Self-similar sets. V. Integer matrices and fractal tilings of $\textbf {R}^ n$
HTML articles powered by AMS MathViewer
- by Christoph Bandt
- Proc. Amer. Math. Soc. 112 (1991), 549-562
- DOI: https://doi.org/10.1090/S0002-9939-1991-1036982-1
- PDF | 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 M. Gardner, In which "monster" curves force redefinition of the word "curve," Scientific American 235 (1976), 124-133.
- 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 P. Levy, Les courbes planes ou gauches et les surfaces composees de parties semblables au tout, J. Ecole Polytechnique III 7-8 (1938/39), 227-291.
- 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
Bibliographic Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 112 (1991), 549-562
- MSC: Primary 58F08; Secondary 28A80, 58F12
- DOI: https://doi.org/10.1090/S0002-9939-1991-1036982-1
- MathSciNet review: 1036982