A non-mixing Arnold flow on a surface

TL;DR

This paper constructs a smooth, area-preserving flow on a genus 2 surface with a unique ergodic component that is not mixing, highlighting complex dynamical behavior in low-dimensional systems.

Contribution

It introduces a novel example of a non-mixing, uniquely ergodic flow on a genus 2 surface with specific boundary properties.

Findings

Flow is uniquely ergodic but not mixing.

Flow is bounded by separatrices of saddles.

Constructed flow is smooth and area-preserving.

Abstract

We construct a smooth area preserving flow on a genus 2 surface with exactly one open uniquely ergodic component, that is asymmetrically bounded by separatrices of non-degenerate saddles and that is nevertheless not mixing.