We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of invariant tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily weak rescalings.

中文翻译：

---

表面强磁系统的范式：Zoll 系统的俘获区域和刚度

我们证明了封闭、定向表面上强磁场的范式，并使用它推导出相关流动的两个动力学结果。首先，我们展示了不变环面和捕获区域的存在，前提是自然非共振条件成立。其次，我们证明流动不可能是 Zoll，除非（i）黎曼度量具有恒定曲率并且磁函数是恒定的，或者（ii）磁函数消失并且度量是 Zoll。我们通过在平坦的双环面上展示奇异的磁场来补充第二个结果，从而产生用于任意弱重新缩放的 Zoll 流。