Liv\v{s}ic Theorems for Non-Commutative Groups including Diffeomorphism Groups and Results on the Existence of Conformal Structures for Anosov Systems

TL;DR

This paper extends Livsic theorems to non-commutative groups, including diffeomorphism groups, and explores their implications for the existence of conformal structures in Anosov systems.

Contribution

It provides a new proof of Livsic theorems for non-commutative groups and applies these methods to conformal structures in Anosov systems.

Findings

Extended Livsic theorems to diffeomorphism groups

Developed localization methods for cocycle analysis

Established new results on conformal structures for Anosov systems

Abstract

The celebrated Livsic theorem states that given M a manifold, a Lie group G, a transitive Anosov diffeomorphism f on M and a Holder function \eta: M \mapsto G whose range is sufficiently close to the identity, it is sufficient for the existence of \phi :M \mapsto G satisfying \eta(x) = \phi(f(x)) \phi(x)^{-1} that a condition -- obviously necessary -- on the cocycle generated by \eta restricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of diffeomorphisms of a compact manifold. This has applications to rigidity theory. The localization procedure we develop can be applied to obtain some new results on the existence of conformal structures for Anosov systems.