Weak mixing for interval exchange transformations and translation flows

TL;DR

This paper demonstrates that most interval exchange transformations are either weakly mixing or irrational rotations, and that typical translation flows on higher genus surfaces are weakly mixing, advancing understanding of their dynamical properties.

Contribution

It establishes the typical weak mixing behavior for interval exchange transformations and translation flows on surfaces of genus at least two.

Findings

Most interval exchange transformations are either weakly mixing or irrational rotations.

Typical translation flows on surfaces of genus ≥ 2 are weakly mixing.

Provides a dichotomy in the dynamical behavior of these systems.

Abstract

We show that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation flow on a surface of genus $g \geq 2$ (with prescribed singularity types) is weakly mixing.