03/01: Bassam Fayad

On KAM rigidity of parabolic affine actions on the torus

Abstract: A celebrated result of KAM theory is Arnold’s proof that a perturbed Diophantine rotation of the circle is reducible if the rotation number is preserved.

We say that an action is KAM rigid if any of its small perturbations is reducible, under a preservation of some Diophantine data, even when the model is not necessarily a toral translation.

In earlier work with Danijela Damjanovic, we proved that an affine action on the torus that has a higher rank linear part except for a rank one factor that is Identity is KAM rigid. That result combined the mechanisms of KAM rigidity with the mechanisms of local rigidity of partially hyperbolic higher rank affine actions on tori proved by Damjanovic and Katok.

For affine actions with parabolic generators the situation is completely different due to the absence of cohomological stability above an individual parabolic element, except for the step 2 case as shown by Katok.

We will see that KAM rigidity holds for typical abelian actions by step 2 unipotent matrices on the torus, and in some cases even when only one generator is of step 2.

This is a joint work with Danijela Damjanovic and Maria Saprykina.