Multiple mixing for a class of conservative surface flows

TL;DR

This paper proves higher order mixing for a class of conservative surface flows under certain arithmetic conditions, using a novel application of the Ratner property to establish multiple mixing.

Contribution

It introduces a new method applying the Ratner property to conservative surface flows, demonstrating their higher order mixing under specific arithmetic conditions.

Findings

Flows are mixing of all orders under certain conditions

Generalization of Ratner property applied outside horocycle flows

First use of Ratner property to prove multiple mixing in this context

Abstract

Arnol'd and Kochergin mixing conservative flows on surfaces stand as the main and almost only natural class of mixing transformations for which higher order mixing has not been established, nor disproved. Under suitable arithmetic conditions on their unique rotation vector, of full Lebesgue measure in the first case and of full Hausdorff dimension in the second, we show that these flows are mixing of any order. For this, we show that they display a generalization of the so called Ratner property on slow divergence of nearby orbits, that implies strong restrictions on their joinings, that in turn yield higher order mixing. This is the first case in which the Ratner property is used to prove multiple mixing outside its original context of horocycle flows and we expect our approach will have further applications.