Mixing Properties of Commuting Nilmanifold Automorphisms

TL;DR

This paper investigates the mixing behaviors of commuting automorphisms on nilmanifolds, establishing conditions for mixing of all orders and demonstrating exponential mixing, with applications to cocycle rigidity.

Contribution

It proves that commuting automorphisms with ergodic elements are mixing of all orders and exhibits exponential mixing, advancing understanding of nilmanifold dynamics.

Findings

Automorphisms are mixing of all orders under ergodicity.

Exponential 2-mixing and 3-mixing are established.

Smooth cocycle rigidity is proved for higher-rank abelian groups.

Abstract

We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every nontrivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential 2-mixing and 3-mixing. As an application we prove smooth cocycle rigidity for higher-rank abelian groups of nilmanifold automorphisms.