We prove an almost sure invariance principle (approximation by d-dimensional Brownian motion) for vector-valued Hölder observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom A diffeomorphisms and flows as well as systems modeled by Young towers with moderate tail decay rates.
In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a two-dimensional Brownian motion.
"A vector-valued almost sure invariance principle for hyperbolic dynamical systems." Ann. Probab. 37 (2) 478 - 505, March 2009. https://doi.org/10.1214/08-AOP410