Symbolic Dynamics

Here is an introduction to Symbolic Dynamics or, if you like books, there is Symbolic Dynamics and Coding.

One of the things that makes symbolic dynamics useful/interesting are Markov Partitions. A Markov Partition is a finite partition of the state space for a dynamical system. For a point x in the state space, look at it’s orbit and write down which element in the partition each point of the orbit belongs to. This infinite sequence of partition elements uniquely defines x. Doing this for all points in the state space yields a mapping from the dynamical system to a shift space. For more details, see the paper Symbolic Dynamics and Markov Partitions.