Fast approximation of Lyapunov exponents: Beyond the locally constant case

We study the problem of estimating the maximal Lyapunov exponent of dominated cocycles. In particular we are concerned with cocycles over Gibbs states on shifts of finite type for which both the function defining the cocycle and the potential defining the Gibbs state may depend on infinitely many coordinates but are still very regular. We show that when the $n$th variation of both the cocycle and the potential is $O(e^{-cn^{2}})$ for some $c>h_{\text{top}}$ then using periodic points of period less then $n$ the Lyapunov exponent can be approximated to an accuracy $O(n^{-kn})$ for some explicit $k>0$.