Generation of Mandelbrot and Julia sets by using M-iteration process

In this manuscript, we introduce the M-iteration process for generating Mandelbrot and Julia sets. We establish an escape criterion for a polynomial of the form x k + 1 + c in the complex plane corresponding to the M-iteration process. Next, we present some graphical examples of Mandelbrot and Julia sets generated using the proven escape criterion and the escape-time algorithm. We also compare the images generated with the M, Mann, and Picard–Mann iterations. Moreover, we study the dependency between the iterations’ parameters and three numerical measures (the average escape time, non-escaping area index, and box-counting dimension) used in the literature. The results show that fractal images generated using the M-iteration are entirely different from those generated using the other two analysed iteration schemes. Moreover, the dependencies are highly non-linear and vary between the iterations. • Generalization of Mandelbrot and Julia sets using the M-iteration process. • A new escape criterion for the generalized sets has been derived. • Non-trivial and interesting images of the sets have been obtained. • Analysis of the impact of the iteration parameters on the sets has been performed.