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

Horseshoes and Lyapunov exponents for Banach cocycles over nonuniformly hyperbolic systems