Non-stationary smooth geometric structures for contracting measurable cocycles

We implement a differential-geometric approach to normal forms for contracting measurable cocycles to Diff(q) (R-n, 0), q >= 2. We obtain resonance polynomial normal forms for the contracting cocycle and its centralizer, via C(q )changes of coordinates. These are interpreted as non-stationary invariant differential-geometric structures. We also consider the case of contracted foliations in a manifold, and obtain C-q homogeneous structures on leaves for an action of the group of subresonance polynomial diffeomorphisms together with translations.