Abstract

In 1996, A. Norton and D. Sullivan asked the following question: If $f:\mathbb{T}^2\rightarrow \mathbb{T}^2$ is a diffeomorphism, $h:\mathbb{T}^2\rightarrow \mathbb{T}^2$ is a continuous map homotopic to the identity, and $h f=T_{\rho } h$, where $\rho \in \mathbb{R}^2$ is a totally irrational vector and $T_{\rho }:\mathbb{T}^2\rightarrow \mathbb{T}^2,\, z\mapsto z+\rho $ is a translation, are there natural geometric conditions (e.g., smoothness) on $f$ that force $h$ to be a homeomorphism? In [ 22], the 1st author and Z. Zhang gave a negative answer to the above question in the $C^{\infty }$ category: in general, not even the infinite smoothness condition can force $h$ to be a homeomorphism. In this article, we give a negative answer in the $C^{\omega }$ category (see also [ 22, Question 3]): we construct a real analytic conservative and minimal totally irrational pseudo-rotation of $\mathbb{T}^2$ that is semi-conjugate to a translation but not conjugate to a translation.

中文翻译：

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分析案例中的诺顿-沙利文问题

摘要

1996 年，A. Norton 和 D. Sullivan 提出了以下问题：如果 $f:\mathbb{T}^2\rightarrow \mathbb{T}^2$ 是微分同胚，则 $h:\mathbb{T}^2 \rightarrow \mathbb{T}^2$ 是恒等式的连续映射同伦，$hf=T_{\rho } h$，其中 $\rho \in \mathbb{R}^2$ 是完全无理向量而 $T_{\rho }:\mathbb{T}^2\rightarrow \mathbb{T}^2,\, z\mapsto z+\rho $ 是平移，$ 上是否存在自然几何条件（例如平滑度） f$ 迫使 $h$ 成为同胚？在[22]中，第一作者和Z.Zhang在$C^{\infty}$范畴内对上述问题给出了否定回答：一般来说，即使是无限平滑条件也不能迫使$h$成为同胚. 在本文中，我们在 $C^{\omega }$ 类别中给出否定答案（另见 [22，问题 3]）：