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We develop a new method for proving local differentiable rigidity for actions of higher rank abelian groups. Unlike earlier methods it does not require previous knowledge of structural stability and instead uses a version of the KAM (Kolmogorov-Arnold-Moser) iterative scheme. As an application we show C∞ local rigidity for Zk (k ≥ 2) partially hyperbolic actions by toral automorphisms. We also prove the existence of irreducible genuinely partially hyperbolic higher rank actions by automorphisms on any torus TN for any even N ≥ 6.

Local rigidity of actions of higher rank abelian groups and KAM method