A question of Norton-Sullivan in the analytic case

Abstract

In 1996, A. Norton and D. Sullivan asked the following question: If $f:{\mathbb{T}}^2\to {\mathbb{T}}^2$ is a diffeomorphism, $h:{\mathbb{T}}^2\to {\mathbb{T}}^2$ is a continuous map homotopic to the identity, and $hf=T_{\rho }h$ where $\rho \in {\mathbb{R}}^2$ is a totally irrational vector and $T_{\rho }:{\mathbb{T}}^2\to {\mathbb{T}}^2,z\mapsto z+\rho$ 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 ].

12 pages. arXiv admin note: text overlap with arXiv:1708.02529. In the proof of our main theorem, we use the same approximation by conjugation scheme in our preceding article [J. Wang and Z. Zhang, GAFA 2018, arXiv:1708.02529]. The different is that we have to replace the $C^{\infty }$ conjugacies in the proof of [J. Wang and Z. Zhang, GAFA 2018] by the $C^{\omega }$ conjugacies in this article