Approximation by Brownian motion for Gibbs measures and flows under a function

Let [formula-omitted] denote a flow built under a Hölder-continuous function l over the base (Σ, μ) where Σ is a topological Markov chain and μ some (ψ-mining) Gibbs measure. For a certain class of functions f with finite 2 + δ-moments it is shown that there exists a Brownian motion B(t) with respect to μ and σ² > 0 such that μ-a.e. [formula-omitted] for some 0 < λ < 5δ/588. One can also approximate in the same way by a Brownian motion B*(t) with respect to the probability [formula-omitted]. From this, the central limit theorem, the weak invariance principle, the law of the iterated logarithm and related probabilistic results follow immediately. In particular, the result of Ratner ([6]) is extended.