Georgia Institute of Technology College of Sciences

The classical Livshits theorem characterizes coboundaries over a transitive Anosov flow as precisely those functions which integrate to zero over all periodic orbits of the flow. I will present a variant of this theorem which uses a weaker assumption and gives a weaker conclusion that the function is an ``abelian coboundary.” Such weaker version corresponds to studying the cohomological equation on infinite abelian covers e.g., for geodesic flows on abelian covers of hyperbolic surfaces. I will also discuss a connection to the marked length spectrum rigidity of Riemannian metrics. Joint work with Federico Rodriguez Hertz.