Self-similar sets. V. Integer matrices and fractal tilings of 𝑅ⁿ

Bandt, Christoph

doi:10.1090/S0002-9939-1991-1036982-1

Self-similar sets. V. Integer matrices and fractal tilings of $\textbf {R}^ n$
HTML articles powered by AMS MathViewer

by Christoph Bandt PDF

Proc. Amer. Math. Soc. 112 (1991), 549-562 Request permission

Abstract:

Tilings with self-similar tiles and "just-touching fractals" are constructed from matrices of integers, extending work by Levy, Mandelbrot, Dekking, Bedford, and others.

References

Christoph Bandt, Self-similar sets. I. Topological Markov chains and mixed self-similar sets, Math. Nachr. 142 (1989), 107–123. MR 1017373, DOI 10.1002/mana.19891420107
Christoph Bandt, Self-similar sets. III. Constructions with sofic systems, Monatsh. Math. 108 (1989), no. 2-3, 89–102. MR 1026611, DOI 10.1007/BF01308664
Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields 79 (1988), no. 4, 543–623. MR 966175, DOI 10.1007/BF00318785
Michael F. Barnsley, Fractals everywhere, 2nd ed., Academic Press Professional, Boston, MA, 1993. Revised with the assistance of and with a foreword by Hawley Rising, III. MR 1231795
Tim Bedford, Generating special Markov partitions for hyperbolic toral automorphisms using fractals, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 325–333. MR 863197, DOI 10.1017/S0143385700003527
F. M. Dekking, Recurrent sets, Adv. in Math. 44 (1982), no. 1, 78–104. MR 654549, DOI 10.1016/0001-8708(82)90066-4
K. J. Falconer, The Hausdorff dimension of some fractals and attractors of overlapping construction, J. Statist. Phys. 47 (1987), no. 1-2, 123–132. MR 892926, DOI 10.1007/BF01009037

In which "monster" curves force redefinition of the word "curve,"

235

William J. Gilbert, Radix representations of quadratic fields, J. Math. Anal. Appl. 83 (1981), no. 1, 264–274. MR 632342, DOI 10.1016/0022-247X(81)90262-6
William J. Gilbert, Geometry of radix representations, The geometric vein, Springer, New York-Berlin, 1981, pp. 129–139. MR 661773
William J. Gilbert, The fractal dimension of sets derived from complex bases, Canad. Math. Bull. 29 (1986), no. 4, 495–500. MR 860860, DOI 10.4153/CMB-1986-078-1
Jack Giles Jr., Infinite-level replicating dissections of plane figures, J. Combin. Theory Ser. A 26 (1979), no. 3, 319–327. MR 535163, DOI 10.1016/0097-3165(79)90110-9
Branko Grünbaum and G. C. Shephard, Tilings and patterns, W. H. Freeman and Company, New York, 1987. MR 857454
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055

Les courbes planes ou gauches et les surfaces composees de parties semblables au tout

7-8

Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 83 (1990), no. 420, iv+128. MR 988082, DOI 10.1090/memo/0420
Benoit B. Mandelbrot, The fractal geometry of nature, Schriftenreihe für den Referenten. [Series for the Referee], W. H. Freeman and Co., San Francisco, Calif., 1982. MR 665254
R. Daniel Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), no. 2, 811–829. MR 961615, DOI 10.1090/S0002-9947-1988-0961615-4
Wm. Douglas Withers, Folding polynomials and their dynamics, Amer. Math. Monthly 95 (1988), no. 5, 399–413. MR 937529, DOI 10.2307/2322475