Georgia Institute of Technology College of Sciences

The classical Livsic theorem says that a H\"older cocycle over a transitive Anosov diffeomorphism/flow is a coboundary if and only if it satisfies the periodic obstruction on all periodic orbits. It is natural to ask whether satisfying the periodic obstruction for all closed orbits of period at most $T$ is enough to conclude that the cocycle is, in some quantitative sense, close to being a coboundary. We show that for transitive Anosov flows, this is indeed enough to find an approximate solution to the cohomological equation with error decaying exponentially in $T$, improving on the polynomial rates obtained first by S. Katok for contact flows in dimension 3, and then later Gouëzel and Lefeuvre in higher dimensions. This is joint work with Jonathan DeWitt, Spencer Durham, and James Marshall Reber.