  We prove an almost sure invariance principle (approximation by d-dimensional
Brownian motion) for vector-valued Holder observables of large classes of
nonuniformly hyperbolic dynamical systems. These systems include Axiom~A
diffeomorphisms and flows as well as systems modelled by Young towers with
moderate tail decay rates. In particular, the position variable of the planar
periodic Lorentz gas with finite horizon approximates a 2-dimensional Brownian
motion.
