KAM rigidity for partially hyperbolic affine Z^k actions on the torus with a rank one factor

Abstract

We show that ergodic affine abelian discrete actions on the torus, that have a rank-one factor in their linear part, are locally rigid in a KAM sense if and only if the rank one factor is trivial and the action is higher-rank transversally to this factor. Since it has been proved by Damjanovic and Katok that affine actions with higher-rank linear part are locally rigid, our result completes the local rigidity picture for affine actions on the torus.

References in corpus (1)

• Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions

Cited by in corpus (1)

• Rigidity of some abelian-by-cyclic solvable group actions on ${\mathbb{T}}^N$