瓷砖台球和Dynnikov的螺旋面
Tiling billiards and Dynnikov's helicoid
摘要：
这里有两个问题。首先，了解动态的瓷砖台球在一个循环的四边形周期性瓷砖。其次，描述属$3$的中心对称次表面$S \subset \mathbb{T}^3$的平面截面的连通分量的拓扑。在这份说明中，我们表明，这两个问题是相关的，通过最近提出的Ivan Dynnikov的螺旋结构。第二个问题是谢尔盖·诺维科夫（Sergei Novikov）提出的经典问题的一个特例。正如我们在本说明的最后所示，对一大类瓷砖台球（周期性局部可折叠的瓷砖台球）和诺维科夫高等属问题之间关系的探索似乎很有希望。
英文摘要：
Here are two problems. First, understand the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describe the topology of connected components of plane sections of a centrally symmetric subsurface $S \subset \mathbb{T}^3$ of genus $3$. In this note we show that these two problems are related via a helicoidal construction proposed recently by Ivan Dynnikov. The second problem is a particular case of a classical question formulated by Sergei Novikov. The exploration of the relationship between a large class of tiling billiards (periodic locally foldable tiling billiards) and Novikov's problem in higher genus seems promising, as we show in the end of this note.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
