[Paper Review] A van der Corput lemma and weak mixing over groups

Conrad Beyers, Rocco Duvenhage+1 more|ArXiv.org|Dec 2, 2005

Limits and Structures in Graph Theory10 references3 citations

TL;DR

This paper establishes weak mixing of all orders for measure-preserving dynamical systems over abelian second countable topological groups with an invariant measure, using a generalized van der Corput lemma for Hilbert space-valued functions on such groups. The key contribution is proving that $M$-weak mixing relative to a uniformly space-filling net implies weak mixing of all orders when homomorphisms in $M$ are distinct and translational.

ABSTRACT

We study weak mixing of all orders for weakly mixing measure preserving dynamical systems, where the dynamics is given by the action of an abelian second countable topological group which has an invariant measure under the group operation. One of the main technical tools we use is a van der Corput lemma for Hilbert space valued functions on a second countable topological group.

Motivation & Objective

• To extend the concept of weak mixing of all orders beyond $\mathbb{Z}$ and $\mathbb{R}$ to more general abelian second countable topological groups with invariant measures.
• To develop a van der Corput lemma for Hilbert space-valued functions on such groups as a central technical tool.
• To provide a direct generalization of the $\frac{1}{N}\sum_{n=1}^N$ averaging form, avoiding reliance on invariant means.
• To establish conditions under which $M$-weak mixing implies weak mixing of all orders via homomorphisms of the group.
• To characterize weak mixing of all orders in terms of convergence of multiple ergodic averages involving distinct homomorphisms.

Proposed method

• Introduce a generalized van der Corput lemma for Hilbert space-valued functions on second countable topological groups, using right-invariant measures.
• Define space-filling nets (Følner nets) and uniformly space-filling nets to generalize the averaging process beyond $\mathbb{Z}$ and $\mathbb{R}$.
• Use the $L^2$-norm convergence of averages $\frac{1}{\mu(\Lambda_n)}\int_{\Lambda_n} u_g \, dg$ to analyze weak mixing behavior.
• Apply the generalized van der Corput lemma to control the norm of multiple correlation averages involving distinct homomorphisms.
• Utilize induction on the number of sets in the correlation to prove weak mixing of all orders.
• Employ the condition that $\{\Lambda_n^{-1}\Lambda_n\}$ is uniformly space-filling to ensure the required convergence properties.

Experimental results

Research questions

• RQ1 Under what conditions does $M$-weak mixing relative to a uniformly space-filling net imply weak mixing of all orders for group actions on a probability space?
• RQ2 How can the classical van der Corput lemma be generalized to Hilbert space-valued functions on abelian second countable topological groups with invariant measures?
• RQ3 What role does the structure of the set $M$ of homomorphisms play in ensuring weak mixing of all orders?
• RQ4 When can the assumption of $M$-weak mixing relative to $\{\Lambda_n^{-1}\Lambda_n\}$ be dropped or replaced by another uniformly space-filling sequence?
• RQ5 In what cases does the standard $\frac{1}{N}\sum_{n=1}^N$ form generalize to non-abelian or non-discrete group actions?

Key findings

• The paper proves that if a measure-preserving system is $M$-weakly mixing relative to a uniformly space-filling net $\{\Lambda_n\}$, then it is weakly mixing of all orders with respect to distinct homomorphisms in $M$.
• The generalized van der Corput lemma (Theorem 2.7′) ensures that $\lim_{n\to\infty}\left\|\frac{1}{\mu(\Lambda_n)}\int_{\Lambda_n} u_g \, dg\right\|_{L^2} = 0$ under suitable conditions, enabling control of multiple correlations.
• For $G = \mathbb{Z}$ with $\Lambda_n = \{-n,\dots,n\}$, the system is weakly mixing of all orders if it is weakly mixing relative to $\{\Lambda_n\}$, recovering the classical result.
• For $G = \mathbb{R}^q$, taking $\Lambda_m$ as the open ball of radius $m$ yields a uniformly space-filling net with $\Lambda_m^{-1}\Lambda_m = \Lambda_{2m}$, satisfying the conditions of Theorem 4.4.
• If $\{\Lambda_n^{-1}\Lambda_n\}$ is uniformly space-filling, the assumption of $M$-weak mixing relative to $\{\Lambda_n^{-1}\Lambda_n\}$ can be dropped due to Corollary 3.10.
• The result extends to $M$ being the set of non-zero diagonal $q\times q$ matrices acting on $\mathbb{R}^q$, which form a translational set of homomorphisms, ensuring weak mixing of all orders.