Gottschalk-Hedlund theorem revisited (Q2043301)

Ergodic optimization studies invariant measures in topological dynamical systems that minimize the integral of a given continuous function across all invariant measures. Associated to this is the problem of solving a cohomological equation to find a coboundary and a transfer function. In a rich and complex history, many results show that certain types of well-behaved dynamical systems with properties related to hyperbolicity do have such solutions, but also that there are other families of systems, like rotations with specific Diophantine properties, that do not. Here the KAM approach is used to give new ways to approach the cohomological equation under quite weak hypotheses. An application is to give an improvement of the classical Gottschalk-Hedlund theorem giving conditions equivalent to the existence of a continuous coboundary. A ``discounted'' version of the KAM approach is also considered, giving rise to numerical approximation schemes which are not guaranteed to converge, with an identification of when the coboundaries do arise as a limit.