论文标题
Pentagram地图和无穷小单曲的极限点
The limit point of the pentagram map and infinitesimal monodromy
论文作者
论文摘要
Pentagram地图将平面Polygon $ P $带到Polygon $ p'$，其顶点是连续短对角线的交点$ P $。在此地图下，凸多边形的轨道是一系列多边形序列，该序列将指数收敛到点。此外，正如Glick最近证明的那样，该极限点的坐标可以作为与多边形相关的某些操作员的特征向量计算。在本文中，我们表明，Glick的操作员可以解释为多边形的无穷小单曲。也就是说，存在某种自然的无限无限扰动，多边形再次是多边形，但通常不封闭。 Glick的操作员的措施是这种扰动的多边形在多大程度上没有关闭。
The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. The orbit of a convex polygon under this map is a sequence of polygons which converges exponentially to a point. Furthermore, as recently proved by Glick, coordinates of that limit point can be computed as an eigenvector of a certain operator associated with the polygon. In the present paper we show that Glick's operator can be interpreted as the infinitesimal monodromy of the polygon. Namely, there exists a certain natural infinitesimal perturbation of a polygon, which is again a polygon but in general not closed; what Glick's operator measures is the extent to which this perturbed polygon does not close up.
