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