Slow chaos in surface flows

Abstract

AbstractFlows on surfaces describe many systems of physical origin and are one of the most fundamental examples of dynamical systems, studied since Poincará. In the last decade, there have been a lot of advances in our understanding of the chaotic properties of smooth area-preserving flows (a class which include locally Hamiltonian flows), thanks to the connection to Teichmueller dynamics and, very recently, to the influence of the work of Marina Ratner in homogeneous dynamics. We motivate and survey some of the recent breakthroughs on their mixing and spectral properties and the mechanisms, such as shearing, on which they are based, which exploit analytic, arithmetic and geometric techniques.