Classification and Structure of Cocycles of Amenable Ergodic Equivalence Relations

Let T be an ergodic Lebesgue space transformation of an arbitrary type, α, β the nontransient cocycles of the associated ergodic equivalence relation with values in a locally compact separable group G, r the Radon-Nikodym cocycle. Suppose that the Mackey actions of G × R associated with cocycles α0 = α × r and β0 = β × r are isomorphic. It is proved that in this case α ∘ θ is cohomologous to β for some θ ∈ N[T]. The structure of an arbitrary cocycle is described.