Homeomorphisms of solenoids

In this talk, we consider dynamical systems which come from laminations. We will start with reviewing some basic examples of laminations, and how they arise naturally. We will then discuss a special class of laminations, called (weak) solenoids. The problem we consider is to describe the homeomorphisms between solenoids, and especially the self-homeomorphisms of solenoidal spaces. When a solenoid has leaves which have dimension two or greater, it turns out that this problem leads to very interesting questions about the actions of groups on Cantor sets. We discuss the techniques for the study of such equicontinuous minimal actions on a Cantor set. We introduce a new invariant, which allows us to determine when a solenoid is homogeneous or not, and quantify the degree of non-homogeneity of a solenoid. These results are based on a series of joint works with Alex Clark, Jessica Dyer and Olga Lukina.