Multidimensional local central limit theorem of some non-uniformly hyperbolic systems

We consider Young’s nonuniformly hyperbolic system (X, T, ν) where ν is the SRB measure corresponding to the system (X, T), and show that if the components of a Hölder observable f: X → ? ^d are cohomologously independent, then f satisfies the multidimensional central limit theorem. Moreover if f is aperiodic, then f satisfies the local multidimensional central limit theorem.