Gernot Greschonig, Ulrich Haböck
Nilpotent Extensions of Minimal Homeomorphisms
Preprint series: ESI preprints

MSC:

34C35 Dynamical systems, See also {54H20, 58Fxx, 70-XX}

54H15 Transformation groups and semigroups, See also {20M20, 22-XX, 57Sxx}

54H20 Topological dynamics, See also {28Dxx, 34C35, 58Fxx}

58F99 None of the above but in this section

Abstract: 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.