This course introduces the rich interplay between topological dynamics and the structure of C*-algebras arising from dynamical systems. Constructing a C*-algebra from a dynamical system allows one to use C*-algebraic tools to study such systems, while at the same time providing interesting examples of C*-algebras where the underlying dynamics provides information on their structures. In the first lecture, I will introduce C*-algebras and give an overview of this relationship.

The next two lectures will consider more specific examples. We will begin by considering zero-dimensional examples, namely, shifts of finite type and Cantor minimal systems, and see how the resulting C*-algebraic constructions capture dynamical invariants. Next, we will move on to their higher-dimensional analogues by constructing C*-algebras from Smale spaces and from minimal homeomorphisms on higher-dimensional spaces. With these examples, we will look at what structural properties arise from the dynamics and make connections to the classification programme for C*-algebras.