Let T:X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f:X → R^d is a Hlder continuous function with ∫_X^(fdm) = 0.If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ~2:=σ~2 (f ) such that S^fn √ n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ~2 . Moreover, there exists a real number A > 0 such that, for any integer n ≥ 1,Π( m*( 1√ nS f n ),N (0,σ~2 ) ≤A√n, where m*(1√ n S^fn)denotes the distribution of 1√ n S^fn with respect to m, and Π is the Prokhorov metric.
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