Shearing and mixing in parabolic flows

Abstract

Parabolic flows are dynamical systems in which nearby trajectories diverge with polynomial speed. A classical example is the horocycle flow on a surface of constant negative curvature. Other important classes of examples are smooth area-preserving flows on surfaces, whose study is connected with Teichmueller dynamics, and Heisenberg nilflows. We survey some of the chaotic properties of these flows and some recent results on time changes of the above mentioned classes of examples. We focus in particular on mixing and we explain the shearing mechanism which is responsible for mixing in parabolic dynamics.