Livšic Theorem for Banach Rings

Abstract

We prove the Livšic Theorem for Hölder continuous cocycles with values in Banach rings. We consider a transitive homeomorphism $\sigma :X\to X$ that satisfies the Anosov Closing Lemma, and a Hölder continuous map $a:X\to B^{\times }$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. We show that it is a coboundary with a Hölder continuous transition function if and only if $a({\sigma }^{n-1}p)\dots a(\sigma p)a(p)=e$ for each periodic point $p={\sigma }^{n}p$.