Quaternion Julia sets reimagined: The Garodia-Uddin iteration approach

In recent years, significant research has been conducted on using various iteration schemes derived from fixed-point theory to generate Mandelbrot and Julia sets in the complex space. Building on these advancements, this work explores the application of the Garodia-Uddin iteration scheme to construct Julia sets of q↦qk+c in the quaternion space. Specifically, we establish the escape criterion for the Garodia-Uddin orbit and analyze the symmetry of the Julia set for the even values of k. Additionally, we provide and discuss 2D and 3D graphical examples of sets generated from the Garodia-Uddin iteration scheme. Furthermore, we investigate the effect of a key parameter in the proposed approach on the average escape time, non-escaping area index, and fractal dimension for 2D cross sections of quaternion Julia sets of varying degrees. Finally, the Julia sets obtained are compared to the ones which come from the Picard-Mann iteration scheme.