2n边形上鞍连接的斜率间隙分布

Veech曲面的斜率间隙分布

Slope Gap Distributions of Veech Surfaces

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。

The gap distribution of slopes on the golden L

We give an explicit formula for the limiting gap distribution of slopes of saddle connections on the golden L, or any translation surface in its SL(2, R)-orbit, in particular the double pentagon. This is the first explicit computation of the distribution of gaps for a flat surface that is not a torus cover.

The distribution of gaps for saddle connections on the octagon

We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the translation surface has multiple cusps. We also show how to parametrize a Poincaré section for the horocycle flow on $SL(2,\mathbb{R})/SL(X,ω)$ associated to an arbitrary Veech surface $(X, ω)$. As a corollary, we show that the associated gap distribution is piecewise real analytic.

Gaps of saddle connection directions for some branched covers of tori

We compute the gap distribution of directions of saddle connections for two classes of translation surfaces. One class will be the translation surfaces arising from gluing two identical tori along a slit. These yield the first explicit computations of gap distributions for non-lattice translation surfaces. We show that this distribution has support at 0 and quadratic tail decay. We also construct examples of translation surfaces in any genus $d>1$ that have the same gap distribution as the gap distribution of two identical tori glued along a slit. The second class we consider are twice-marked …

双七边形的斜率间隙分布及确定获胜向量的算法

Slope gap distribution of the double heptagon and an algorithm for determining winning vectors

本文提出寻找获胜完整向量的算法，用于计算任意Veech曲面的斜率间隙分布，并应用于正则双七边形，扩展了金色L曲面的相关工作。

平移曲面上的点过程与间隙统计

Point Processes and Gap Statistics for Translation Surfaces

该研究从点过程视角探究平移曲面上鞍连接方向的行为，定义相关方向与斜率点过程，证明几乎所有平移曲面上方向点过程经缩放后依分布收敛于斜率点过程，还得到间隙分布的极限性质及主层中斜率点过程收敛于泊松点过程的结论。