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

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 $2$.