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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 \mapstoG whose range is sufficiently close to the identity, it is sufficient for theexistence of \phi :M \mapsto G satisfying \eta(x) = \phi(f(x)) \phi(x)^{-1}that a condition -- obviously necessary -- on the cocycle generated by \etarestricted to periodic orbits is satisfied. In this paper we present a newproof of the main result. These methods allow us to treat cocycles takingvalues in the group of diffeomorphisms of a compact manifold. This hasapplications to rigidity theory. The localization procedure we develop can beapplied to obtain some new results on the existence of conformal structures forAnosov systems.

Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems