Weak mixing for area preserving flows on surfaces

TL;DR

This paper proves that certain smooth, area-preserving flows on surfaces with fixed points exhibit weak mixing behavior on their quasi-minimal components, under analyticity assumptions near saddle points.

Contribution

It establishes weak mixing for a broad class of area-preserving flows on surfaces with fixed points, assuming analyticity near saddles, extending understanding of their ergodic properties.

Findings

Flow is weakly mixing on each quasi-minimal component.

Analyticity near saddle fixed points is crucial for the result.

Applicable to flows with finitely many fixed points on surfaces.

Abstract

Let $(\phi_t)$ be an area-preserving smooth flow on a compact, connected, orientable surface $\mathcal M$ with at least one but finitely many fixed points. Assume that $(\phi_t)$ is analytic (up to a canonical change of coordinates) in the neighborhood of each saddle fixed point. We show that the flow $(\phi_t)$ is weakly mixing on each of its (finitely many) quasi-minimal components.