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We implement a differential-geometric approach to normal forms forcontracting measurable cocycles to $\mbox{Diff}^q({\bf R}^n, {\bf 0})$, $q \geq2$. We obtain resonance polynomial normal forms for the contracting cocycle andits centralizer, via $C^q$ changes of coordinates. These are interpreted asnonstationary invariant differential-geometric structures. We also consider thecase of contracted foliations in a manifold, and obtain $C^q$ homogeneousstructures on leaves for an action of the group of subresonance polynomialdiffeomorphisms together with translations.

Nonstationary smooth geometric structures for contracting measurable cocycles