Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global rigidity of smooth ${\mathbb Z}\ltimes_λ{\mathbb R}$-actions on ${\mathbb T}^2$

Changguang Dong, Yi Shi

Abstract

For $λ>1$, we consider the locally free ${\mathbb Z}\ltimes_λ{\mathbb R}$ actions on ${\mathbb T}^2$. We show that, if the action is $C^r$ with $r\geq2$, then it is $C^{r-ε}$-conjugate to an affine action generated by a hyperbolic automorphism and a linear translation flow along expanding eigen-direction of the automorphism. In contrast, there exists a $C^{1+α}$-action which is semi-conjugate, but not topologically conjugate to an affine action.

Global rigidity of smooth ${\mathbb Z}\ltimes_λ{\mathbb R}$-actions on ${\mathbb T}^2$

Abstract

For , we consider the locally free actions on . We show that, if the action is with , then it is -conjugate to an affine action generated by a hyperbolic automorphism and a linear translation flow along expanding eigen-direction of the automorphism. In contrast, there exists a -action which is semi-conjugate, but not topologically conjugate to an affine action.
Paper Structure (4 sections, 8 theorems, 50 equations)

This paper contains 4 sections, 8 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Invariant foliation and linear action
  3. Topological conjugacy on $\mathbb T^2$
  4. Smooth conjugacy

Key Result

Theorem A

Let $\lambda>1$ and $r\geq2$. Suppose $\rho:\mathbb Z\ltimes_\lambda\mathbb R\to \operatorname{Diff}^r(\mathbb T^2)$ is a locally free action, then it is $C^{r-\epsilon}$-smoothly conjugate to an affine action, for any $\epsilon>0$. More precisely, there exist such that $\rho$ is $C^{r-\epsilon}$-smoothly conjugate to the group action generated by $\{A,v_{at}\}$ for some $a\in\mathbb R\setminus\{

Theorems & Definitions (25)

  • Theorem A
  • Example 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1: Fr
  • Proposition 3.1
  • ...and 15 more