We prove that, for a $C^\infty$-generic contact form $\lambda$ adapted to a given contact distribution on a closed three-manifold, there exists a sequence of periodic Reeb orbits which is equidistributed with respect to $d \lambda$. This is a quantitative refinement of the $C^\infty$-generic density theorem for three-dimensional Reeb flows, which was previously proved by the author. The proof is based on the volume theorem in embedded contact homology (ECH) by Cristofaro–Gardiner, Hutchings, Ramos, and inspired by the argument of Marques–Neves–Song, who proved a similar equidistribution result for minimal hypersurfaces.We also discuss a question about generic behavior of periodic Reeb orbits “representing” ECH homology classes, and give a partial affirmative answer to a toy model version of this question which concerns boundaries of star-shaped toric domains.

中文翻译：

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$C^\infty$-通用三维Reeb流的等分布周期轨道

我们证明，对于适用于封闭三流形上给定接触分布的 $C^\infty$-泛型接触形式 $\lambda$，存在一系列周期性 Reeb 轨道，该轨道相对于 $d\拉姆达$。这是对三维 Reeb 流的 $C^\infty$-泛型密度定理的量化改进，作者之前已经证明了这一点。该证明基于 Cristofaro-Gardiner、Hutchings、Ramos 的嵌入接触同源 (ECH) 中的体积定理，并受到 Marques-Neves-Song 论证的启发，后者证明了最小超曲面的类似等分布结果。我们还讨论了关于“代表”ECH同源类的周期性Reeb轨道的一般行为的问题，