Multiple mixing for parabolic systems

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.

Figures (2)

• Figure 1: Shearing phenomenon: the pushed geodesic segment approximate an orbit segment parametrized with linear speed.
• Figure 2: Condition (G4) for LUS flows: for any valid choice of $L$, the relative measure of the blue set $A_L$ in $P_a$ is approximately $\varepsilon / \Delta S_a$, where $\Delta S_a := S^{+}(a,t_{i_a},x_a)-S^{-}(a,t_{i_a},x_a)$.

Theorems & Definitions (120)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Definition 3.1
• Lemma 3.2
• proof
• Definition 3.3: Globally uniformly shearing flows
• Lemma 3.4: Uniform $k$-mixing