We prove a Liv\v{ s}ic-type theorem for H\"older 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 $p\in \text{ Fix}(f^n)$ and $n\in \mathbb{ N}$, there exists a measurable map $P:M\to GL(d,\mathbb{ R})$ satisfying $A(x)=P(f(x))P(x)^{ -1}$ for $\mu$-almost every $x\in M$. 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.
Categories: math.DS