Local rigidity of partially hyperbolic actions I. KAM method and \({\mathbb Z^k}\) actions on the torus (Q624921)

The paper studies differentiable rigidity for algebraic partially hyperbolic actions of \(\mathbb{Z}^k\times \mathbb{R}^l\), \(k+l\geq 2\), in the setting: the group \(\mathbb{Z}^k\times \mathbb{R}^l\) contains a subgroup \(L\) isomorphic to \(\mathbb{Z}^2\) such that for the suspension of the restriction of the action to \(L\) every element other than identity acts ergodically with respect to the standard invariant measure obtained from the Haar measure. The authors show \(\mathbb C^\infty\) local rigidity for \(\mathbb{Z}^k(k\geq 2)\) higher rank partially hyperbolic actions by toral automorphisms, using a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme. They also prove the existence of irreducible genuinely partially hyperbolic higher rank actions on any torus \(\mathbb{T}^N\) for any even \(N\geq 6\).