Exponential multiple mixing for commuting automorphisms of a nilmanifold

Abstract

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

arXiv comment14 pages, minor corrections of typos based on referee's report. Final accepted version to appear in Ergodic Theory Dynam. Systems