Local rigidity of actions of higher rank Abelian groups and Kam method

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-infinity local rigidity for Z(k) (k greater than or equal to 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 T-N for any even N greater than or equal to 6.