Symbolic dynamics and Markov partitions

Adler, Roy

doi:10.1090/S0273-0979-98-00737-X

Symbolic dynamics and Markov partitions

Author: Roy L. Adler
Journal: Bull. Amer. Math. Soc. 35 (1998), 1-56
MSC (1991): Primary 58F03, 58F08, 34C35
DOI: https://doi.org/10.1090/S0273-0979-98-00737-X
MathSciNet review: 1477538
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The decimal expansion of real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by symbolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by symbolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely, hyperbolic automorphisms of the two dimensional torus. While there are some results in higher dimensions, this area remains a fertile one for research.

[ATW] Roy Adler, Charles Tresser, and Patrick A. Worfolk, Topological conjugacy of linear endomorphisms of the 2-torus, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1633–1652. MR 1407693, https://doi.org/10.1090/S0002-9947-97-01895-3
[AW1] R. L. Adler and B. Weiss, Entropy, a complete metric invariant for automorphisms of the torus, Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 1573–1576. MR 0212156
[AW2] 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
[Be] K. Berg, On the conjugacy problem for K-systems, Ph.D. Thesis, University of Minnesota, 1967.
[Bo1] Rufus Bowen, Markov partitions for Axiom 𝐴 diffeomorphisms, Amer. J. Math. 92 (1970), 725–747. MR 0277003, https://doi.org/10.2307/2373370
[Bo2] Rufus Bowen, On Axiom A diffeomorphisms, American Mathematical Society, Providence, R.I., 1978. Regional Conference Series in Mathematics, No. 35. MR 0482842
[Bo3] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
[Bo4] Rufus Bowen, Markov partitions are not smooth, Proc. Amer. Math. Soc. 71 (1978), no. 1, 130–132. MR 0474415, https://doi.org/10.1090/S0002-9939-1978-0474415-8
[C] Elise Cawley, Smooth Markov partitions and toral automorphisms, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 633–651. MR 1145614, https://doi.org/10.1017/S0143385700006404
[FS] A. Fathi and M. Shub, Some dynamics of pseudo-Anosov diffeomorphisms, Astérique 66-67 (1979), 181-207.
[F] David Fried, Finitely presented dynamical systems, Ergodic Theory Dynam. Systems 7 (1987), no. 4, 489–507. MR 922362, https://doi.org/10.1017/S014338570000417X
[H] J. Hadamard, Les surfaces á courbures opposées et leurs lignes geodésiques, Journal de Mathematiques. 5 série IV (1898), 27-73.
[HK] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374
[KV] R. Kenyon and A. Vershik, Arithmetic construction of sofic partitions of hyperbolic toral automorphisms, Prepublication ou Rapport de Recherce no. 178, Ecole Normale Superieure de Lyon, 1995.
[LM] Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. MR 1369092
[Pr1] B. Praggastis, Markov partitions for hyperbolic toral automorphisms, Ph.D. Thesis, University of Washington, 1994.
[Pr2] -, Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc. (to appear).
[S] Michael Shub, Global stability of dynamical systems, Springer-Verlag, New York, 1987. With the collaboration of Albert Fathi and Rémi Langevin; Translated from the French by Joseph Christy. MR 869255
[Si1] Ja. G. Sinaĭ, Markov partitions and \cyr U-diffeomorphisms, Funkcional. Anal. i Priložen 2 (1968), no. 1, 64–89 (Russian). MR 0233038
[Si2] Ja. G. Sinaĭ, Construction of Markov partitionings, Funkcional. Anal. i Priložen. 2 (1968), no. 3, 70–80 (Loose errata) (Russian). MR 0250352
[Sm] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 0228014, https://doi.org/10.1090/S0002-9904-1967-11798-1
[T] William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, https://doi.org/10.1090/S0273-0979-1988-15685-6
[W] R. F. Williams, The “𝐷𝐴” maps of Smale and structural stability, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 329–334. MR 0264705