预分形Koch雪花台球的相容周期杂化轨道序列
Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards
摘要：
Koch雪花KS是一条无处可微的曲线。具有边界KS的台球桌Omega（KS）先验地没有很好地定义。也就是说，人们不能先验地确定台球在桌子的边界中受到碰撞所经过的最小路径。正是这个问题使得欧米茄（KS）成为一个有趣但难以分析的表。在本文中，我们处理这个问题的近似（从内部）欧米茄（KS）由定义良好的（prefractal）合理多边形台球表欧米茄（KS_n）。首先证明了由有理台球Ω（KS_n）确定的平面S（KS_n）是单穿孔六角环面的分支覆盖。这样的结果，当结合的结果[Gut 2]，使我们能够定义一个序列的兼容轨道的prefractal台球。定义了一个预分形台球Omega（KS_n）的杂化轨道，并证明了预分形台球的每一个稠密轨道都是Omega（KS_n）的稠密杂化轨道.这个结果是关键，在获得一个拓扑二分法的一系列兼容的轨道。此外，我们确定了相容轨道序列是相容周期混合轨道序列的一个充分条件。然后，我们研究相容的周期混合轨道序列的极限行为。我们表明，平凡的限制，特别是（最终）常数序列的兼容混合轨道构成一个轨道的欧米茄（KS）。此外，我们表明，工会的两个适当选择的非平凡的多边形路径连接两个难以捉摸的极限点的科赫雪花。最后，我们将讨论如何可能将我们的结果推广到其他分形台球桌和如何理解的Veech组的prefractal台球的结构可能有助于确定“分形平面”自然与台球流。
英文摘要：
The Koch snowflake KS is a nowhere differentiable curve. The billiard table Omega(KS) with boundary KS is, a priori, not well defined. That is, one cannot a priori determine the minimal path traversed by a billiard ball subject to a collision in the boundary of the table. It is this problem which makes Omega(KS) such an interesting, yet difficult, table to analyze. In this paper, we approach this problem by approximating (from the inside) Omega(KS) by well-defined (prefractal) rational polygonal billiard tables Omega(KS_n). We first show that the flat surface S(KS_n) determined from the rational billiard Omega(KS_n) is a branched cover of the singly punctured hexagonal torus. Such a result, when combined with the results of [Gut2], allows us to define a sequence of compatible orbits of prefractal billiards. We define a hybrid orbit of a prefractal billiard Omega(KS_n) and show that every dense orbit of a prefractal billiard is a dense hybrid orbit of Omega(KS_n). This result is key in obtaining a topological dichotomy for a sequence of compatible orbits. Furthermore, we determine a sufficient condition for a sequence of compatible orbits to be a sequence of compatible periodic hybrid orbits. We then examine the limiting behavior of a sequence of compatible periodic hybrid orbits. We show that the trivial limit of particular (eventually) constant sequences of compatible hybrid orbits constitutes an orbit of Omega(KS). In addition, we show that the union of two suitably chosen nontrivial polygonal paths connects two elusive limit points of the Koch snowflake. Finally, we discuss how it may be possible for our results to be generalized to other fractal billiard tables and how understanding the structures of the Veech groups of the prefractal billiards may help in determining `fractal flat surfaces' naturally associated with the billiard flows.
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