伞形台球的遍历性
Ergodicity in umbrella billiards
摘要：
我们研究了一个三参数的台球桌与圆弧边界的家庭。这些伞形台球可以被看作是两个参数的月亮和不对称柠檬台球的推广，其中后者的类包括新参数为$0$的实例。像这两个先前研究的类，对于某些参数伞台球表现出混沌行为的证据，尽管未能满足某些标准的散焦或分散，两个最好理解的机制，产生遍历性和双曲性。对于对应于非遍历柠檬和月亮台球的一些参数，新参数的小的增加将椭圆$2$ -周期点变换为高阶椭圆点的级联。随着新参数的增加，这些可能会稳定或消散。我们的特点的周期点，并提出新的遍历的例子的证据。
英文摘要：
We investigate a three-parameter family of billiard tables with circular arc boundaries. These umbrella-shaped billiards may be viewed as a generalization of two-parameter moon and asymmetric lemon billiards, in which the latter classes comprise instances where the new parameter is $0$. Like those two previously studied classes, for certain parameters umbrella billiards exhibit evidence of chaotic behavior despite failing to meet certain criteria for defocusing or dispersing, the two most well understood mechanisms for generating ergodicity and hyperbolicity. For some parameters corresponding to non-ergodic lemon and moon billiards, small increases in the new parameter transform elliptic $2$-periodic points into a cascade of higher order elliptic points. These may either stabilize or dissipate as the new parameter is increased. We characterize the periodic points and present evidence of new ergodic examples.
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