Multiple mixing for parabolic systems

TL;DR

This paper introduces locally uniformly shearing systems (LUS) as a framework to understand mixing in parabolic systems, proving that all LUS flows are mixing of all orders and applying this to specific smooth flows.

Contribution

The paper defines LUS and polynomial LUS, proving they imply mixing of all orders and applying these results to Kochergin flows and unipotent flows.

Findings

All LUS flows are mixing of all orders.

Polynomially mixing systems that are polynomial LUS are polynomially mixing of all orders.

Kochergin flows and smooth time-changes of unipotent flows are polynomially mixing of all orders.

Abstract

The famous Rokhlin Problem asks whether mixing implies higher order mixing. So far, all the known examples of zero entropy, mixing dynamical systems enjoy a variant of the mixing via shearing mechanism. In this paper we introduce the notion of locally uniformly shearing systems (LUS) which is a rigorous way of describing the mixing via shearing mechanism. We prove that all LUS flows are mixing of all orders. We then show that mixing smooth flows on surfaces and smooth time-changes of unipotent flow are LUS. We also introduce the notion of quantitative LUS. We show that polynomially mixing systems that are polynomially LUS are in fact polynomially mixing of all orders. As a consequence we show that Kochergin flows on $\mathbb{T}^2$ (for a.e. irrational frequency) as well as smooth time-changes of unipotent flows are polynomially mixing of all orders.