双环面上伪转动的刚性和一个Norton-Sullivan问题
The rigidity of pseudo-rotations on the two-torus and a question of Norton-Sullivan
摘要：
证明了在一定的有界性条件下，具有一般旋转向量的$\mathbb{T}^2$上的$C^{r}$保守无理伪旋转是$C^{r-1}$ -刚性的.我们还得到了具有类似性质的Hölder伪旋转的$C^0$ -刚性。这些结果是[B]中主要结果的部分推广。布拉姆，发明。（2015），第2，561-580期; A.阿维拉湾Fayad，P. Le Calvez，D. Xu和Z. Zhang，arXiv：1509.06906v1].然后，我们使用这些结果来研究保守的不合理的伪旋转$\mathbb{T}^2$与一般的旋转向量，是半共轭的平移通过半共轭同伦的身份。我们证明了任何这样的同构的保守中心化子同构于$\mathbb{R}^2/\mathbb{Z}^2$的不可数子群。在与亚历克诺顿和丹尼斯沙利文的问题，我们描述了拓扑线性化的地图在这类使用的保守中心化群的拓扑。在极小情形下，我们得到了所有完全无理向量拓扑线性化的精确刻画。我们还构造了一个$C^\infty$保守的和最小的完全无理的伪旋转同构，它是半共轭的平移，但拓扑非线性化。这就否定了诺顿和沙利文在$C^{\infty}$范畴中提出的问题。
英文摘要：
We show that under certain boundedness condition, a $C^{r}$ conservative irrational pseudo-rotations on $\mathbb{T}^2$ with a generic rotation vector is $C^{r-1}$-rigid. We also obtain $C^0$-rigidity for Hölder pseudo-rotations with similar properties. These provide a partial generalisation of the main results in [B. Bramham, Invent. Math. (2015), no. 2, 561-580; A. Avila, B. Fayad, P. Le Calvez, D. Xu and Z. Zhang, arXiv: 1509.06906v1]. We then use these results to study conservative irrational pseudo-rotations on $\mathbb{T}^2$ with a generic rotation vector that is semi-conjugate to a translation via a semi-conjugacy homotopic to the identity. We show that the conservative centralizers of any such diffeomorphism is isomorphic to a uncountable subgroup of $\mathbb{R}^2/\mathbb{Z}^2$. In connection with a question of Alec Norton and Dennis Sullivan, we describe the topologically linearizable maps within this class using the topology of the conservative centralizer group. In the minimal case, we obtain a precise characterization of topological linearizability for all totally irrational vectors. We also construct a $C^\infty$ conservative and minimal totally irrational pseudo-rotation diffeomorphism that is semi-conjugate to a translation, but is topologically nonlinearizable. This gives a negative answer to the question of Norton and Sullivan in the $C^{\infty}$ category.
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