AI 中文总结
该研究为双环面上定量刚性的Sarnak猜想相关论文，构造了不满足有界平均运动的半无理及全无理微分同胚例子，证明其数量为连续统多个。
AI 中文摘要
本注为论文《双环面上的刚性与Sarnak猜想》提供例子。对每个满足0<δ<1/2的δ，我们给出两类半无理C^∞微分同胚的构造，它们属于二维环面T²，既满足该论文定理1和定理2的假设，又不具有有界平均运动；同时给出满足定理1的全无理对应构造。第一种构造是显式的Anosov-Katok极限，第二种是光滑特殊流的时间1映射后接Moser归一化。两种情形下，该映射都保持Lebesgue面积，其旋转集为单点集，提升后的位移一致为O(n^δ)但无界。半无理版本的旋转集为{(α,0)}，满足定理1和定理2；全无理版本的旋转集为{(α,α²)}和{(α²,α)}，满足定理1（定理2按定义限于半无理情形）。我们还证明每类例子恰有连续统多个，其中包含连续统多个拓扑共轭类。
英文摘要
This note supplies genuinely non-fibred examples for the manuscript: Rigidity on the Two-Torus and Sarnak's Conjecture. For every $0<δ<\tfrac12$, we construct $C^\infty$ Lebesgue-area-preserving pseudo-rotations of $\mathbb{T}^2$ which satisfy the $(C,δ)$-deviation condition but do not have bounded mean motion. We give both semi-irrational and totally irrational rotation vectors and two realizations: a controlled weakly mixing Anosov--Katok construction and an explicit weakly mixing special flow construction. Weak mixing is used as a conjugacy-invariant obstruction to every continuous circle-rotation factor. Consequently, none of the resulting maps is topologically conjugate, by a linear or nonlinear change of coordinates, to a skew product over a circle rotation. In the special-flow realization, the same lacunary Fourier series simultaneously gives weak mixing, the sharp upper bound $O(n^δ)$, and unbounded deviations; in fact no smaller deviation exponent is possible. The semi-irrational examples meet the assumptions of Theorems~1 and~2 of the cited manuscript, whereas the totally irrational examples meet those of Theorem~1. Each construction produces continuum many maps and continuum many topological conjugacy classes of each rotation type.
CommentsChatGPT was used to assist in generating concrete examples following instructions on the use of the Anosov--Katok method and the construction of special flows. All mathematical content was verified by the authors
