  In this paper we study topological cocycles for minimal homeomorphisms on a
compact metric space. We introduce a notion of an essential range for
topological cocycles with values in a locally compact group, and we show that
this notion coincides with the well known topological essential range if the
group is abelian. We define then a regularity condition for cocycles and prove
several results on the essential ranges and the orbit closures of the skew
product of regular cocycles. Furthermore we show that recurrent cocycles for a
minimal rotation on a locally connected compact group are always regular,
supposed that their ranges are in a nilpotent group, and then their essential
ranges are almost connected.

Dynamical Systems