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The Periodic Orbit Conjecture for Steady Euler Flows
Qualitative Theory of Dynamical Systems ( IF 1.9 ) Pub Date : 2021-05-20 , DOI: 10.1007/s12346-021-00490-w
Robert Cardona

The periodic orbit conjecture states that, on closed manifolds, the set of lengths of the orbits of a non-vanishing vector field all whose orbits are closed admits an upper bound. This conjecture is known to be false in general due to a counterexample by Sullivan. However, it is satisfied under the geometric condition of being geodesible. In this work, we use the recent characterization of Eulerisable flows (or more generally flows admitting a strongly adapted one-form) to prove that the conjecture remains true for this larger class of vector fields.



中文翻译:

稳态Euler流的周期轨道猜想。

周期轨道猜想指出,在闭合流形上,所有消失的矢量场的轨道长度的集合均具有上限,该轨道的所有轨道都是闭合的。由于沙利文(Sullivan)的反例,这种推测通常被认为是错误的。但是,在可测地线的几何条件下是令人满意的。在这项工作中,我们使用Eulerisable流(或更普遍地说,是允许强烈适应一种形式的流)的最新特征来证明这种猜想对于这种更大的向量场仍然成立。

更新日期:2021-05-20
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