Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves
Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves
复制标题
具有有限数量不变曲线上的标准位移动力学的外台球
作者：
M. Bialy;A. Mironov;Lior Shalom
摘要我们给出了一个优美的凸平面曲线的显式例子，使得外台球具有给定有限条不变曲线。此外，这些曲线上的动力学是标准位移。这个例子可以看作是所谓的古特金台球桌的外部模拟。我们在两条不变曲线之间的区域中检验了这些台球的全可积性。接下来，我们对该区域的动态进行了计算机模拟。乍一看，动力学看起来是规则的，但通过放大图像，我们可以看到双曲周期轨道附近的混沌行为成分。我们认为这是一个有用的几何例子，说明扭曲映射的规则行为和混沌行为共存。
Abstract We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard shift. This example can be considered as an outer analog of the so-called Gutkin billiard tables. We test total integrability of these billiards, in the region between the two invariant curves. Next, we provide computer simulations on the dynamics in this region. At first glance, the dynamics looks regular but by magnifying the picture we see components of chaotic behavior near the hyperbolic periodic orbits. We believe this is a useful geometric example for coexistence of regular and chaotic behavior of twist maps.