A New Perspective on Escape-Time Fractals: M-iteration with s -convexity in Complex Sets

Abstract

Fractals, with their intricate structures and self-similar patterns, continue to fascinate mathematicians and scientists alike. In this paper, we introduce a novel generalization for generating Mandelbrot, Julia, and Multicorn sets by extending the M-iteration process through incorporation of the s-convex combination. We first establish an escape criterion for complex polynomials of the form xk+1 + c, which forms the basis of our escape-time algorithms. These algorithms enable the construction of diverse sets of fractals under the proposed iteration scheme, unveiling new structural variations. Additionally, we present graphical representations to visualize these complex sets and conduct a detailed analysis of the interplay between the iteration parameters and two key numerical measures: the average escape time and the non-escaping area index. Our findings reveal highly nonlinear dependencies, shedding new light on the intricate dynamics of fractal evolution.