模曲面上球与环中闭测地线的方差
The variance of closed geodesics in balls and annuli on the modular surface
摘要：
我们渐近估计分布的闭测地线在小随机球或环上的模块表面$Γ\backslash\mathbb{H}$的方差。提出了一个概率模型，其中闭测地线建模使用随机测线段，我们严格分析了这个模型使用混合的测地流在$Γ\backslash\mathbb{H}$。这导致了一个猜想的渐近行为的方差，这不像在以前探讨的情况下，不等于期望值。我们证明了这一猜想的小球和annuli，解决了一个问题留下的汉弗莱斯和Radziwill开放。
英文摘要：
We asymptotically estimate the variance for the distribution of closed geodesics in small random balls or annuli on the modular surface $Γ\backslash\mathbb{H}$. A probabilistic model in which closed geodesics are modeled using random geodesic segments is proposed, and we rigorously analyze this model using mixing of the geodesic flow in $Γ\backslash\mathbb{H}$. This leads to a conjecture for the asymptotic behavior of the variance, which unlike in previously explored cases is not equal to the expected value. We prove this conjecture for small balls and annuli, resolving a question left open by Humphries and Radziwill.
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