An Introduction to the Dynamics of Compact Group Automorphisms after Klaus Schmidt
Levin Ceglie
Abstract
This thesis provides an introduction to the theory of algebraic dynamical systems based on the work of Klaus Schmidt. We begin by establishing fundamental dynamical properties through the lens of actions of countable groups on compact metrizable groups by continuous automorphisms, which we call quasi-algebraic actions. Specifically, we introduce the notions of topological transitivity, ergodicity, mixing, and expansiveness, and illustrate them through four running examples. With these foundations in place, we explore the rigidity of quasi-algebraic actions, structural constraints imposed by dynamical conditions, and provide a spectral characterization of ergodicity and mixing. The culmination of this work is an “algebraic dictionary” that establishes a correspondence between algebraic -actions and countable modules over a Laurent polynomial ring, translating dynamical properties into algebraic ones and providing a unified framework for their study.