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
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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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