论文标题
Weil-模块化表面上的彼得森测量学
Weil--Petersson geodesics on the modular surface
论文作者
论文摘要
我们考虑模块表面上的Weil-Petersson（WP）度量。我们将WP测量学提升到模块化表面的通用覆盖层，并分析升力的几何特性，作为通用盖上双曲线度量的路径。对于通用覆盖厚部分中的任何一对不同点，我们证明连接这对的WP和双曲线测量片段，厚部分中的同伴 - 旅行以及在尖距离旅行期间这些段之间的所有偏差。此外，我们对游览期间的偏差进行了定量分析。 我们利用旅行的同伴来得出从基础点进行复发的WP和双曲线测量射线之间的对应关系。我们表明，通信可以晋升为方向圆上的同构。通过分析典型的WP测量线的尖绕，我们表明同态形态学将圆上的Lebesgue量度推向了一个单数尺寸。就持续的分数系数而言，奇异性归结为我们证明的比较，即，平均系数沿典型的WP射线界定，但沿典型的双曲线射线无限。
We consider the Weil--Petersson (WP) metric on the modular surface. We lift WP geodesics to the universal cover of the modular surface and analyse geometric properties of the lifts as paths in the hyperbolic metric on the universal cover. For any pair of distinct points in the thick part of the universal cover, we prove that the WP and hyperbolic geodesic segments that connect the pair, fellow-travel in the thick part and all deviations between these segments arise during cusp excursions. Furthermore, we give a quantitative analysis of the deviation during an excursion. We leverage the fellow traveling to derive a correspondence between recurrent WP and hyperbolic geodesic rays from a base-point. We show that the correspondence can be promoted to a homeomorphism on the circle of directions. By analysing cuspidal winding of a typical WP geodesic ray, we show that the homeomorphism pushes forward a Lebesgue measure on the circle to a singular measure. In terms of continued fraction coefficients, the singularity boils down to a comparison that we prove, namely, the average coefficient is bounded along a typical WP ray but unbounded along a typical hyperbolic ray.
