A Livv{s}ic theorem for matrix cocycles over non-uniformly hyperbolic systems

We prove a Livv{s}ic-type theorem for Holder continuous and matrix-valued cocycles over non-uniformly hyperbolic systems. More precisely, we prove that whenever $(f,mu)$ is a non-uniformly hyperbolic system and $A:M to GL(d,mathbb{R}) $ is an $alpha$-H{o}lder continuous map satisfying $ A(f^{n-1}(p))ldots A(p)=text{Id}$ for every $pin text{Fix}(f^n)$ and $nin mathbb{N}$, there exists a measurable map $P:Mto GL(d,mathbb{R})$ satisfying $A(x)=P(f(x))P(x)^{-1}$ for $mu$-almost every $xin M$. Moreover, we prove that whenever the measure $mu$ has local product structure the transfer map $P$ is $alpha$-H{o}lder continuous in sets with arbitrary large measure.