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

Abstract

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. © 2004 Danijela Damjanović and Anatole Katok. © 2004 American Mathematical Society.