Weak equivalence of cocycles and Mackey action in generic dynamics

Abstract

LetR be an equivalence relation generated by a countable ergodic homeomorphism group of a perfect Polish spaceX. We consider cocycles taking values in Polish groups onR modulo meager subsets ofX. Two cocycles are called weakly equivalent if they are cohomologous up to an automorphism ofR. The notion of generic associated Mackey action is introduced, which is an invariant of weak equivalence for cocycles. Regular cocycles with values in an arbitrary Polish group and transient cocycles with values in an arbitrary countable group are completely classified up to weak equivalence.