We study the dynamics of magnetic flows on Heisenberg groups, investigating the extent to which properties of the underlying Riemannian geometry are reflected in the magnetic flow. Much of the analysis, including a calculation of the Mañé critical value, is carried out for \((2n+1)\)-dimensional Heisenberg groups endowed with any left invariant metric and any exact, left-invariant magnetic field. In the three-dimensional Heisenberg case, we obtain a complete analysis of left-invariant, exact magnetic flows. This is interesting in and of itself, because of the difficulty of determining geodesic information on manifolds in general. We use this analysis to establish two primary results. We first show that the vectors tangent to periodic magnetic geodesics are dense for sufficiently large energy levels and that the lower bound for these energy levels coincides with the Mañé critical value. We then show that the marked magnetic length spectrum of left-invariant magnetic systems on compact quotients of the Heisenberg group determines the Riemannian metric. Both results confirm that this class of magnetic flows carries significant information about the underlying geometry. Finally, we provide an example to show that extending this analysis of magnetic flows to the Heisenberg-type setting is considerably more difficult.

我们研究了海森堡群上磁流的动力学，研究了基本黎曼几何的特性在磁流中反映的程度。大部分分析，包括计算 Mañé 临界值，都是针对\((2n+1)\)维海森堡群被赋予任何左不变度量和任何精确的左不变磁场。在三维海森堡案例中，我们获得了对左不变的精确磁流的完整分析。这本身就很有趣，因为通常很难确定流形上的测地线信息。我们使用此分析来确定两个主要结果。我们首先表明，对于足够大的能级，与周期性磁测地线相切的矢量是密集的，并且这些能级的下限与 Mañé 临界值一致。然后，我们证明了海森堡群紧商上的左不变磁系统的标记磁长度谱决定了黎曼度量。这两个结果都证实了这类磁流携带有关基础几何形状的重要信息。最后，我们提供了一个例子来表明将这种磁流分析扩展到海森堡类型的设置要困难得多。