Exponential Mixing of Nilmanifold Automorphisms

TL;DR

This paper proves that ergodic automorphisms of compact nilmanifolds exhibit exponential mixing and higher-order mixing, enabling the derivation of probabilistic limit theorems and regularity results for solutions of cohomological equations.

Contribution

It establishes exponential mixing for all ergodic automorphisms of compact nilmanifolds, a significant advancement in understanding their dynamical behavior.

Findings

Ergodic automorphisms are exponentially mixing.

Higher-order exponential mixing is proven.

Probabilistic limit theorems are derived.

Abstract

We study dynamical properties of automorphisms of compact nilmanifolds and prove that every ergodic automorphism is exponentially mixing and exponentially mixing of higher orders. This allows to establish probabilistic limit theorems and regularity of solutions of the cohomological equation for such automorphisms. Our method is based on the quantitative equidistribution results for polynomial maps combined with Diophantine estimates.