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

Qinbo Chen, Danijela Damjanović

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.

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 () actions on the torus. We prove a 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.
Paper Structure (32 sections, 207 equations)

This paper contains 32 sections, 207 equations.

Table of Contents

  1. Introduction
  2. Background on rigidity for smooth actions
  3. The main results
  4. Volume preserving actions
  5. The method.
  6. Strategy of the proof
  7. Further discussion
  8. Organization of the paper
  9. Notation
  10. Preliminaries
  11. Ergodic toral automorphisms and partial hyperbolicity
  12. Higher rank actions on tori
  13. Fréchet spaces and tame linear maps
  14. Partially hyperbolic actions of product type
  15. The linearized conjugacy equations
  16. ...and 17 more sections

Theorems & Definitions (17)

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  • ...and 7 more