On the topological essential range and regularity of cocycles over compact and generic systems

TL;DR

This paper explores the properties of topological essential range and regularity for continuous cocycles over minimal systems, linking them with generic definitions and analyzing recurrent cocycles over minimal rotations.

Contribution

It introduces alternative generic definitions of cocycle regularity using Mackey actions and provides a detailed description of recurrent cocycles over minimal rotations.

Findings

Relations between topological and generic notions of regularity

Descriptions of recurrent cocycles over minimal rotations

Consequences for cocycles with discrete group values

Abstract

We consider the notions of topological essential range and regularity for continuous cocycles over minimal $\Z$-systems introduced in \cite{GH} and discuss relations with their generic counterparts. The alternative generic definitions can be given by using the notion of generic Mackey action associated with a cocycle. We further present a description of recurrent cocycles over minimal rotations with values in discrete groups and derive several consequences.