  The decimal expansion 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.
