A smooth mixing flow on a surface with non-degenerate fixed points

TL;DR

This paper constructs a smooth, area-preserving flow on a genus 5 surface that is mixing and has finitely many non-degenerate fixed points, solving a long-standing open problem in dynamical systems.

Contribution

It provides the first example of such a flow with these properties on a closed surface, addressing a four-decade-old open problem.

Findings

Flow is smooth, area-preserving, and mixing.

Flow has finitely many non-degenerate fixed points.

No saddle connections on the surface.

Abstract

We construct a smooth, area preserving, mixing flow with finitely many non-degenerate fixed points and no saddle connections on a closed surface of genus 5. This resolves a problem that has been open for four decades.