We prove that any ergodic \(SL_2({\mathbb {R}})\)-invariant probability measure on a stratum of translation surfaces satisfies strong regularity: the measure of the set of surfaces with two non-parallel saddle connections of length at most \(\epsilon _1,\epsilon _2\) is \(O(\epsilon _1^2 \cdot \epsilon _2^2)\). We prove a more general theorem which works for any number of short saddle connections. The proof uses the multi-scale compactification of strata recently introduced by Bainbridge–Chen–Gendron–Grushevsky-Möller and the algebraicity result of Filip.

中文翻译：

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测量具有短鞍座连接的平移表面的界限

我们证明平移曲面层上的任何遍历\(SL_2({\mathbb {R}})\) -不变概率测度满足强规律性：具有两个长度为的非平行鞍座连接的曲面集的测度大多数\(\epsilon _1,\epsilon _2\)是\(O(\epsilon _1^2 \cdot \epsilon _2^2)\)。我们证明了一个适用于任意数量的短鞍连接的更一般的定理。该证明使用 Bainbridge–Chen–Gendron–Grushevsky-Möller 最近引入的地层多尺度紧化和 Filip 的代数结果。