After the fundamental work of Livschitz in [1, 2], various research directions emerged, among which the following stand out: (i) the study of cocycles with values in groups and semigroups beyond R, as well as the investigation of corresponding regularity results; (ii) the analysis of how a certain degree of regularity (Ck for k = 1, 2, . . . , ∞, ?) of the cocycle can confer corresponding regularity to the solution of the cohomological equation; and (iii) the study of higher-dimensional cohomology naturally associated with the action of groups such as Zk or Rk. The aim of this article is to present, as self-contained as possible, a review of the natural generalizations of the notions of cocycles and cochains, as well as their corresponding results, in the study of cohomological equations.

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Cocycle,Cohomological equation,Coboundary,Cochain,Livschitz theorem,Lie group,Anosov flow,Anosov diffeomorphism

• Mathematics - Natural Sciences