具有高阶极点的平面上鞍形连接的计数
Counting saddle connections in flat surfaces with poles of higher order
摘要：
在紧致黎曼曲面上对应于$k$ -微分的平坦曲面是有限面积的，只要没有$k$阶或更高阶的极点。我们用\textit{flat surfaces with poles of higher order}表示那些具有由$k$ -微分定义的平坦结构的曲面，该微分具有至少一个阶为至少$k$的极点。具有高阶极点的平面具有与通常的有限面积平面不同的几何和动力学性质。特别地，它们可以具有有限数量的鞍形连接。我们给出了鞍形连接个数的上、下界和相关量。在$k=1\ or\ 2$的情况下，我们提供了一个组合的地层，可以有一个无限数量的鞍形连接的表征。
英文摘要：
Flat surfaces that correspond to $k$-differentials on compact Riemann surfaces are of finite area provided there is no pole of order $k$ or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat structures defined by a $k$-differential with at least one pole of order at least $k$. Flat surfaces with poles of higher order have different geometrical and dynamical properties than usual flat surfaces of finite area. In particular, they can have a finite number of saddle connections. We give lower and upper bounds for the number of saddle connections and related quantities. In the case $k=1\ or\ 2$, we provide a combinatorial characterization of the strata for which there can be an infinite number of saddle connections.
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