Davide Ravotti

Maharam-Pollicott-Ruelle resonances and self-similar translation flows on Abelian covers

Translation flows provide a natural class of zero-entropy dynamical systems on flat surfaces obtained by gluing polygons edge-to-edge. In certain special cases, called self-similar, the flow repeats its behavior in a precise way when the surface is stretched and folded by a symmetry known as a pseudo-Anosov map. When this happens, we can understand the flow by studying this symmetry. A powerful method for doing this, pioneered in a work by Giulietti and Liverani, uses a tool called transfer operator. By analyzing how the transfer operator acts on a carefully chosen space of distributions, it is possible to deduce fine dynamical and statistical properties of self-similar flows on compact surfaces. This strategy was successfully carried out by Faure, Gouëzel, and Lanneau.
In this talk, I will describe how we extend this approach to self-similar translation flows on certain non-compact surfaces, namely Abelian covers. By introducing a family of weighted transfer operators, we deduce several results on the dynamics of the translation flows with respect to its ergodic measures (the so-called Maharam measures), including an asymptotic description of their time-averages. More broadly, we investigate the interplay between the spectral properties of these operators, the corresponding “Maharam” distributions, and geometric quantities naturally associated with the cover. The talk is based on a work-in-progress, joint with Mauro Artigiani, Roberto Castorrini, and Yuriy Tumarkin.