Periodic approximation of Lyapunov exponents for Banach cocycles

We consider group-valued cocycles over dynamical systems. The base system is a homeomorphism of a metric space satisfying a closing property, for example a hyperbolic dynamical system or a subshift of finite type. The cocycle A takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. We prove that upper and lower Lyapunov exponents of A with respect to an ergodic invariant measure can be approximated in terms of the norms of the values of A on periodic orbits of f. We also show that these exponents cannot always be approximated by the exponents of A with respect to measures on periodic orbits. Our arguments include a result of independent interest on construction and properties of a Lyapunov norm for the infinite-dimensional setting. As a corollary, we obtain estimates of the growth of the norm and of the quasiconformal distortion of the cocycle in terms of the growth at the periodic points of f.