The convergence of a sequence of iterated polygons: a discrete combinatorial analysis (Q2011066)

Let \(\lambda\in(0,1)\) be an arbitrarily fixed parameter. One considers the dynamical system \(T\) that associates to each planar polygon \(\mathcal{P}\) (obtained by joining the points \(v_{1},\dots,v_{N},\) in this order) the polygon \(\mathcal{Q}\) whose vertices \(w_{1},\dots,w_{N}\) are defined as follows: \(w_{1}=(1-\lambda)v_{1}+\lambda v_{2},\) \(w_{2}=(1-\lambda)v_{2}+\lambda v_{3},\dots,\) and \(w_{N}=(1-\lambda)v_{N}+\lambda v_{0}.\) It is proved that each trajectory has a limit point, the centroid of the starting polygon. A similar result holds in the context of spaces with global nonpositive curvature (CAT(0) spaces). For the entire collection see [Zbl 1415.39001].