We prove several general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $\operatorname {SL}(2, \mathbb {R})$ -action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $\operatorname {SL}(2, \mathbb {R})$ -action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the ‘weak convergence’ of push-forwards of horocycle measures under the geodesic flow and a short proof of weaker versions of theorems of Chaika and Eskin on Birkhoff genericity and Oseledets regularity in almost all directions for the Teichmüller geodesic flow.

中文翻译：

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horocycle 不变测度的测地线推进的限制

我们证明了任何连续的 horocycle 和测地线子群的遍历平均值的几个一般条件收敛结果$\operatorname {SL}(2, \mathbb {R})$-作用于局部紧致空间。这些结果是由 Eskin、Mirzakhani 和 Mohammadi 的定理推动的$\operatorname {SL}(2, \mathbb {R})$-对阿贝尔微分模空间的作用。根据我们的论点，我们可以从这些定理中推导出测地线流动下大周期测量推进的“弱收敛”的改进版本，以及在几乎Teichmüller 测地线流的所有方向。