A Question of Norton–Sullivan in the Analytic Case. (23rd June 2020)

• Record Type: Journal Article
• Authors: Wang, Jian; Yang, Hui
• Is Part Of: International mathematics research notices. Volume 2021:Number 22(2021)
• Page Start: 17201
• Page End: 17219
• Publication Date: 2020-06-23
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnz231
• Languages: English
• ISSNs: 1073-7928

Abstract

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.