Local rigidity of partially hyperbolic actions I. KAM method and ${\mathbb Z^k}$ actions on the torus

Abstract

We show $\Bbb C^\infty$ local rigidity for $\mathbb{Z}^k$ $(k\ge 2)$ higher rank partially hyperbolic actions by toral automorphisms, using a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme. We also prove the existence of irreducible genuinely partially hyperbolic higher rank actions on any torus $\mathbb{T}^N$ for any even $N\ge 6$.