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 Holder continuous and takes values in GL(d,R)
Nearby in the stack
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