Fractal generation of Julia, Mandelbrot, and multicorn sets with an AA-Iterative algorithm

Fractals play an important role in understanding natural phenomena, such as rainbows, trees, mountains, plants, lightning, ferns, and snowflakes. In this paper, a novel AA-Iteration process is introduced that converges faster than traditional methods for generating fractals, along with an improved escape criterion for polynomials of the form xk+a. Utilizing this criterion, three algorithms have been developed to generate key classes of fractals, including Julia sets, Mandelbrot sets, and multicorns. The proposed approach extends existing literature by investigating how variations in the iteration parameters α, β, and γ lead to marked differences in the fractal images, with AA-iteration results differing distinctly from those produced by Picard-Thakur and Thakur iterations. Furthermore, numerical measures such as Average Escape Time (AET), Nonescaping Area Index (NAI), and Fractal Dimension (FD) are analyzed, revealing a nonlinear and complex dependency on the iteration parameters. Finally, to showcase practical utility of our AA-Iterative algorithm escape criterion by to a synthetic two dimensional velocity field, effectively partitioning laminar and turbulent regions with superior accuracy and speed compared to classical algorithms.