Liv?ic Theorem For Matrix Cocycles Over Nonuniformly Hyperbolic Systems

In this paper, we mainly prove the liv?ic theorem for GL(m,) cocycles over a nonuniformly hyperbolic system.Let f be a C1+γdiffeomorphism of a compact manifold M. Let A : M GL(m,)be a α-H?lder continuous function.Firstly, we prove that the Lyapunov exponents of with respect to an ergodic hyperbolic measure can be approximated by the Lyapunov exponets of at periodic points. Compared with Kalinin [6], this result needn’t the system has the closing property in the whole space.Then we prove the Liv?ic theorem for a nonuniformly hyperbolic system: Suppose that μ is an f-invarint hyperbolic measure, and A satisfies A(fn 1x) A(f x)A(x) =Id, fnp = p, n +. then there exists a measurable Borel function C : M GL(m,), such that for μ-almost every x M, A(x) = C(f x) C(x) 1. This generalizes the result of Katok.