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

TL;DR

This paper establishes $C^ abla$ local rigidity for certain isometric toral extensions of partially hyperbolic $ abla$ actions on the torus, using a generalized KAM scheme, under intersection and volume-preserving conditions.

Contribution

It provides the first $C^ abla$ local rigidity results for isometric extensions of partially hyperbolic toral actions, extending previous rigidity theories.

Findings

Proves $C^ abla$ local rigidity under intersection property.

Establishes local rigidity within volume-preserving action class.

Uses a generalized KAM iterative scheme for the proof.

Abstract

This paper studies local rigidity for some isometric toral extensions of partially hyperbolic $\mathbb{Z}^k$ ($k\geqslant 2$) actions on the torus. We prove a $C^\infty$ 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 KAM iterative scheme.