双台球，扭曲地图，和冲击振荡器
Dual billiards, twist maps, and impact oscillators
摘要：
本文将扭曲映射理论的技巧应用于平面上严格凸闭曲线G上对偶台球产生的环面映射。证明了当G上存在一点其曲率半径为零或不连续时，在G附近不存在不变圆。此外，当曲率半径不是$C^1$时，存在轨道收敛到G点的例子。如果曲率半径的导数是有界的，这样的轨道就不可能存在。本文的最后一节涉及的影响振荡器的动力学是相同的双台球地图。附录是关于台球和对偶台球中不变圆反问题的联系的一个注记。
英文摘要：
In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex closed curve G in the plane. It is shown that there do not exist invariant circles near G when there is a point on G where the radius of curvature vanishes or is discontinuous. In addition, when the radius of curvature is not $C^1$ there are examples with orbits that converge to a point of G. If the derivative of the radius of curvature is bounded, such orbits cannot exist. The final section of the paper concerns an impact oscillator whose dynamics are the same as a dual billiards map. The appendix is a remark on the connection of the inverse problems for invariant circles in billiards and dual billiards.
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