Veech曲面的斜率间隙分布
Slope Gap Distributions of Veech Surfaces
AI总结：
本文证明了Veech平移曲面上鞍连接斜率的极限间隙分布始终是分段实解析的且非解析点有限，并通过Poincaré截面的显式参数化与首次返回映射的有限性结果，证明其尾部始终具有二次衰减。
摘要：
平移曲面的斜率间隙分布是衡量曲面上鞍连接方向的随机性的一种度量。已知Veech曲面（一种高度对称的平移曲面）的间隙分布是分段实解析的。然而，除此之外，目前关于斜率间隙分布的一般行为所知甚少，包括非解析点的数量或尾部情况。我们证明了Veech平移曲面上鞍连接斜率的极限间隙分布始终是分段实解析的，且具有有限个非解析点。我们通过对与任意Veech曲面SL(X,ω)相关联的SL(2,R)/SL(X,ω)上的horocycle流取一个Poincaré截面的显式参数化，并建立此流下首次返回映射的一个关键有限性结果来做到这一点。我们利用该有限性结果证明了Veech曲面的斜率间隙分布的尾部始终具有二次衰减。
英文摘要：
The slope gap distribution of a translation surface is a measure of how random the directions of the saddle connections on the surface are. It is known that Veech surfaces, a highly symmetric type of translation surface, have gap distributions that are piecewise real analytic. Beyond that, however, very little is currently known about the general behavior of the slope gap distribution, including the number of points of non-analyticity or the tail. We show that the limiting gap distribution of slopes of saddle connections on a Veech translation surface is always piecewise real-analytic with \emph{finitely} many points of non-analyticity. We do so by taking an explicit parameterization of a Poincaré section to the horocycle flow on $\text{SL}(2,\mathbb{R})/\text{SL}(X,ω)$ associated to an arbitrary Veech surface $\text{SL}(X,ω)$ and establishing a key finiteness result for the first return map under this flow. We use the finiteness result to show that the tail of the slope gap distribution of Veech surfaces always has quadratic decay.
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