LIVSIC THEOREM FOR BANACH RINGS

The Livsic Theorem for Holder continuous cocycles with values in Banach rings is proved. We consider a transitive homeomorphism sigma : X -> X that satisfies the Anosov Closing Lemma and a Holder continuous map a : X -> B-x from a compact metric space X to the set of invertible elements of some Banach ring B. The map a (x) is a coboundary with a Holder continuous transition function if and only if a (sigma(n - 1) p) : : : a (sigma p) a (p) is the identity for each periodic point p = sigma(n) p