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

We consider the notions of topological essential range and regularity for continuous cocycles over minimal $\mathbb{Z}$-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.