A Livšic theorem for matrix cocycles over non-uniformly hyperbolic systems

Abstract

We prove a Livšic-type theorem for Hölder 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 $\alpha$-Hölder continuous map satisfying $A(f^{n-1}(p))\dots A(p)=\text{Id}$ 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 $\alpha$-Hölder continuous in sets with arbitrary large measure.