On local rigidity of partially hyperbolic affine ℤ k actions

The following dichotomy for affine ℤ k Zᵏ actions on the torus 𝕋 d {T}ᵈ , k , d ∈ ℕ k,d∈{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.