Given a simple closed curve $$\gamma $$ γ on a connected, oriented, closed surface S of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on S having a simple closed geodesic of length L of the same topological type as $$\gamma $$ γ equidistributes with respect to a natural probability measure as $$L \rightarrow \infty $$ L → ∞ . We prove several generalizations of Mirzakhani’s result and discuss some of the technical aspects ommited in her original work. The dynamics of the earthquake flow play a fundamental role in the arguments in this paper.

中文翻译：

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双曲曲面模空间上膨胀平层族的等分分布

给定一个简单的闭合曲线 $$\gamma $$ γ 在一个连接的、有向的、负欧拉特征的闭合曲面 S 上，Mirzakhani 证明了 S 上双曲结构模空间中的点集具有长度为 L与 $$\gamma $$ 相同拓扑类型的 γ 等分布关于自然概率测度为 $$L \rightarrow \infty $$ L → ∞ 。我们证明了 Mirzakhani 结果的几个概括，并讨论了她原始工作中遗漏的一些技术方面。地震流的动力学在本文的论点中起着重要作用。