LAUSR

We prove a Livšic-type theorem for Hölder continuous and matrix-valued cocycles over non-uniformly hyperbolic systems. More precisely, we prove that whenever $$(f,\mu )$$(f,μ) is a non-uniformly hyperbolic system and $$A:M \rightarrow GL(d,\mathbb {R}) $$A:M→GL(d,R) is an $$\alpha $$α-Hölder continuous map satisfying $$ A(f^{n-1}(p))\ldots A(p)=\text {Id}$$A(fn-1(p))…A(p)=Id for every $$p\in \text {Fix}(f^n)$$p∈Fix(fn) and $$n\in \mathbb {N}$$n∈N, there exists a measurable map $$P:M\rightarrow GL(d,\mathbb {R})$$P:M→GL(d,R) satisfying $$A(x)=P(f(x))P(x)^{-1}$$A(x)=P(f(x))P(x)-1 for $$\mu $$μ-almost every $$x\in M$$x∈M. Moreover, we prove that whenever the measure $$\mu $$μ has local product structure the transfer map P is $$\alpha $$α-Hölder continuous in sets with arbitrary large measure.