On the dynamics of Stirling's iterative root-finding method for rational functions

TL;DR

This work investigates the dynamics of Stirling's iterative root-finding method $St_f(z)$ when applied to rational and polynomial functions, including Möbius maps. It establishes that the method does not satisfy the Scaling theorem and derives the fixed-point structure: for a rational $R(z)=\frac{P(z)}{Q(z)}$ with simple zeros, the zeros of $R$ are superattracting fixed points of $St_R$, while finite extraneous fixed points are rationally indifferent; for polynomials with simple zeros, the Julia set $J(St_P)$ is connected. The paper further analyzes quadratic unicritical polynomials, revealing symmetry of the dynamical plane and symmetric free critical orbits, with infinity acting as a rationally indifferent fixed point; in the Möbius-map setting, it shows case-dependent dynamics with disconnected Julia sets, bounds on Herman rings, and examples illustrating the variety of Fatou components. Overall, the results delineate how Stirling's method differs from classical Newton dynamics while highlighting persistent structural features across function classes, and they situate these dynamics relative to known root-finding dynamics.

Abstract

We study the dynamics of Stirling's iterative root-finding method $St_f(z)$ for rational and polynomial functions. It is seen that the Scaling theorem is not satisfied by Stirling's iterative root-finding method. We prove that for a rational function $R(z)$ with simple zeroes, the zeroes are the superattracting fixed points of $St_{R}(z)$ and all the extraneous fixed points of $St_{R}(z)$ are rationally indifferent. For a polynomial $p(z)$ with simple zeroes, we show that the Julia set of $St_p(z)$ is connected. Also, the symmetry of the dynamical plane and free critical orbits of Stirling's iterative method for quadratic unicritical polynomials are discussed. The dynamics of this root-finding method applied to Möbius map is investigated here. We have shown that the possible number of Herman rings of this method for Möbius map is at most $2$.

Figures (3)

• Figure 1: The dynamical plane of $St_f(z)$ for $f(z)=\frac{2z-1}{z}$
• Figure 2: The Julia set of $St_{f_1}(z)$ for $\beta=-4$ and $\beta=4$
• Figure 3: $(a)$ The disconnected Julia set of $St_M(z)$ for $a=0$ and $(b)$ The Julia set $St_M(z)$ for $a\neq 0$

Theorems & Definitions (25)

• Lemma 1.1: Riemann-Hurwitz formula
• Lemma 2.1
• proof
• Theorem 2.1
• proof
• Remark 2.1
• Example 2.1
• Theorem 2.2
• proof
• Remark 2.2