Generalized Mandelbrot Sets of a Family of Polynomials P n z = z n + z + c ; n ≥ 2

Abstract

In this paper, we study the general Mandelbrot set of the family of polynomials $\left\{P_n\left(z\right)=z^n+z+c;\left(n\ge 2\right)\right\}$ , denoted by GM( $P_n$ ). We construct the general Mandelbrot set for these polynomials by the escaping method. We determine the boundaries, areas, fractals, and symmetry of the previous polynomials. On the other hand, we study some topological properties of $\text{GM}\left(P_n\right)$ . We prove that $\text{GM}\left(P_n\right)$ is bounded and closed; hence, it is compact. Also, we characterize the general Mandelbrot set as a union of basins of attraction. Finally, we make a comparison between the properties of famous Mandelbrot set $M\left(z^2+c\right)$ and our general Mandelbrot sets.