Livsic theorem for low-dimensional diffeomorphism cocycles

TL;DR

This paper extends Livsic's theorem to cocycles valued in groups of low-dimensional diffeomorphisms, providing conditions for coboundaries without localization assumptions and in low regularity settings.

Contribution

It introduces a Livsic type theorem for low-dimensional diffeomorphism cocycles, broadening applicability to low regularity and removing localization constraints.

Findings

Proves Livsic theorem for low-dimensional diffeomorphism cocycles

Provides necessary and sufficient conditions for coboundaries in any dimension

Achieves results without localization assumptions and in low regularity

Abstract

We prove a Livsic type theorem for cocycles taking values in groups of diffeomorphisms of low-dimensional manifolds. The results hold without any localization assumption and in very low regularity. We also obtain a general result (in any dimension) which gives necessary and sufficient conditions to be a coboundary.