|
Symbolic dynamics and Markov partitions
Author(s):
Roy
L.
Adler
Journal:
Bull. Amer. Math. Soc.
35
(1998),
1-56.
MSC (1991):
Primary 58F03, 58F08, 34C35
Retrieve article in:
PDF
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.
References:
- [ATW]
- Roy Adler, Charles Tresser, Patrick A. Worfolk, Topological conjugacy of linear endomorphisms of the 2-torus, Trans. Amer. Math. Soc. 349 (1997), 1633-1652. MR 97m:58181
- [AW1]
- R.L. Adler and B. Weiss, Entropy, a complete metric invariant for automorphisms of the torus, Proc. Natl. Acad. Sci. 57 (1967), 1573-1576. MR 35:3031
- [AW2]
- Roy L. Adler and Benjamin Weiss, Similarity of automorphisms of the torus, Memoirs American Math. Soc. 98 (1970). MR 41:1966
- [Be]
- K. Berg, On the conjugacy problem for K-systems, Ph.D. Thesis, University of Minnesota, 1967.
- [Bo1]
- R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725-747. MR 43:2740
- [Bo2]
- -, On Axiom A Diffeomorphisms, Regional Conference Series in Mathematics, No. 35, American Mathematical Society, Providence, Rhode Island, 1978. MR 58:2888
- [Bo3]
- -, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin, Heidelberg, New York, 1975. MR 56:1364
- [Bo4]
- -, Markov partitions are not smooth, Proc. Amer. Math. Soc. 71 (1970), 130-132. MR 57:14055
- [C]
- E. Cawley, Smooth Markov partitions and toral automorphisms, Ergodic Th. and Dynam. Sys. 11 (1991), 633-651. MR 92k:58199
- [FS]
- A. Fathi and M. Shub, Some dynamics of pseudo-Anosov diffeomorphisms, Astérique 66-67 (1979), 181-207.
- [F]
- D. Fried, Finitely presented dynamical systems, Ergodic Th. and Dynam Sys. 7 (1987), 489-507. MR 89h:58157
- [H]
- J. Hadamard, Les surfaces á courbures opposées et leurs lignes geodésiques, Journal de Mathematiques. 5 série IV (1898), 27-73.
- [HK]
- B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, New York, Melbourne, 1995. MR 96c:58055
- [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]
- D. Lind and B Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, New York, Melbourne, 1995. MR 97a:58050
- [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]
- M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1987. MR 87m:58086
- [Si1]
- Ya. G. Sinai, Markov partitions and c-diffeomorphisms, Functional Analysis and Its Applications 2 (1968), 61-82. MR 38:1361
- [Si2]
- -, Construction of Markov partitions, Functional Analysis and Its Applications 2 (1968), 245-253. MR 40:3591
- [Sm]
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 37:3598
- [T]
- W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417-431. MR 89k:57023
- [W]
- R. F. Williams, The ``DA" maps of Smale and structural stability, Proc. Symp. in Pure Math., vol. 14, Amer. Math. Soc., Providence, RI, 1970, pp. 329-334. MR 41:9296
Similar Articles:
Retrieve articles in Bulletin of the American Mathematical Society
with MSC
(1991):
58F03, 58F08, 34C35
Retrieve articles in all Journals with MSC
(1991):
58F03, 58F08, 34C35
Additional Information:
Roy
L.
Adler
Affiliation:
Mathematical Sciences Department, IBM, Thomas J. Watson Research Center, Yorktown Heights, New York 10598
Email:
adler@watson.ibm.com
DOI:
10.1090/S0273-0979-98-00737-X
PII:
S 0273-0979(98)00737-X
Received by editor(s):
July 8, 1997
Additional Notes:
Appeared as MSRI Preprint No. 1996-053.
Copyright of article:
Copyright
1998,
American Mathematical Society
|
Comments: webmaster@ams.org