Analytical investigation reveals dichotomy in rigidity of parabolic affine actions, suggesting critical insights into dynamics.
We show the following dichotomy for a linear parabolic Z² Z 2 -action ρ L ρ L on the torus with at least one step-2 generator: (i) Any affine Z² Z 2 -action with linear part ρ L ρ L has a ℤ-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine Z² Z 2 -action with linear part ρ L ρ L is KAM-rigid under volume preserving perturbations.
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Damjanović et al. (2026) studied this question.
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