KAM-rigidity for parabolic affine Abelian actions

Abstract We show the following dichotomy for a linear parabolic ${\mathbb{Z}}^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -action $\rho _{L}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:math> on the torus with at least one step-2 generator: (i) Any affine ${\mathbb{Z}}^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -action with linear part $\rho _{L}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:math> has a ℤ-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine ${\mathbb{Z}}^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -action with linear part $\rho _{L}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:math> is KAM-rigid under volume preserving perturbations.