Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces

TL;DR

This paper proves that typical area-preserving flows on higher genus surfaces are mixing of all orders, using a variation of Ratner's property, and extends results to Arnold's flows with almost every singularity configuration.

Contribution

It establishes multiple mixing for flows on higher genus surfaces and introduces a switchable Ratner property for special flows over rotations.

Findings

Flow restricted to mixing minimal components is mixing of all orders.

The switchable Ratner property describes parabolic behavior and aids in joinings classification.

Almost every Arnold's flow with specific singularities is mixing of all orders.

Abstract

We consider typical area preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is mixing of all orders, thus answering affimatively to Rohlin's multiple mixing question in this context. The main tool is a variation of the Ratner property (a property originally proved by Ratner for the horocycle flow), i.e. the switchable Ratner property introduced by Fayad and Kanigowski for special flows over rotations. This property, which is of independent interest, provides a quantitative description of the parabolic behaviour of these flows and has implications to joinings classification. The main result is formulated in the language of special flows over interval exchange transformations with asymmetric logarithmic singularities. We also prove a strengthening of one of Fayad and Kanigowski's main results, by showing that Arnold's flows are mixing of all orders for almost every location of the singularities.