Topological properties of the immediate basins of attraction for the secant method

TL;DR

This paper investigates the topological structure of immediate basins of attraction for the secant method applied to polynomials, revealing the existence of 4-cycles on their boundaries and conditions for simple connectivity.

Contribution

It provides new insights into the topology of immediate basins of attraction for the secant method, including the existence of boundary 4-cycles and criteria for simple connectivity.

Findings

Existence of a 4-cycle in the boundary of the immediate basin

Conditions under which the immediate basin is simply connected

Topological properties of basins for roots of real polynomials

Abstract

We study the discrete dynamical system defined on a subset of $R^2$ given by the iterates of the secant method applied to a real polynomial $p$. Each simple real root $\alpha$ of $p$ has associated its basin of attraction $\mathcal A(\alpha)$ formed by the set of points converging towards the fixed point $(\alpha,\alpha)$ of $S$. We denote by $\mathcal A^*(\alpha)$ its immediate basin of attraction, that is, the connected component of $\mathcal A(\alpha)$ which contains $(\alpha,\alpha)$. We focus on some topological properties of $\mathcal A^*(\alpha)$, when $\alpha$ is an internal real root of $p$. More precisely, we show the existence of a 4-cycle in $\partial \mathcal A^*(\alpha)$ and we give conditions on $p$ to guarantee the simple connectivity of $\mathcal A^*(\alpha)$.