Multiple Fractional Cohomological Equations and Quantitative Mixing on Nilmanifolds

TL;DR

This paper introduces a new analytic approach using fractional cohomological equations to prove super-exponential mixing rates for automorphisms on nilmanifolds, extending mixing results beyond tori.

Contribution

It develops a novel method based on fractional cohomological equations to establish super-exponential mixing of all orders on nilmanifolds, a first in the field.

Findings

Proves solvability of multiple fractional cohomological equations in a specific spectral range.

Establishes exponential decay of second-order correlations with partial regularity.

Demonstrates super-exponential mixing of all orders for irrational automorphisms.

Abstract

We develop a new analytic method for quantitative mixing of automorphisms on nilmanifolds. The method is based on the introduction and solvability of \emph{multiple fractional cohomological equations of Type~$I$} (sum type). We prove that these equations are solvable in a cohomology-free range governed by the spectral behavior at the edge \(0\), with estimates in partial Sobolev/H\"older norms along (weak) stable/unstable subgroup directions only. As consequences, we obtain exponential decay of order-two correlations under partial regularity, without transverse derivatives, and quantitative mixing of all orders (a quantitative Rokhlin theorem) with rates explicit in the dynamical data. In particular, we show that irrational automorphisms exhibit super-exponential mixing of all orders for $C^\infty$ observables. To our knowledge, these are the first examples of super-exponential mixing beyond the torus, and the first examples of all-orders super-exponential mixing.