Abstract
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], respectivelyQ[λe 2πı/m].
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References
R. Adler, B. Weiss, Entropy is a complete metric invariant for automorphisms of the torus, Proc. Nat. Acad. Sci. 57:6 (1967), 1573–1576.
C. Bandt, Self-similar sets 5. Integer matrices and fractal tilings of ℝn, Proc. AMS. 112:2 (1991), 549–562.
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer Lecture Notes in Math 470 (1975), 78–83.
F.M. Dekking, Recurrent sets, Adv. in Math. 44 (1982), 78–104.
F.M. Dekking, Replicating superfigures and endomorphisms of free groups, J. Combin. Th. Ser. A 32 (1982), 315–320.
M. Gardner, Extraordinary nonperiodic tiling that enriches the theory of tiles, Scientific American (January 1977), 116–119.
W. Gilbert, Radix representations of quadratic fields, Journal of Math. Anal. and Appl. 83 (1981), 264–274.
S. Golomb, Replicating figures in the plane, Math. Gaz. 48 (1964), 403–412.
K. Gröchenig, A. Haas, Self-similar lattice tilings, J. Fourier Analysis, to appear.
R. Kenyon, Self-similar tilings, Thesis, Princeton Univ., 1990.
R. Kenyon, Inflationary similarity-tilings, Comment. Math. Helv., 69 (1994), 169–198.
J. Lagarias, Y. Wang, Integral self-affine tiles in ℝn. I: Standard and nonstandard digit sets, J. London Math. Soc., to appear.
D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Erg. Th. Dyn. Sys. 4 (1984), 283–300.
B. Praggastis, Markov partitions for hyperbolic toral automorphisms, Thesis, Univ. of Washington, Seattle (1992).
C. Radin, M. Wolff, Space tilings and local isomorphism, Geometriae Dedicata 42 (1992), 355–360.
Y. Sinai, Constructions of Markov partitions, Func. Anal. and its Appl. 2:2 (1968), 70–80.
R.S. Strichartz, Wavelets and self-affine tilings, Constructive Approximation 9 (1993), 327–346.
W.P. Thurston, Groups, tilings, and finite state automata, Lecture notes, AMS colloquium lectures, (1990).
A.M. Vershik, Arithmetic isomorphism of hyperbolic toral automorphisms and sofic shifts, Func. Anal. and its Appl. 26:3 (1992), 170–173.
A. Vince, Replicating tessellations, SIAM J. Disc. Math. 6 (1993), 501–521.
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Research at MSRI is supported in part by NSF grant DMS-9022140.
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Kenyon, R. The construction of self-similar tilings. Geometric and Functional Analysis 6, 471–488 (1996). https://doi.org/10.1007/BF02249260
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DOI: https://doi.org/10.1007/BF02249260