Danijela Damjanovic, KTH Royal Institute of Technology
For two commuting parabolic toral automorphisms, such that one of them is of step 2, we obtain a dichotomy: either the action these two maps generate cannot be linear part of a higher rank affine action, or a full measure set of affine actions with such a linear part enjoys local rigidity in the same sense as Diophantine toral translations do. (In this context an affine action is higher rank if the only rank-one factor is a Kronecker factor i.e. a factor generated by an irrational toral translation). Main part of the result is a generalisation of classical KAM theorem for Diophantine toral translations to this parabolic setting. This is joint work with B. Fayad and M. Saprykina.