A theory of essential values of cocycles over minimal rotations with values in locally compact Abelian groups, especially , is developed. Criteria for such a cocycle to be conservative are given. The group of essential values of a cocycle is described.
The main results of this paper are: 1. No topologically transitive cocycle -extension of minimal rotation on the unit circle by a continuous real-valued bounded variation ℤ-cocycle admits minimal subsets. 2. A minimal rotation on a compact metric monothetic group does not admit a topologically transitive real-valued cocycle if and only if the group is finite.
We show that semisimple actions of l.c.s.c. Abelian groups and cocycles with values in such groups can be used to build new examples of semisimple automorphisms (ℤ-actions) which are relatively weakly mixing extensions of irrational rotations.
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