Operator algebras and symbolic dynamical systems associated to symbolic substitutions

Abstract

In this thesis, we shall look at symbolic dynamical systems and operator algebras that are associated with these systems. We shall focus on minimal shift dynamical systems generated by symbolic substitutions. By first characterizing the shift spaces associated to primitive substitutions, we shall see that all minimal dynamical systems generated by symbolic substitutions are conjugate to proper primitive substitutions. In doing this, we will look at ordered Bratteli diagrams and strongly maximal TAF-algebras and we will see how we can associate these to a type of dynamical system called a Cantor minimal system, of which infinite minimal shift spaces are an example. We also shall see how we can associate a semi-crossed product algebra to a topological dynamical system and how isomorphism of two semi-crossed product algebras is equivalent to conjugacy of their associated dynamical systems. The semi-crossed product algebra is more general in that it can be associated to any dynamical system, whereas TAF-algebras can only be associated to Cantor minimal systems.