On local rigidity of partially hyperbolic affine ℤk actions

The following dichotomy for affine \mathbb{Z}^{k} actions on the torus {\mathbb{T}}^{d}, k,d\in{\mathbb{N}}, is shown to hold: (i) The linear part of the action has no rank-one factors, and then the affine action is locally rigid. (ii) The linear part of the action has a rank-one factor, and then the affine action is locally rigid in a probabilistic sense if and only if the rank-one factors are trivial. Local rigidity in a probabilistic sense means that rigidity holds for a set of full measure of translation vectors in the rank-one factors.