Solutions of cohomological equations have strong implications in their relations with thermodynamic formalism, among other areas. In the seventies, Livsic proved that whenever $f: M \to M$ is an Anosov diffeomorphism, $G$ is an abelian group and $U: M \to G$ is a cocycle, the solutions
of the equations of the form $U \circ f - U = 0$ (cohomological equations) are completely determined by the values of $U$ on the periodic points of $f$. In the last decades there have been many successful attempts to understand the possible solutions of Livsic theorem for non-abelian groups. In this talk I will discuss this problem in the context of linear cocycles and relate it with the problem of symmetries for the dynamics. |