## Numeration systems and Markov partitions from self similar tilings

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- by Brenda Praggastis PDF
- Trans. Amer. Math. Soc.
**351**(1999), 3315-3349 Request permission

## Abstract:

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.## References

- Roy L. Adler and Brian Marcus,
*Topological entropy and equivalence of dynamical systems*, Mem. Amer. Math. Soc.**20**(1979), no. 219, iv+84. MR**533691**, DOI 10.1090/memo/0219 - 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. MR**0257315** - Timothy Bedford,
*Crinkly Curves, Markov Partitions and Dimension*, Ph.D. Thesis, Warwick University, 1984. - Richard Kershner,
*The number of circles covering a set*, Amer. J. Math.**61**(1939), 665–671. MR**43**, DOI 10.2307/2371320 - Rufus Bowen,
*Markov partitions are not smooth*, Proc. Amer. Math. Soc.**71**(1978), no. 1, 130–132. MR**474415**, DOI 10.1090/S0002-9939-1978-0474415-8 - Elise Cawley,
*Smooth Markov partitions and toral automorphisms*, Ergodic Theory Dynam. Systems**11**(1991), no. 4, 633–651. MR**1145614**, DOI 10.1017/S0143385700006404 - Christiane Frougny and Boris Solomyak,
*Finite beta-expansions*, Ergodic Theory Dynam. Systems**12**(1992), no. 4, 713–723. MR**1200339**, DOI 10.1017/S0143385700007057 - 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 - Richard Kenyon,
*Inflationary tilings with a similarity structure*, Comment. Math. Helv.**69**(1994), no. 2, 169–198. MR**1282366**, DOI 10.1007/BF02564481 - Richard W. Kenyon,
*Self-similar tilings, Technical report*, Geometry Supercomputer Project, University of Minnesota, 1990. Research Report GCG 21. - Donald E. Knuth,
*The art of computer programming. Volume 3*, Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973. Sorting and searching. MR**0445948** - D. A. Lind,
*Dynamical properties of quasihyperbolic toral automorphisms*, Ergodic Theory Dynam. Systems**2**(1982), no. 1, 49–68. MR**684244**, DOI 10.1017/s0143385700009573 - Douglas Lind and Brian Marcus,
*An introduction to symbolic dynamics and coding*, Cambridge University Press, Cambridge, 1995. MR**1369092**, DOI 10.1017/CBO9780511626302 - W. Parry,
*On the $\beta$-expansions of real numbers*, Acta Math. Acad. Sci. Hungar.**11**(1960), 401–416 (English, with Russian summary). MR**142719**, DOI 10.1007/BF02020954 - Brenda Praggastis,
*Markov Partitions for Hyperbolic Toral Automorphisms*, Ph.D. thesis, University of Washington, 1994. - G. Rauzy,
*Nombres algébriques et substitutions*, Bull. Soc. Math. France**110**(1982), no. 2, 147–178 (French, with English summary). MR**667748**, DOI 10.24033/bsmf.1957 - William P. Thurston,
*Groups, tilings, and finite state automata*, Lecture notes distributed in conjunction with the Colloquium Series, 1989. In*AMS Colloquium lectures*.

## Additional Information

**Brenda Praggastis**- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- Email: praggast@sprynet.com
- Received by editor(s): October 2, 1996
- Published electronically: April 8, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 3315-3349 - MSC (1991): Primary 58F03, 34C35, 54H20
- DOI: https://doi.org/10.1090/S0002-9947-99-02360-0
- MathSciNet review: 1615950