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

TL;DR

This paper establishes a KAM-type local rigidity result for ergodic affine abelian actions on the torus with a rank-one factor, showing rigidity occurs only when the rank-one factor is trivial and the action is higher-rank transversally.

Contribution

It completes the local rigidity classification for affine torus actions by identifying conditions under which actions with a rank-one factor are rigid.

Findings

Rigidity holds iff the rank-one factor is trivial.

Higher-rank transversality is necessary for rigidity.

The result extends previous higher-rank rigidity theorems.

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.