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Nilpotent extensions of minimal homeomorphisms

Published online by Cambridge University Press:  22 September 2005

GERNOT GRESCHONIG  and
ULRICH HABÖCK
GERNOT GRESCHONIG
Affiliation:
Faculty of Mathematics, University of Vienna, Nordbergstraße 15, A-1090 Vienna, Austria (e-mail: gernot.greschonig@univie.ac.at, ulrich.haboeck@univie.ac.at)
ULRICH HABÖCK
Affiliation:
Faculty of Mathematics, University of Vienna, Nordbergstraße 15, A-1090 Vienna, Austria (e-mail: gernot.greschonig@univie.ac.at, ulrich.haboeck@univie.ac.at)
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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 then define 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, assuming that their ranges are in a nilpotent group, and then their essential ranges are almost connected.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 25 , Issue 6 , December 2005 , pp. 1829 - 1845
Copyright
2005 Cambridge University Press

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