Dynamics of Equicontinuous Group Actions on Cantor Sets

In this thesis, we consider the class of minimal equicontinuous Cantor dynamical systems. We show that every such system can be represented by a group chain inverse limit system, and conversely that every group chain yeilds a minimal equicontinuous Cantor dynamical system. This gives us a concrete representation of minimal equicontinuous Cantor dynamical systems, which makes them easier to work with. We use this representation to classify such systems as regular, weakly regular, or irregular, extending work by Fokkink and Oversteegen. We show that such systems can be equivalently classified as regular, weakly regular, or irregular according to the number of orbits of the action by the Autormorphism group, or equivalently according to the number of equivalence classes of group chains associated to the system. We give examples of group chains of each level of regularity. We introduce a new invariant of such systems, called the discriminant group, and show that its cardinality is related to the classification of the system as regular, weakly regular, or irregular.