Abstract: 

Livsic's famous theorem for hyperbolic systems states that a Holder continuous function which sums/integrates to zero around all closed orbits is necessarily a couboudary. Recently F. Rodriguez-Hertz and A. Gogolev proved an interesting Abelian Livsic Theorem whereby they assume the hypothesis on fewer closed orbits and still obtain a (slightly modified) conclusion. In this talk I will present a slightly different viewpoint, some related results and generalizations, and discuss various associated questions. This is joint work with Richard Sharp.