A study of Mandelbrot and Julia Sets via Picard-Thakur iteration with s-convexity.

Abstract

Nowadays, many researchers are employing various iterative techniques to analyse the dynamics of fractal patterns. In this paper, we explore the formation of Mandelbrot and Julia sets using the Picard-Thakur iteration process, extended with s-convexity. To achieve this, we establish an escape criterion using a complex polynomial of the form [Formula: see text], where k ≥ 1 and x, c ∈ ℂ. Based on our proposed algorithms, we provide graphical illustrations of the Mandelbrot and Julia sets. Additionally, we extend our research to examine the relationship between the sizes of Mandelbrot and Julia sets and the iteration parameters, utilising some well-known methods from the literature.