Derived system and dual sequence of a barypolygonal sequence -- Part 1

Abstract

This study continues three recent papers in which barypolygonal sequences have been defined and their properties of convergence demonstrated. Any barypolygonal sequence $\mathcal{B}$ of a finite set $\mathcal{A}$ comprising $p\ge 2$ points of any finite dimensional affine space can be used in order to define recurrently a definite sequence of barypolygonal sequences starting with $\mathcal{B}$. This sequence $(\mathcal{B}^{(m)})_{M\in N}$, called sequence of B's derivatives, is determined by real sequences that are solutions of a non linear recurrent system $(S)$: the barypolygonal derived system of $\mathcal{B}$. Each term of the sequence $(\mathcal{B}^{(m)})_{M\in N}$ converges toward a point $G_m$. The sequence $(G_m )_{m\in N}$ is the dual sequence of B. The convergence of the latter and the properties of the derived system are here investigated for any $p$ if $\mathcal{B}$ is regular and in any case if $p\in{2;3}$.