二维环面的刚性与萨纳克猜想
Quantitative Rigidity of pseudo-rotations on the two-torus and Sarnak's conjecture
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中文总结 AI 辅助
该研究针对二维环面的伪旋转，通过定量自由圆盘估计建立了满足$(C,\delta)$偏差条件的不同类型伪旋转的刚性结果，进而证明这些类均满足萨纳克猜想。
中文摘要 AI 辅助
针对二维环面的伪旋转，我们建立了相对于其旋转向量满足$(C,\delta)$偏差条件的定量刚性结果。核心方法是定量自由圆盘估计，该估计可将轨道偏差的界转化为迭代映射与恒等映射间距离的显式控制。在上述$(C,\delta)$偏差条件下，我们证明：Hölder连续的超刘维尔无理伪旋转具有指数衰减率的$C^0$刚性；强非布里奥型的$C^k$半无理伪旋转具有超多项式衰减率的$C^{k-1}$刚性。此外，在该偏差条件及足够大的无理测度下，我们证明圆旋转上的$\mathbb{T}^2$上Hölder连续斜积具有多项式衰减率的$C^0$刚性。由此可得，所有这些类均满足萨纳克猜想。
英文摘要
We establish quantitative rigidity results for pseudo-rotations of the two-torus under a $(C,δ)$-deviation condition relative to their rotation vectors. The main ingredient is a quantitative free-disk estimate that converts bounds on orbit deviation into explicit control of the distance between the iterates and the identity map. Under such a $(C,δ)$-deviation condition, we show that Hölder continuous super-Liouvillean irrational pseudo-rotations are $C^0$-rigid with an exponential decay rate and that $C^k$ semi-irrational pseudo-rotations of strong non-Brjuno type exhibit $C^{k-1}$-rigidity with a superpolynomial decay rate. Moreover, under this deviation condition and sufficiently large irrationality measure, we show that Hölder continuous skew products on $\mathbb{T}^2$ over circle rotations are $C^0$-rigid with a polynomial decay rate. As a consequence, all these classes satisfy Sarnak's conjecture.
发表机构
- College of Mathematics, Sichuan University（四川大学数学学院）
- School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）
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