Dynamics and “arithmetics” of higher genus surface flows

Abstract

We survey some recent advances in the study of (area-preserving) flows on surfaces, in particular on the typical dynamical, ergodic, and spectral properties of smooth area-preserving (or locally Hamiltonian) flows, as well as recent breakthroughs on linearization and rigidity questions in higher genus. We focus in particular on the Diophantine-like conditions which are required to prove such results, which can be thought of as a generalization of arithmetic conditions for flows on tori and circle diffeomorphisms. We will explain how these conditions on higher genus flows and their Poincaré sections (namely generalized interval exchange maps) can be imposed by controlling a renormalization dynamics, but are of more subtle nature than in genus one since they often exploit features which originate from the nonuniform hyperbolicity of the renormalization.