Dynamics and 'arithmetics' of higher genus surface flows

TL;DR

This paper reviews recent progress in understanding the dynamics, ergodic properties, and spectral behavior of higher genus surface flows, emphasizing Diophantine conditions and renormalization techniques.

Contribution

It highlights new developments in linearization and rigidity for higher genus flows, extending arithmetic conditions beyond genus one cases.

Findings

Advances in ergodic and spectral analysis of surface flows

Identification of Diophantine-like conditions for higher genus flows

Use of renormalization dynamics to impose conditions

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 Poincare' 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 non-uniform hyperbolicity of the renormalization.