Horseshoes and Lyapunov exponents for Banach cocycles over nonuniformly hyperbolic systems

Let $f$ be a $C^r$$(r>1)$ diffeomorphism of a compact Riemannian manifold $M$, preserving an ergodic hyperbolic measure $\mu$ with positive entropy, and let $\mathcal{A}$ be a H\"older continuous cocycle of injective bounded linear operators acting on a Banach space $X$. We prove that there is a sequence of horseshoes for $f$ and dominated splittings for $\mathcal{A}$ on the horseshoes, such that not only the measure theoretic entropy of $f$ but also the Lyapunov exponents of $\mathcal{A}$ with respect to $\mu$ can be approximated by the topological entropy of $f$ and the Lyapunov exponents of $\mathcal{A}$ on the horseshoes, respectively.