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We consider the notions of topological essential range and regularity forcontinuous cocycles over minimal $\Z$-systems introduced in \cite{GH} anddiscuss relations with their generic counterparts. The alternative genericdefinitions can be given by using the notion of generic Mackey actionassociated with a cocycle. We further present a description of recurrentcocycles over minimal rotations with values in discrete groups and deriveseveral consequences.

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