In this short note, we prove that singular Reeb vector fields associated with generic \(b\)-contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) \(2N\) or an infinite number of escape orbits, where \(N\) denotes the number of connected components of the critical set. In case where the first Betti number of a connected component of the critical surface is positive, there exist infinitely many escape orbits. A similar result holds in the case of \(b\)-Beltrami vector fields that are not \(b\)-Reeb. The proof is based on a more detailed analysis of the main result in [19].

中文翻译：

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从 $$2N$$ 到无限多个逃逸轨道

在这篇简短的文章中，我们证明了与具有紧凑嵌入式临界表面的三维流形上的通用\(b\)接触形式相关的奇异 Reeb 矢量场具有（至少）\(2N\)或无限数量的逃逸轨道，其中\(N\)表示关键集的连通分量的数量。当临界面连通分量的第一个贝蒂数为正时，存在无穷多个逃逸轨道。类似的结果在\(b\) -Beltrami 向量场不是\(b\) -Reeb 的情况下成立。该证明基于对[19]中主要结果的更详细分析。