Abelian covers of some translation flows

Translation flows on flat surfaces are classical examples of slowly chaotic, parabolic flows. Substantial progress has been made in recent years in understanding the ergodic properties of translation flows on compact surfaces, and renormalization has proved to be a fundamental tool in this area.
In this talk, we will be interested in the case of non-compact surfaces and in particular we focus on infinite abelian covers of compact ones. I will outline a method to study the ergodic integrals of self-similar translation flows on this type of surfaces, namely translation flows which are renormalized by a pseudo-Anosov diffeomorphism.
This is based on a joint work with Henk Bruin, Charles Fougeron, and Dalia Terhesiu.