标题： 非毕晓普周期轨道在对称台球中的出现 显示英文标题

标题： Non-Birkhoff periodic orbits in symmetric billiards

摘要： 我们研究了对称凸平面台球中的非Birkhoff周期轨道。 我们的主要结果提供了一个关于存在具有预定最小周期、旋转数和时空对称性的此类轨道的定量准则。 我们利用这一准则找到了对称台球拥有无限多非Birkhoff周期轨道的充分条件。 由此可知，圆形台球的任意小解析扰动都具有任何有理旋转数且周期任意长的非Birkhoff周期轨道。 我们还将一个已知的椭圆台球的结果推广到了其他$\mathbb{D}_2$-对称台球上。 最后，我们提供了可以用来数值计算和可视化我们证明存在的非Birkhoff周期轨道的Matlab代码。 显示英文摘要

摘要： We study non-Birkhoff periodic orbits in symmetric convex planar billiards. Our main result provides a quantitative criterion for the existence of such orbits with prescribed minimal period, rotation number, and spatiotemporal symmetry. We exploit this criterion to find sufficient conditions for a symmetric billiard to possess infinitely many non-Birkhoff periodic orbits. It follows that arbitrarily small analytical perturbations of the circular billiard have non-Birkhoff periodic orbits of any rational rotation number and with arbitrarily long periods. We also generalize a known result for elliptical billiards to other $\mathbb{D}_2$-symmetric billiards. Lastly, we provide Matlab codes which can be used to numerically compute and visualize the non-Birkhoff periodic orbits whose existence we prove analytically.