This article presents a detailed treatment of Livschitz theorem for hyperbolic diffeomorphisms. Based on the periodic data, this theorem provides a necessary and sufficient condition so that cohomological equations have sufficiently regular solutions. It is one of the main tools to obtain global data of cohomological nature from the periodic data. Since it is crucial in the statement of Livschitz theorem, it is also presented the Anosov closing lemma.

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Cocycles,Cohomological equations,Periodic data,Livschitz theorem,Anosov closing lemma