A question of Norton-Sullivan and the rigidity of pseudo-rotations on the two-torus

Jian Wang
20/11/2018 , 15:30 | Sala 228

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 $hf = T_{\rho}h$ 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 this talk,we give a negative answer to this question with respect to the regularity.We also show that under certain boundedness condition,a $C^r$(resp. Hölder)conservative irrational pseudo-rotation on $\mathbb{T}^2$ with a generic rotation vector is $C^{r-1}$-rigid(resp. $C^0$-rigid). These provide a partial generalization of the main results in[Bramham,Invent. Math. 199 (2), 561-580, 2015;A. Avila,B. Fayad,P. Le Calvez,D. Xu,Z. Zhang,arXiv: 1509.06906v1]. These are joint works with Zhiyuan Zhang and Hui Yang.