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Local rigidity of higher rank partially hyperbolic algebraic actions

Zhenqi Jenny Wang

Abstract

We give a complete solution to the local classification program of higher rank partially hyperbolic algebraic actions. We show $C^\infty$ local rigidity of abelian ergodic algebraic actions for symmetric space examples, twisted symmetric space examples and automorphisms on nilmanifolds. The method is a combination of representation theory, harmonic analysis and a KAM iteration. A striking feature of the method is no specific information from representation theory is needed. It is the first time local rigidity for non-accessible partially hyperbolic actions has ever been obtained other than torus examples. Even for Anosov actions, our results are new: it is the first time twisted spaces with non-abelian nilradical have been treated in the literature.

Local rigidity of higher rank partially hyperbolic algebraic actions

Abstract

We give a complete solution to the local classification program of higher rank partially hyperbolic algebraic actions. We show local rigidity of abelian ergodic algebraic actions for symmetric space examples, twisted symmetric space examples and automorphisms on nilmanifolds. The method is a combination of representation theory, harmonic analysis and a KAM iteration. A striking feature of the method is no specific information from representation theory is needed. It is the first time local rigidity for non-accessible partially hyperbolic actions has ever been obtained other than torus examples. Even for Anosov actions, our results are new: it is the first time twisted spaces with non-abelian nilradical have been treated in the literature.

Paper Structure

This paper contains 95 sections, 48 theorems, 746 equations.

Table of Contents

  1. Introduction
  2. Abelian algebraic actions
  3. Rigidity of actions and related notions
  4. History and motivation
  5. Result of the paper
  6. Proof strategy
  7. Outline of this paper
  8. Preliminaries on unitary representation theory
  9. Sobolev space and elliptic regularity theorem
  10. Useful results
  11. Directional smoothing operators
  12. Truncation by Fourier transform
  13. Notations
  14. Main results
  15. Basic facts of group algebra
  16. ...and 80 more sections

Key Result

Theorem 1.3

If $G$ is standard and $\alpha_A$ is a higher-rank partially hyperbolic action on $\mathcal{X}$, then there is $\ell_0\in\mathbb{N}$ such that $\alpha_A$ is $C^{\infty,\ell_0,\infty}$ locally rigid.

Theorems & Definitions (90)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 80 more