Symbolic dynamics generally studies properties of invariant sets and measures which are defined on infinite sequences of characters of a fixed (finite or countably infinite) alphabet.
One example of such a system arises from a substitution. A (deterministic) substitution consists of a finite alphabet along with a set of transformation rules. A classical example is the Fibonacci substitution, which is composed of the transformations taking a to ab and b to a. Random substitutions are a generalization of deterministic substitutions where a transformation rule is chosen randomly from a list of rules. Deterministic and random substitutions have associated dynamical systems and invariant measures, which capture certain “asymptotic statistics” of the substitution.