Exponential multiple mixing for commuting automorphisms of a nilmanifold

TL;DR

This paper proves that actions of $Z^l$ by automorphisms on compact nilmanifolds exhibit exponential multiple mixing, extending previous results and demonstrating strong statistical properties of such dynamical systems.

Contribution

It establishes exponential $n$-mixing for $Z^l$-actions on nilmanifolds under ergodicity assumptions, generalizing earlier work by Gorodnik and Spatzier.

Findings

Proves exponential $n$-mixing for $Z^l$-actions on nilmanifolds.

Extends previous results to higher-dimensional actions.

Shows ergodicity implies strong mixing properties.

Abstract

Let $l\in \mathbb{N}_{\geq 1}$ and $\alpha : \mathbb{Z}^l\rightarrow \text{Aut}(\mathscr{N})$ be an action of $\mathbb{Z}^l$ by automorphisms on a compact nilmanifold $\mathscr{N}$. We assume the action of every $\alpha(z)$ is ergodic for $z\in \mathbb{Z}^l\smallsetminus\{0\}$ and show that $\alpha$ satisfies exponential $n$-mixing for any integer $n\geq 2$. This extends results of Gorodnik and Spatzier [Acta Math., 215 (2015)].