Livsic Theorem for Cocycles with Value in Gl (N,qp)

We extend the well-known results of Livsic theorem on the regularity of measurable solutions to GL(n, Qp) valued cocycles, where Qp is the p-adic field, which is a non-archimedean field. To prove the main result, we give the approximation of Lyapunov exponents by the Lyapunov exponents at periodic points and prove the uniformly boundedness of the cocycle if it is uniformly bounded at periodic points under the non-archimedean condition.