Abstract

Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this article, we study counting problems that result from focusing on properties of the square torus one by one. After drawing insights from experimental evidence, we consider the implications between these properties and as well as the frequencies of these properties in each stratum of translation surfaces.

中文翻译：

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方形平铺表面的统计：对称性和短环

摘要

方块曲面是一类平移曲面，在几何学和动力学中特别受关注，因为作为方形环面的覆盖面，它们具有一些简单性和结构。在本文中，我们将重点研究正方形环面的性质而产生的计数问题。在从实验证据中得出见解后，我们考虑了这些属性之间的含义以及这些属性在翻译表面的每个层中的频率。