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

TL;DR

This paper investigates the convergence properties of barypolygonal sequences and their derived dual sequences, extending previous work and analyzing the behavior for sequences with two or three points in finite-dimensional affine spaces.

Contribution

It introduces the derived system and dual sequence of barypolygonal sequences, providing new convergence results and properties for these sequences in finite-dimensional affine spaces.

Findings

Sequences converge to a point G_m.

Dual sequences also converge under certain conditions.

Properties of the derived system are characterized.

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 $p\in{2;3}$.