Exploring fractal geometry through Das–Debata iteration: A new perspective on Mandelbrot and Julia Sets

Mandelbrot sets, and Julia sets are two key concepts in fractal geometry. A useful tool to generate these fractal sets is the escape time algorithm, which is an iterative scheme based on a complex function paired with some well-known fixed point iteration methods. The existing literature primarily generates Mandelbrot and Julia sets through the use of a complex polynomial or a complex rational function. This work presents a novel technique in light of this fractal generation process. We employ an iterative method that involves two operators, which was introduced to identify common fixed points, i.e., the Das–Debata iteration. The escape criterion for the Das–Debata iteration is derived by utilizing a complex polynomial and rational function and by altering the positions of the functions in the iteration process. Some illustrative examples of Mandelbrot and Julia sets obtained through the proposed iterative method are provided. We compare the generated fractals by the two orders of the functions in the iterative method. To determine the dependence of fractal sets on the iteration parameters, we analyze two numerical measures: the average escape time and the non-escaping area index. In the two considered orders of the functions, it is revealed that their dependence is nonlinear.