Abstract. We study causes of stretching of Birkhoff sums and study their action in the mixing of various surface flows. In so doing, we succeed in amplifying the result of Khanin and Sinai about mixing in the Arnold's example of flow with nonsingular fixed points on a two-dimensional torus.

Introduction

There are three known kinds of mixing flows on two-dimensional surfaces: continuous flows without fixed points on a torus, smooth flows with singular fixed points, and smooth flows with nonsingular fixed points (Arnold's example). However, in the last case mixing arises not on the whole torus but only on an ergodic component.

We suggest a special flow St, constructed over a circle rotation or an interval exchange transformation (which we denote by T) and under some positive “roof” function, as an ergodic relative of a Borel measure-preserving flow on a two-dimensional surface. In such special flows the only possible cause of mixing is the difference in the times that various points take to get from the “floor” to the “roof”. This can cause, as time passes, a small rectangle to be strongly stretched and almost uniformly distributed along trajectories and hence over the phase space.