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

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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.

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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:
https://doi.org/10.1090/S0273-0979-98-00737-X

Received by editor(s):
July 8, 1997

Additional Notes:
Appeared as MSRI Preprint No. 1996-053.

Article copyright:
© Copyright 1998
American Mathematical Society