On the boundaries of self-similar tiles in ℝ¹

He, Xing-Gang

doi:10.1090/S0002-9939-06-08643-6

On the boundaries of self-similar tiles in $\mathbb {R}^1$
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Proc. Amer. Math. Soc. 134 (2006), 3163-3170 Request permission

Abstract:

The aim of this note is to study the construction of the boundary of a self-similar tile, which is generated by an iterated function system $\{\phi _i(x)=\frac {1}{N} (x+d_i)\}_{i=1}^N$. We will show that the boundary has complicated structure (no simple points) in general; however, it is a regular fractal set.

References

Christoph Bandt, Self-similar sets. V. Integer matrices and fractal tilings of $\textbf {R}^n$, Proc. Amer. Math. Soc. 112 (1991), no. 2, 549–562. MR 1036982, DOI 10.1090/S0002-9939-1991-1036982-1
Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
Karlheinz Gröchenig and Andrew Haas, Self-similar lattice tilings, J. Fourier Anal. Appl. 1 (1994), no. 2, 131–170. MR 1348740, DOI 10.1007/s00041-001-4007-6
X.G. He and K.S. Lau, On a generalized dimension of self-affine fractals, Preprint.
Xing-Gang He, Ka-Sing Lau, and Hui Rao, Self-affine sets and graph-directed systems, Constr. Approx. 19 (2003), no. 3, 373–397. MR 1979057, DOI 10.1007/s00365-002-0515-0
Richard Kenyon, Self-replicating tilings, Symbolic dynamics and its applications (New Haven, CT, 1991) Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 239–263. MR 1185093, DOI 10.1090/conm/135/1185093
Ka-Sing Lau and Hui Rao, On one-dimensional self-similar tilings and $pq$-tiles, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1401–1414. MR 1946397, DOI 10.1090/S0002-9947-02-03207-5
Jeffrey C. Lagarias and Yang Wang, Tiling the line with translates of one tile, Invent. Math. 124 (1996), no. 1-3, 341–365. MR 1369421, DOI 10.1007/s002220050056
Jeffrey C. Lagarias and Yang Wang, Self-affine tiles in $\textbf {R}^n$, Adv. Math. 121 (1996), no. 1, 21–49. MR 1399601, DOI 10.1006/aima.1996.0045
Jeffrey C. Lagarias and Yang Wang, Integral self-affine tiles in $\mathbf R^n$. I. Standard and nonstandard digit sets, J. London Math. Soc. (2) 54 (1996), no. 1, 161–179. MR 1395075, DOI 10.1112/jlms/54.1.161
Jeffrey C. Lagarias and Yang Wang, Integral self-affine tiles in $\textbf {R}^n$. II. Lattice tilings, J. Fourier Anal. Appl. 3 (1997), no. 1, 83–102. MR 1428817, DOI 10.1007/s00041-001-4051-2
A. M. Odlyzko, Nonnegative digit sets in positional number systems, Proc. London Math. Soc. (3) 37 (1978), no. 2, 213–229. MR 507604, DOI 10.1112/plms/s3-37.2.213
Yuval Peres, Wilhelm Schlag, and Boris Solomyak, Sixty years of Bernoulli convolutions, Fractal geometry and stochastics, II (Greifswald/Koserow, 1998) Progr. Probab., vol. 46, Birkhäuser, Basel, 2000, pp. 39–65. MR 1785620
E. Seneta, Nonnegative matrices and Markov chains, 2nd ed., Springer Series in Statistics, Springer-Verlag, New York, 1981. MR 719544, DOI 10.1007/0-387-32792-4
Robert S. Strichartz and Yang Wang, Geometry of self-affine tiles. I, Indiana Univ. Math. J. 48 (1999), no. 1, 1–23. MR 1722192, DOI 10.1512/iumj.1999.48.1616
W. Thurston, Group tilings, and finite state automata, AMS Colloquium Lecture Notes, 1989.
Andrew Vince, Digit tiling of Euclidean space, Directions in mathematical quasicrystals, CRM Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, pp. 329–370. MR 1798999, DOI 10.1112/s0024610700008711
Y. Xu, Fractals and Tilings, Ph.D. Thesis, University of Pittsburgh 2000.