AI 中文总结
研究具有无限谱的光滑场强旋转性质失效问题，通过表明原始证明的收缩论证适用于满足特定条件的场，揭示反例违反条件，找到阻止收缩方法扩展到有限谱设置外的障碍。
AI 中文摘要
本注记确定了控制收缩方法扩展的基本机制。虽然由有限三角多项式定义的每个周期向量场与线性漂移的偏差有界，但对于光滑周期向量场，此性质通常不成立。我们表明原始证明中的收缩论证适用于任何满足自然一致可和性条件的场，而反例违反了该条件，从而揭示了阻止该方法扩展到有限谱设置之外的障碍。
英文摘要
The present note identifies the fundamental mechanism governing the extension of the contraction method. Although every periodic vector field defined by a finite trigonometric polynomial admits a bounded deviation from a linear drift, this property fails in general for smooth periodic vector fields. We show that the contraction argument underlying the original proof extends to any field satisfying a natural uniform summability condition, and that the counterexample violates this condition, thereby revealing the obstruction that prevents the method from extending beyond the finite-spectrum setting. For a smooth periodic vector field on the $n$-torus, the contraction method for establishing strong rotation vectors extends only to those asymptotic directions $ρ\in \mathbb{R}^n$ for which a certain spectral sum remains uniformly bounded along a sequence of rational approximations. We introduce the "arithmetic cone" $\mathfrak{C}(f)$, defined as the set of all $ρ$ admitting such an approximation. We establish its basic algebraic property: it is a cone. We prove that, under a uniform contraction condition, every element of $\mathfrak{C}(f)$ yields a strong rotation vector for the dynamics. The construction reveals a precise link between the Fourier asymptotics of $f$ and the arithmetic of admissible rotation directions. In the second part, we introduce the class of "spectrally admissible" fields $\mathcal{A}_{spec}$, for which the cone of the augmented field equals the whole space, and we show that it contains all finite trigonometric polynomials.
