Local rigidity of higher rank non-abelian action on tours

TL;DR

This paper establishes local smooth rigidity for higher rank ergodic nilpotent actions on tori, extending previous results and providing new examples, using a generalized KAM iterative scheme.

Contribution

It proves local smooth rigidity for higher rank ergodic nilpotent actions on tori and constructs examples with rank-one factors, broadening the scope of rigidity results.

Findings

Rigidity holds for actions on any torus T_N with N ≥ 6.

Examples of actions with rank-one factors exhibiting smooth rigidity.

Generalization of the KAM iterative scheme to this context.

Abstract

In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms and prove the existence of such action on any torus TN for any even N ? 6. We also give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme.