论文标题
计数鞍座连接
Counting pairs of saddle connections
论文作者
论文摘要
我们表明，几乎在每个翻译表面上，与跨产品的有界大小的鞍座连接的数量具有渐近生长，例如$ c r^2 $，其中常数$ c $仅取决于地层的区域和连接的组件。证明技术结合了对鞍座连接计数的经典结果和siegel-deech变换为$ l^2 $的关键结果。为了捕获有关鞍座连接对的信息，我们认为对横乘积的有界幅度对，因为这对组合可以通过在地球流量下具有均衡的纤维组近似。在晶格表面的情况下，跨产品的小界面幅度相当于计数平行的马鞍连接对，在$ c r^2 $中，$ c r^2 $，其中$ c $在这种情况下取决于给定的晶格表面。
We show that for almost every translation surface the number of pairs of saddle connections with bounded magnitude of the cross product has asymptotic growth like $c R^2$ where the constant $c$ depends only on the area and the connected component of the stratum. The proof techniques combine classical results for counting saddle connections with the crucial result that the Siegel--Veech transform is in $L^2$. In order to capture information about pairs of saddle connections, we consider pairs with bounded magnitude of the cross product since the set of such pairs can be approximated by a fibered set which is equivariant under geodesic flow. In the case of lattice surfaces, small bounded magnitude of the cross product is equivalent to counting parallel pairs of saddle connections, which also have a quadratic growth of $c R^2$ where $c$ depends in this case on the given lattice surface.
