Uniform enveloping semigroupoids for groupoid actions

TL;DR

This paper introduces a novel approach to analyze topological dynamical systems using uniform enveloping semigroupoids of groupoid actions, extending classical results beyond minimal systems.

Contribution

It develops a new framework for describing extensions of dynamical systems as groupoid actions and constructs a uniform enveloping semigroupoid to handle nonminimal orbit structures.

Findings

Groupoid actions can be characterized by their uniform enveloping semigroupoids.

A groupoid action is (pseudo)isometric iff its uniform enveloping semigroupoid is a compact groupoid.

Provides an operator theoretic characterization via a Peter-Weyl-type theorem for representations of compact, transitive groupoids.

Abstract

We establish new characterizations for (pseudo)isometric extensions of topological dynamical systems. For such extensions, we also extend results about relatively invariant measures and Fourier analysis that were previously only known in the minimal case to a significantly larger class, including all transitive systems. To bypass the reliance on minimality of the classical approaches to isometric extensions via the Ellis semigroup, we show that extensions of topological dynamical systems can be described as groupoid actions and then adapt the concept of enveloping semigroups to construct a uniform enveloping semigroupoid for groupoid actions. This approach allows to deal with the more complex orbit structures of nonminimal systems. We study uniform enveloping semigroupoids of general groupoid actions and translate the results back to the special case of extensions of dynamical systems. In particular, we show that, under appropriate assumptions, a groupoid action is (pseudo)isometric if and only if the uniform enveloping semigroupoid is actually a compact groupoid. We also provide an operator theoretic characterization based on an abstract Peter-Weyl-type theorem for representations of compact, transitive groupoids on Banach bundles which is of independent interest.