Some Properties of the Mandelbrot Sets M(Q_α)

Abstract

The aim of this work is to study the concept of the parameter plane, so we are interested in introducing the Mandebrot set,as well as the Mandelbrot set is a fractals ,whereas, if we zoom in on any small piece in the border regions of the shape, we notice that it is similar to the original shape, that is the Mandelbrot set, with special topological specifications, and we introduce of some properties of the Mandelbrot sets for the map in the form , where is a complex constant, and we prove that the conjecture is the Duadi-Hubbard theorem . By studying of the parameter plane that the Mandelbrot set is a component form in one of its details in the dynamical plane it is the Julia set , in other words every small detail of the Mandelbrot set represents the Julia set, so there is more than one characteristic of the Julia set, we will limit ourselves only to presenting the connection of the Julia set for .