Partial sums of excursions along random geodesics and volume asymptotics for thin parts of moduli spaces of quadratic differentials
Partial sums of excursions along random geodesics and volume asymptotics for thin parts of moduli spaces of quadratic differentials
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二次微分模空间薄部分沿随机测地线和体积渐近线的偏移的部分和
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发表时间：
2014
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通讯作者：
Vaibhav Gadre
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作者：
Vaibhav Gadre
对于SL（2，R）中的非均匀格，我们考虑了相应有限面积双曲曲面或轨道上随机测地线尖邻域的偏移。我们证明了一个包含这些漂移的部分和的强律。推广了连分式的Diamond和valer定理。在Teichmuller环境下，我们考虑了二次微分模空间上SL（2，R）作用的不变测度。通过Eskin和Mirzakhani的工作，这些度量在二次微分层的仿射不变子流形上得到了支持。对于一个关于SL（2，R）不变测度的Teichmuller测地线随机，我们研究了它在相关仿射不变子流形的薄部分上的偏移。在不变测度的正则性假设下，我们证明了涉及这些漂移的某些部分和的类似强定律。这些定律的极限与薄壁部分的体积渐近有关。根据西格尔-维奇理论，这些是由各种西格尔-维奇常数给出的。作为一个直接的结果，我们表明，相对于Masur-Veech测量，度量这个词在Teichmuller测地线上的增长速度比tlogt更快。
For a non-uniform lattice in SL(2,R), we consider excursions in cusp neighborhoods of a random geodesic on the corresponding finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. In the Teichmuller setting, we consider invariant measures for the SL(2,R) action on the moduli spaces of quadratic differentials. By the work of Eskin and Mirzakhani, these measures are supported on affine invariant submanifolds of a stratum of quadratic differentials. For a Teichmuller geodesic random with respect to a SL(2,R)-invariant measure, we study its excursions in thin parts of the associated affine invariant submanifold. Under a regularity hypothesis for the invariant measure, we prove similar strong laws for certain partial sums involving these excursions. The limits in these laws are related to the volume asymptotic of the thin parts. By Siegel-Veech theory, these are given by various Siegel-Veech constants. As a direct consequence, we show that the word metric grows faster than T log T along Teichmuller geodesics random with respect to the Masur-Veech measure.