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, μ) is a non-uniformly hyperbolic system and A: M→ GL(d, R) is an α-Hölder continuous map satisfying A(fn-1(p)) … A(p) = Id for every p∈ Fix (fn) and n∈ N, there exists a measurable map P: M→ GL(d, R) satisfying A(x) = P(f(x)) P(x) - 1 for μ-almost every x∈ M. Moreover, we prove that whenever the measure μ has local product structure the transfer map P is α-Hölder continuous in sets with arbitrary large measure.