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

TL;DR

This paper analyzes the complex dynamics of Stirling's iterative root-finding method for rational and polynomial functions, revealing properties of fixed points, Julia sets, and the method's behavior on M"{o}bius maps.

Contribution

It provides new insights into the dynamical behavior of Stirling's method, including fixed point classification and the structure of Julia sets, which were previously not well understood.

Findings

Zeroes are superattracting fixed points for rational functions.

Julia set of Stirling's method for polynomials is connected.

Number of Herman rings for M"{o}bius maps is at most two.

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 M\"{o}bius map is investigated here. We have shown that the possible number of Herman rings of this method for M\"{o}bius map is at most $2$.