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.