发表机构
Max-Planck-Institut für Mathematik in den Naturwissenschaften（马克斯·普朗克自然科学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究环面上非欧几里得几何产生的平面铺层上的辛铺层台球动力系统，通过定义相关仿射结构的开子集，分析其动力学并证明发散轨迹的稳定性结果，引入适配非欧几里得仿射铺层的稀薄性概念。
AI 中文摘要
2023年，Richard Schwartz引入了一种新的动力系统，它是两种常见台球——铺层台球与辛台球的结合。本文研究在由环面上非欧几里得几何产生的平面铺层上运行的该动力系统。我们通过Oliver Baues和William Goldman的工作，回顾环面上平坦共形结构的仿射类似物，并定义该形变空间的一个开子集，对应平面完全仿射铺层的标记。我们精确给出该定义，并对铺层的对称性提出代数条件以定义它。随后我们分析在这类铺层上运行的辛铺层台球的动力学，研究该系统的长期动力学，以证明关于发散轨迹的稳定性结果。这种发散根据来自铺层对称群的几何不变量定义。我们认为，从某种意义上说，这种发散是因为在铺层中向远处移动时，非欧几里得铺层的“瓦片”变得“稀薄”所致。为此，我们引入一种适配于非欧几里得仿射铺层的稀薄性概念。
英文摘要
In 2023, Richard Schwartz introduced a new dynamical system which is a marriage of two types of familiar billiards, tiling billiards and symplectic billiards. In this paper we investigate this dynamical system played on tilings of the plane which arise from non-Euclidean geometries on the torus. We review the affine analogue of the flat conformal structures on the torus through the work of Oliver Baues and William Goldman, and define an open subset of this deformation space corresponding to markings of complete affine tilings of the plane. We make this definition precise, and provide algebraic conditions on the symmetries of the tiling to define it. We then analyze the dynamics of symplectic tiling billiards played on these types of tilings and investigate the long-term dynamics of the system to prove a stability result concerning divergent trajectories. The divergence is defined in terms of geometric invariants arising from the tiling symmetry group. We argue that in some sense this divergence is a consequence of the tiles of a non-Euclidean tiling becoming 'thin' as one moves far away in the tiling. To do so we introduce a notion of thinness that is well adapted to the non-Euclidean affine tilings.
Comments46 pages, 30 figures v2: updated pictures to be color blind friendly, fixed various typos particularly definition of rationality, added more exposition
