A question of Norton-Sullivan in the analytic case

TL;DR

This paper constructs a real-analytic conservative minimal pseudo-rotation on the 2-torus that is semi-conjugate but not conjugate to a translation, providing a negative answer to a question about geometric conditions forcing conjugacy.

Contribution

It demonstrates that even in the real-analytic category, certain pseudo-rotations are not conjugate to translations, answering an open question in the field.

Findings

Constructed a real-analytic pseudo-rotation not conjugate to a translation

Showed semi-conjugacy does not imply conjugacy in the real-analytic setting

Provided a negative answer to a previously open question

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 [ J. Wang and Z. Zhang, GAFA 2018 ], the first 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: 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, which simultaneously answers a question raised in [ J. Wang and Z. Zhang, GAFA 2018 ].