An expansive dynamical system is a pair $(X, \varphi)$ where $X$ is a compact metric space and $\varphi$ is a homeomorphism that is sensitive to initial conditions. An important subclass of expansive dynamical systems are Smale spaces. Building on work of Ruelle and Putnam, Thomsen has constructed $C^*$-algebras associated with such a system. I will discuss the structure of these $C^*$-algebras and compare the results for general expansive systems, Smale spaces, and another important class called synchronizing systems. Knowledge of the definitions of expansive, Smale space, and synchronizing are not required as they will be introduced in the talk.

This talk is based on joint work with Andrew Stocker. This work was partially supported by NSF DMS grant 2000057.