Nonstationary smooth geometric structures for contracting measurable cocycles

TL;DR

This paper develops a differential-geometric framework for normal forms of contracting measurable cocycles, leading to invariant structures and homogeneous foliations, advancing understanding of their geometric and dynamical properties.

Contribution

It introduces a novel approach to normal forms for contracting cocycles using differential geometry, including resonance polynomial normal forms and invariant structures.

Findings

Resonance polynomial normal forms for cocycles and their centralizers.

Invariant differential-geometric structures for nonstationary systems.

Homogeneous structures on leaves of contracted foliations.

Abstract

We implement a differential-geometric approach to normal forms for contracting measurable cocycles to $\mbox{Diff}^q({\bf R}^n, {\bf 0})$, $q \geq 2$. We obtain resonance polynomial normal forms for the contracting cocycle and its centralizer, via $C^q$ changes of coordinates. These are interpreted as nonstationary 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.