A flow is called parabolic if nearby points diverge at a subexponential (often polynomial) speed. Classical examples of parabolic flows are horocycle flows on negatively curved surfaces and area-preserving flows on surfaces. Substantial progress has been made in recent years in understanding the ergodic properties of many parabolic flows on compact or finite volume spaces, and renormalization has proved to be a fundamental tool in this area.
In this talk, I will outline a method to study the ergodic integrals of some parabolic flows on infinite abelian covers of compact spaces, focussing in particular on horocycle flows on Z^d covers of negatively curved surfaces and on some translation flows on infinite periodic translation surfaces.
This is based on current works in progress with Roberto Castorrini and with Henk Bruin, Charles Fougeron, and Dalia Terhesiu.
