Lyapunov exponents of cocycles over non-uniformly hyperbolic systems

We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism f of a compact manifold X preserving a hyperbolic ergodic probability measure µ. The cocycle A over f is Hölder continuous and takes values in GL(d, ℝ) or, more generally, in the group of invertible bounded linear operators on a Banach space. For a GL(d, ℝ)-valued cocycle A we prove that the Lyapunov exponents of A with respect to µ can be approximated by the Lyapunov exponents of A with respect to measures on hyperbolic periodic orbits of f. In the infinite-dimensional setting one can define the upper and lower Lyapunov exponents of A with respect to µ, but they cannot always be approximated by the exponents of A on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of A on hyperbolic periodic orbits of f.