Non-stationary smooth geometric structures for contracting measurable cocycles

We implement a differential-geometric approach to normal forms for contracting measurable cocycles to $\operatorname{Diff}^{q}(\mathbb{R}^{n},\mathbf{0})$, $q\geq 2$. 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.