Zero-Dimensional Dynamical Systems, Formal Languages, and Universality

Abstract.

We measure the complexity of dynamical systems on zero-dimensional compact metric spaces by the complexity of formal languages, which these systems generate on clopen partitions of the state space. We show that in the classes of recursive, context-sensitive, context-free, regular, etc., languages there exist universal dynamical systems which yield, by factor maps, all dynamical systems of the class. Universal systems are not unique, but in every class there exists a smallest universal system.