KAM-rigidity for parabolic affine abelian actions

TL;DR

This paper establishes a dichotomy for parabolic affine actions on tori, showing that such actions are either non-rigid with a trivial factor or are KAM-rigid under volume-preserving perturbations, depending on their structure.

Contribution

It proves a dichotomy for linear parabolic $bZ^2$-actions on tori, characterizing when they are KAM-rigid or not based on their affine extensions.

Findings

Almost every affine action with given linear part is KAM-rigid.

Actions with trivial or identity factors are not KAM-rigid.

The dichotomy depends on the presence of step-2 generators in the action.

Abstract

We show the following dichotomy for a linear parabolic $\mathbb Z^2$-action $\rho_L$ on the torus with at least one step-2 generator: (i) Any affine $\mathbb Z^2$-action with linear part $\rho_L$ has a $\mathbb Z$-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine $\mathbb Z^2$-action with linear part $\rho_L$ is KAM-rigid under volume preserving perturbations.