Exponential mixing, KAM and smooth local rigidity

TL;DR

This paper proves that small smooth perturbations of higher rank ergodic automorphisms on nilmanifolds are smoothly conjugate to the original actions, leveraging exponential mixing and a KAM scheme.

Contribution

It establishes smooth local rigidity for higher rank partially hyperbolic actions on nilmanifolds using a novel KAM approach.

Findings

Small $C^k$ perturbations are smoothly conjugate to original actions.

Exponential mixing ensures convergence of the KAM scheme.

Rigidity results apply to higher rank ergodic automorphisms.

Abstract

Consider actions of $\Z ^r$ by ergodic automorphisms on a compact nilmanifolds for $r \geq 2$. We show that small $C^k$ perturbations of such higher rank partially hyperbolic actions are smoothly conjugate to the original action, using a KAM scheme. The driving force for convergence of this iteration is the exponential mixing of the original action.