Parabolic flows are smooth flows for which nearby points diverge slowly (namely, at polynomial speed) and include, for example, smooth area-preserving flows on surfaces, horocycle flows and nilflows. Apart from a few exceptions, not much is known for general smooth parabolic flows and there is no classification of the characteristic phenomena for this kind of behavior.
We will survey some recent progress in the study of the chaotic features of non-homogeneous parabolic flows. We will explain how a common geometric shearing mechanism plays a key role in understanding their mixing properties.
Some of the results discussed are joint work with Artur Avila, Giovanni Forni and Corinna Ulcigrai, and with Adam Kanigowski.