二维或三维单形的动力系统
Dynamical Systems of Simplices in Dimension 2 or 3
摘要：
设T_0=（A_0，.，A_d）是d-单形，G_0是其质心，S是其外接球，O是S的中心。设（B_i）为S与直线（G_0A_i）的交点，T_1为d-单形（B_1，.，B_d），G_1为质心。通过迭代这个构造，构造了一个具有质心（G_i）的d-单形（T_i）的动力系统。当d=2或3时，我们证明了序列（OG_i）是递减的，并趋于0.我们考虑序列（T_{2i}）_i和（T_{2i+1}）_i ;当d=2时，它们以至少二次速度收敛到两个等边三角形;当d=3时，它们以至少几何速度收敛到两个等腰四面体。在最后一种情况下，我们给出了极限形式的边长的显式表达式。我们还证明了，如果T_0是一个平面循环四边形，那么（T_n）至少以几何速度收敛到一个矩形，或最终以三次速度收敛到一个正方形。证明主要是代数和使用Grobner基础计算。
英文摘要：
Let T_0=(A_0,..,A_d) be a d-simplex, G_0 its centroid, S its circumsphere, O the center of S. Let (B_i) be the points where S intersects the lines (G_0A_i), T_1 the d-simplex (B_1,..,B_d), and G_1 its centroid. By iterating this construction, a dynamical system of d-simplices (T_i) with centroids (G_i) is constructed. For d=2 or 3, we prove that the sequence (OG_i) is decreasing and tends to 0. We consider the sequences (T_{2i})_i and (T_{2i+1})_i ; for d=2 they converge to two equilateral triangles with at least quadratic speed ; for d=3 they converge to two isosceles tetrahedra with at least geometric speed. In this last case, we give an explicit expression of the lengths of the edges of the limit form. We show also that if T_0 is a planar cyclic quadrilateral then (T_n) converges to a rectangle with at least geometric speed or eventually to a square with a speed that is conjectured as cubic. The proofs are largely algebraic and use Grobner basis computations.
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