A classical theorem due to Wadsley implies that, on a contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods of its closed Reeb orbits.

中文翻译：

---

作用谱表征了所有 Reeb 轨道都是闭合的闭合接触 3-流形

Wadsley 的经典定理暗示，在所有 Reeb 轨道都闭合的接触流形上，Reeb 轨道有一个公共周期。在本文中，我们证明，对于封闭的 3 流形上的任何 Reeb 流，以下条件实际上是等效的：（1）每个 Reeb 轨道都是封闭的；(2) 所有闭合的 Reeb 轨道都有一个共同的周期；(3) 动作谱的秩为 1。我们还表明，在固定的封闭 3 流形上，具有 秩为 1 的动作谱的接触形式由一组最小周期确定（直到由微分同胚拉回）其封闭的 Reeb 轨道。