A picture of the ODE's flow in the torus: From everywhere or almost-everywhere asymptotics to homogenization of transport equations

Abstract

In this paper, we study various aspects of the ODE's flow induced by a periodic vector field b in the torus. We present a picture of all logical connections between: the everywhere or the a.e. asymptotics of the flow, the rectification of b, the ergodicity of the flow, the unit set condition for Herman's rotation set, the unit set condition for the set Db composed of the means of b related to the invariant measures being absolutely continuous with respect to Lebesgue's measure, the homogenization of the linear transport equation with oscillating velocity b(x/ε). The main result of the paper is that the a.e. asymptotics of the flow, the unit set condition for Db and the homogenization of the transport equation with divergence free b, are equivalent. Extending the two-dimensional results on Stepanoff flow to any dimension, we show that the flow may be ergodic without satisfying the everywhere asymptotics.