  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.
