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We show C ∞ local rigidity for ℤ k (k ≥ 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 T N for any even N ≥ 6.

Local rigidity of partially hyperbolic actions I. KAM method and ℤkactions on the torus

Local rigidity of partially hyperbolic actions I. KAM method and ℤ^kactions on the torus