Suppose that X is a compact metric space and T is a minimal homeomorphism (i.e. a homeomorphism so that
every T orbit of an element of X is dense in X). A topological cocycle is a dynamical system which is defined by
summing up the iterations of a continuous function on X by the homeomorphism T. This continuous function takes
its values in an abelian locally compact group, in the simplest case it is a real valued function. Furthermore, we
suppose that the homeomorphism T is distal, i.e. two distinct points in X cannot get arbitrarily close to each other
under a common iteration by T.
The aim of the frist part of the reseach project is to obtain a general classification of these systems based on results
of the proposer for a class of distal minimal homeomorphisms on multi dimensional tori. A tool to obtain this
classification will be a structure theory of distal minimal homeomorphisms achieved by Furstenberg. This structure
theory generalises the properties of the class of distal minimal homeomorphisms on multi dimensional tori
mentioned above, and it shows that every distal minimal homeomorphism can be represented by a sequence of
quasi isometric extensions. This representation should be used to prove the regularity of such cocycles. Regularity
implies in the simplest case of a real valued cocycle that it is either a topological coboundary (that is an almost
trivial system) or it is topologically ergodic (then on a topologically large set in X the set of values of the cocycle
for all time iterations is dense in the real numbers).
The second part of the project deals with topological cocycles with values in nonabelian nilpotent groups. Again
regularity properties are the topic of research, but it should be mentioned that topological cocycles with values in
nonabelian groups are a much difficult problem. All the existing results require that the cocycle is defined over a
compact locally connected group with a minimal rotation, and that they take their values in a nilpotent group. As
the connectedness of the space X seems to be inevitable in this case, the topic will be cocycles defined over class
of distal minimal homeomorphisms on multi dimensional tori mentioned above.