LetT:X→XbeanAxiomAdiffeomorphism,mtheGibbsstateforaHldercontinuousfunctionɡ.Assumethatf:X→R~disaHldercontinuousfunctionwith∫_X~(fdm)=0.Ifthecomponentsoffarecohomologouslyindependent,thenthereexistsapositivedefinitesymmetricmatrixσ~2:=σ~2(f)suchthatS~fn√nconvergesindistributionwithrespecttomtoaGaussianrandomvariablewithexpectation0andcovariancematrixσ~2.Moreover,thereexistsarealnumberA>0suchthat,foranyintegern≥1,Π(m*(1√nSfn),N(0,σ~2)≤A√n,wherem*(1√nS~fn)denotesthedistributionof1√nS~fnwithrespecttom,andΠistheProkhorovmetric.
