Abstract

In this paper we investigate the systolic landscape of translation surfaces for fixed genus and fixed angles of their cone points. We furthermore study how the systoles of a translation surface relate to the systoles of its graph of saddle connections. This allows us to develop an algorithm to compute the systolic ratio of origamis in the stratum $\mathcal{H}(1,1)$. We compute the maximal systolic ratio of all origamis in $\mathcal{H}(1,1)$ with up to 67 squares. These computations support a conjecture of Judge and Parlier about the maximal systolic ratio in $\mathcal{H}(1,1)$.

中文翻译：

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平移面的收缩几何

摘要

在本文中，我们研究了固定属及其锥点的固定角度的平移表面的收缩景观。我们进一步研究了平移表面的收缩如何与其鞍连接图的收缩相关。这使我们能够开发一种算法来计算地层中折纸的收缩率$\mathcal{H}(1,1)$. 我们计算所有折纸的最大收缩比$\mathcal{H}(1,1)$最多 67 个方格。这些计算支持 Judge 和 Parlier 关于最大收缩比的猜想$\mathcal{H}(1,1)$.