Symmetry and dynamics of Chebyshev's method

TL;DR

This paper investigates the symmetry and dynamics of Chebyshev's method for root-finding, establishing conditions under which its symmetry set matches that of the polynomial and analyzing the structure of its Julia and Fatou sets.

Contribution

It demonstrates that Chebyshev's method satisfies the Scaling theorem, explores when its symmetry set equals that of the polynomial, and characterizes the Julia and Fatou sets in specific cases.

Findings

Julia set of Chebyshev's method cannot be a line

Symmetry sets coincide for certain polynomial classes

Julia set is connected and locally connected in studied cases

Abstract

The set of all holomorphic Euclidean isometries preserving the Julia set of a rational map $R$ is denoted by $\Sigma R$. It is shown in this article that if a root-finding method $F$ satisfies the Scaling theorem, i.e., for a polynomial $p$, $F_p$ is affine conjugate to $F_{\lambda p \circ T}$ for every nonzero complex number $\lambda $ and every affine map $T$, then for a centered polynomial $p$ of order at least two (which is not a monomial), $\Sigma p\subseteq \Sigma F_p$. As the Chebyshev's method satisfies the Scaling theorem, we have $\Sigma p \subseteq \Sigma {C_p}$, where $p$ is a centered polynomial. The rest part of this article is devoted to explore the situations where the equality holds and in the process, the dynamics of $C_p$ is found. We show that the Julia set $\mathcal{J}(C_p)$ of $ C_p$ can never be a line. If a centered polynomial $p$ is (a) unicritical, (b) having exactly two roots with the same multiplicity, (c) cubic and $\Sigma p$ is non-trivial or (d) quartic, $0$ is a root of $p$ and $\Sigma p $ is non-trivial then it is proved that $\Sigma p = \Sigma C_p$. It is found in all these cases that the Fatou set $\mathcal{F}(C_p)$ is the union of all the attracting basins of $C_p$ corresponding to the roots of $p$ and $\mathcal{J}(C_p)$ is connected. It is observed that $\mathcal{J}(C_p)$ is locally connected in all these cases.