What question did this study set out to answer?

This research aims to explore the KAM-rigidity of parabolic affine ${ ext{Z}}^{2}$-actions on the torus, particularly the implications of their linear components.

March 25, 2026Open Access

KAM-rigidity for parabolic affine Abelian actions

Puntos clave

This research aims to explore the KAM-rigidity of parabolic affine ${ ext{Z}}^{2}$-actions on the torus, particularly the implications of their linear components.
Examined the dynamics of linear parabolic ${ ext{Z}}^{2}$-actions on a torus.
Analyzed the characteristics of affine actions with a linear part $ ho_L$.
Investigated the KAM-rigidity property under specific perturbations.
Identified that affine ${ ext{Z}}^{2}$-actions can either be identity or genuinely parabolic, hence not KAM-rigid.
Established that almost all affine ${ ext{Z}}^{2}$-actions are KAM-rigid when volume preserving perturbations are applied.

Resumen

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

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