Horseshoes and Lyapunov exponents for Banach cocycles over non-uniformly hyperbolic systems

Abstract We extend Katok’s result on ‘the approximation of hyperbolic measures by horseshoes’ to Banach cocycles. More precisely, let f be a $C^r(r>1)$ diffeomorphism compact Riemannian manifold M , preserving an ergodic measure $\mu $ with positive entropy, and $\mathcal {A}$ Hölder continuous cocycle bounded linear operators acting space $\mathfrak {X}$ . prove that there is sequence horseshoes for dominated splittings the horseshoes, such not only theoretic entropy but also Lyapunov exponents respect can approximated topological respectively. As application, we show continuity sub-additive pressure