  We obtain the first results on convergence rates in the Prokhorov metric for
the weak invariance principle (functional central limit theorem) for
deterministic dynamical systems. Our results hold for uniformly
expanding/hyperbolic (Axiom A) systems, as well as nonuniformly
expanding/hyperbolic systems such as dispersing billiards, Henon-like
attractors, Viana maps and intermittent maps. As an application, we obtain
convergence rates for deterministic homogenization in multiscale systems.
