Contributions to Structure Theory in Topological Dynamics and Ergodic Theory

Abstract

This thesis contributes to the structure theory of dynamical systems across four different areas. Building on the foundational work by Ellis on enveloping semigroups, we introduce "enveloping semigroupoids" to capture relative properties of dynamical systems in algebraic terms. Via a systematic development of the representation theory of these groupoids, we provide a conceptually new perspective on existing structure-theoretic results for minimal dynamical systmes and expand these results to systems with more intricate orbit structures. Additionally, we connect the concepts of isometricity and distality in topological dynamics and ergodic theory by showing that isometric extensions in ergodic theory admit canonical (pseudo)isometric topological models, expanding on previous work by Lindenstrauss. Furthermore, we extend the Furstenberg correspondence principle to vector-valued sequences in Hilbert spaces, providing a new framework for understanding statistical properties of vector-valued sequences through operator theory. As a proof of concept, we show how this correspondence principle reduces several versions of the van der Corput inequality to the mean ergodic theorem. The thesis concludes by adapting the Furstenberg-Zimmer structure theorem to accommodate both stationary random walks as well as relatively measure-preserving extensions of nonsingular actions.