Georgia Institute of Technology College of Sciences

We consider a hyperbolic dynamical system (X,f) and a Holder continuous cocycle A over (X,f) with values in GL(d,R), or more generally in the group of invertible bounded linear operators on a Banach space. We discuss approximation of the Lyapunov exponents of A in terms of its periodic data, i.e. its return values along the periodic orbits of f. For a GL(d,R)-valued cocycle A, its Lyapunov exponents with respect to any ergodic f-invariant measure can be approximated by its Lyapunov exponents at periodic orbits of f. In the infinite-dimensional case, the upper and lower Lyapunov exponents of A can be approximated in terms of the norms of the return values of A at periodic points of f. Similar results are obtained in the non-uniformly hyperbolic setting, i.e. for hyperbolic invariant measures. This is joint work with B. Kalinin.