Rigidity properties for some isometric extensions of partially hyperbolic actions on the torus

Abstract

This paper studies local rigidity for some isometric toral extensions of partially hyperbolic Z k \mathbb {Z}^k ( k ⩾ 2 k\geqslant 2 ) actions on the torus. We prove a C ∞ C^\infty local rigidity result for such actions, provided that the smooth perturbations of the actions satisfy the intersection property. We also give a local rigidity result within a class of volume preserving actions. Our method mainly uses a generalization of the Kolmogorov-Arnold-Moser iterative scheme.