Continuous bounded cocycles

TL;DR

This paper proves that in a minimal locally compact groupoid with a continuous Hilbert bundle, every bounded continuous cocycle is a continuous coboundary, extending a classical theorem to the groupoid setting.

Contribution

It generalizes the classical Gottschalk-Hedlund theorem to the context of minimal locally compact groupoids with continuous Hilbert bundles.

Findings

Bounded continuous cocycles are always continuous coboundaries in this setting.

The result extends classical dynamical systems theorems to groupoid frameworks.

Provides a new perspective on the structure of cocycles in groupoid theory.

Abstract

Let $G$ be a minimal locally compact groupoid with compact metrizable unit space and let $E$ be a continuous $G$-Hilbert bundle. We show that a bounded continuous cocycle $c: G\ra r^*E$ is necessarily a continuous coboundary. This is a groupoid version of a classical theorem of Gottschalk and Hedlund.