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Numeration systems and Markov partitions from self similar tilings. (English) Zbl 0984.11008
Summary: Using self similar tilings, we represent the elements of \(\mathbb{R}^n\) as digit expansions with digits in \(\mathbb{R}^n\) being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the integer lattice. We use the digit expansions inherited from these tilings to give a symbolic representation for the toral automorphisms.

MSC:
11A63 Radix representation; digital problems
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
37B10 Symbolic dynamics
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
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[1] Roy L. Adler and Brian Marcus, Topological entropy and equivalence of dynamical systems, Mem. Amer. Math. Soc. 20 (1979), no. 219, iv+84. · Zbl 0412.54050
[2] Roy L. Adler and Benjamin Weiss, Similarity of automorphisms of the torus, Memoirs of the American Mathematical Society, No. 98, American Mathematical Society, Providence, R.I., 1970. · Zbl 0195.06104
[3] Timothy Bedford, Crinkly Curves, Markov Partitions and Dimension, Ph.D. Thesis, Warwick University, 1984.
[4] Rufus Bowen, Markov partitions for Axiom A diffeomorphisms, American Journal of Mathematics, 92:725-747, August, 1970. · Zbl 0208.25901
[5] Rufus Bowen, Markov partitions are not smooth, Proc. Amer. Math. Soc. 71 (1978), no. 1, 130 – 132. · Zbl 0417.58011
[6] Elise Cawley, Smooth Markov partitions and toral automorphisms, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 633 – 651. · Zbl 0754.58028
[7] Christiane Frougny and Boris Solomyak, Finite beta-expansions, Ergodic Theory Dynam. Systems 12 (1992), no. 4, 713 – 723. · Zbl 0814.68065
[8] William J. Gilbert, The fractal dimension of sets derived from complex bases, Canad. Math. Bull. 29 (1986), no. 4, 495 – 500. · Zbl 0564.10007
[9] Richard Kenyon, Inflationary tilings with a similarity structure, Comment. Math. Helv. 69 (1994), no. 2, 169 – 198. · Zbl 0817.52021
[10] Richard W. Kenyon, Self-similar tilings, Technical report, Geometry Supercomputer Project, University of Minnesota, 1990. Research Report GCG 21. · Zbl 0770.52013
[11] Donald E. Knuth, The art of computer programming. Volume 3, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973. Sorting and searching; Addison-Wesley Series in Computer Science and Information Processing. · Zbl 0302.68010
[12] D. A. Lind, Dynamical properties of quasihyperbolic toral automorphisms, Ergodic Theory Dynamical Systems 2 (1982), no. 1, 49 – 68. · Zbl 0507.58034
[13] Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. · Zbl 1106.37301
[14] W. Parry, On the \?-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401 – 416 (English, with Russian summary). · Zbl 0099.28103
[15] Brenda Praggastis, Markov Partitions for Hyperbolic Toral Automorphisms, Ph.D. thesis, University of Washington, 1994. · Zbl 0984.11008
[16] G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), no. 2, 147 – 178 (French, with English summary). · Zbl 0522.10032
[17] William P. Thurston, Groups, tilings, and finite state automata, Lecture notes distributed in conjunction with the Colloquium Series, 1989. In AMS Colloquium lectures.
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