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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics