外接触台球
Outer Contact Billiards
AI总结：
该研究提出外接触台球作为外辛台球的奇维对应物，证明其可生成接触同胚，实射影三维空间中二次曲面对应完全可积且无3周期轨道，仅一个二次表有4周期轨道。
AI中文摘要：
我们引入外接触台球，作为外辛台球的奇维对应物。此前，外台球仅在偶维辛向量空间中被研究。通过对外辛台球进行射影化，我们得到外接触台球，其中仿射中点条件可归结为其射影类似物，即调和共轭。我们证明外接触台球可生成接触同胚。对于实射影三维空间中的二次曲面，我们证明该对应是完全可积的：其定义域由不变二次曲面叶状分割，在每一叶上，动力学由显式线性变换的迭代决定。我们还建立了关于周期轨迹的两个刚性结果：外接触台球不存在3周期轨道，且在二次表中，仅有一个存在4周期轨道。
英文摘要：
We introduce outer contact billiards as an odd dimensional counterpart to outer symplectic billiards. Until now, outer billiards have only been considered in even dimensional symplectic vector spaces. By projectivizing outer symplectic billiards, we obtain outer contact billiards, where the affine midpoint condition descends to its projective analog, namely harmonic conjugation. We prove that outer contact billiards generate contactomorphisms. For quadratic surfaces in $\mathbb{RP}^3$, we show that the correspondence is completely integrable: its domain is foliated by invariant quadrics and, on each leaf, the dynamics is determined by the iteration of an explicit linear transformation. We also establish two rigidity results for periodic trajectories: outer contact billiards admit no 3-periodic orbits and, among quadratic tables, only one admits 4-periodic orbits.
