For every positive, continuous and homogeneous function f on the space of currents on a compact surface \({{{\overline{\Sigma }}}}\), and for every compactly supported filling current \(\alpha \), we compute as \(L \rightarrow \infty \), the number of mapping classes \(\phi \) so that \(f(\phi (\alpha ))\le L\). As an application, when the surface in question is closed, we prove a lattice counting theorem for Teichmüller space equipped with the Thurston metric.

中文翻译：

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测地电流和计数问题

对于紧表面\（{{{\ overline {\ Sigma}}}}} \）上电流空间上的每个正，连续和齐次函数f，以及每个紧要支持的填充电流\（\ alpha \），我们计算为\（L \ rightarrow \ infty \）映射类的数量\（\ phi \）以便\（f（\ phi（\ alpha））\ le L \）。作为应用，当所讨论的曲面闭合时，我们证明了配备了Thurston度量的Teichmüller空间的晶格计数定理。