Recuttings of polygons
Recuttings of polygons
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DOI：
10.1007/bf01085984
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发表时间：
1993-04
影响因子：
0.4
通讯作者：
V. E. Adler
中科院分区：
文献类型：
--
作者：
V. E. Adler
让我们考虑复平面上的N&3个点？91，…，VN，指数模N。在每一步上都可以反映其中之一，比如说Vk，相对于连接其相邻的Vk-1和Vk+L的线的中垂线，直观地说，一个人从多边形v~…截断一个角度。VN，把它倒过来，然后粘回去。由此，定义了一个N值映射(对应)R：Cn-+Cn。问题是研究在这样的重剪作用下顶点的动力学。作为一个例子，让我们来考虑一些微不足道的案例。对于小N，它们提供了动力学的完整描述。1.如果所有边都相等，则顶点保持固定。2.如果所有顶点都位于相同的圆或线上(例如，N=3的情况)，则它们将永远位于该圆或线上。如果重新裁剪是按循环顺序进行的，那么在N-1步之后，我们得到原始的多边形，以某个恒定的角度旋转。两个同心圆或平行线(例如，N=4)中的一个点的数目为偶数，而另一个同心圆或平行线上的奇点为奇点的情况也可以类似地考虑。对于N>；5来说，动态变得更加复杂。数值实验表明，在重切群的作用下，顶点在某点E处以共同的中心填充一个或两个环空。如果重切是以循环的顺序进行的，则填充过程以规则的方式进行，类似于环面上的缠绕。在特殊情况下，顶点沿闭合曲线移动。对于较大的N(N~100)，正多边形在多个位置受扰动时的演化表现为孤子行为。虽然问题还没有完全解决，但我们提出了一种方法，提供了对应的不变量，并建立了与另一对应的关系，该关系可由离散版本的Liouville定理[1]积分。后一种对应关系与Dressing Chain理论[2，3]以及第4和第5 Painlev方程[4，5]的BiCklund变换有关。
Let us consider N> _ 3 points? 91,..., VN, indexed modulo N, on the complex plane. On each step one can reflect one of them, say, vk, with respect to the midperpendicular to the line connecting its neighbors vk-1 and vk+ l• Intuitively, one cuts off an angle from the polygon v~... VN, turns it upside down, and glues it back. Thus, an N-valued mapping (correspondence) R: CN-+ CN is defined. The problem is to investigate the dynamics of the vertices under the action of such recuttings. As an example, let us consider some trivial cases. For small N they provide a complete description of the dynamics. 1. If all sides are equal, then the vertices remain fixed. 2. If all vertices lie on the same circle or line (eg, this is the case for N= 3), then they stay on it forever. If the recuttings are carried out in a cyclic order, then after N-1 steps we obtain the original polygon, turned by some constant angle.3. The case in which the number of points is even and the even vertices lie on one and the odd vertices on the other of two concentric circles or parallel lines (eg, for N= 4) can be considered similarly. For N> _ 5 the dynamics becomes more complicated. Numerical experiments show that under the action of the recutting group the vertices fill one or two annuli with common center at some point E. If the recuttings are carried out in a cyclic order, then the filling process goes in a regular manner, similarly to a winding on the torus. In exceptional cases the vertices move along closed curves. For large N (N~ 100) the evolution of a regular polygon perturbed at several places displays solitonic behavior. Although the problem is not solved completely, we propose a method that provides invariants of the correspondence, and we establish a relationship with another correspondence, which is integrable by the discrete version of Liouville's theorem [1]. The latter correspondence is connected with the theory of dressing chains [2, 3] and with the Bi~ cklund transformations for the 4th and 5th Painlev~ equations [4, 5].