SIAM J. Disc. Math. 6 (1993), 501–521.Google Scholar

Copyright information

© Birkhäuser Verlag 1996

Authors and Affiliations

  • R. Kenyon
    • 1
  1. 1.CNRS UMR 128 Ecole Normale Supérieure de LyonLyonFrance

Personalised recommendations