Exponential multiple mixing for commuting automorphisms of a nilmanifold

Abstract

Let l ∈ N ≥1 and α : Z l → Aut(N ) be an action of Z l by automorphisms on a compact nilmanifold N . We assume the action of every α(z) is ergodic for z ∈ Z l {0} and show that α satisfies exponential n-mixing for any integer n ≥ 2. This extends results of Gorodnik and Spatzier [Acta Math., 215 (2015)].