JULIA SET GENERATOR

The Mandelbrot set is the region of constants $c$, which determines the connectedness of Julia sets. Holds, that if $c$ lies in the Mandelbrot set then the Julia set generated for the same $c$ is connected. On the other hand, if $c$ lies outside the Mandelbrot set then the Julia set is completely disconnected. The property that Julia set comes only in one of two cases and nothing in between is called the fundamental dichotomy.
Because for each $c$ a different Julia set is generated then Mandelbrot set could be considered as the map for the Julia sets.

The method colors each pixel (initial value $z_{0}$) of the screen with the shade of some color depending on the threshold radius $r(c)$. If the magnitude of any $z_{k}$ in the orbit of the initial value $z_{0}$ doesn't exceed $r(c)$ after the maximum number of iterations, which is chosen at the start of the whole process, $z_{0}$ is colored black. However, if $z_{k}$ exceeds $r(c)$ before the maximum number of iterations, the algorithm detects $k$ and determines the shade of color according to $k$. The smallest $k$ matches the lightest shade.

The process of the algorithm of this method is very similar to the escape time algorithm at first, except for the fact that is unnecessary to determine the number of iterations, but the derivative of the iteration function in the value $z_{k}$ after each iteration is calculated and then according to that the pixels of the screen are coloured. For each point outside the set is possible to calculate an estimate of the shortest distance between itself and the boundary of the set by a derivative of the iteration function. The points inside the set are colored black, and for contrast, the points with the smallest distance from the boundary have the lightest shade which decreases as the position of the initial value is further from the set.

In the beginning, it is important to say that in this method Julia set is by the exact definition only the boundary of the whole set.
Julia set is invariant under iteration, which is an important fact in this method. The resulting shape of the set is generated by using the iteration of the inverse function to $f(z) = z^{2} + c$ i.e. $f^{-1}(z) = \pm \sqrt{z\, -\,c}$. It also uses the fact that the Julia set for f−1(c)f−1(c) is an attractor for orbits of the surrounding initial values, and also that the repelling periodic points and their preimages are dense in the Julia set. It means that the Julia set has infinitely many repelling periodic points and moreover in an arbitrarily small neighbourhood of any point in the Julia set exists repelling periodic point.
A periodic point with a period one is a fixed point and thus is sufficient to select a repelling fixed point as the initial value $z_{0}$ of the inverse iteration process. All results obtained by each inverse iteration are rendered on the screen and then at the end, the Julia set boundary shape is created.

The connection between J and M can be practically observed in the interactive window below. After selecting one of the methods and choosing a constant $c$ by mouse click anywhere on the screen with the Mandelbrot set, the Julia set is generated. If the higher accuracy is needed, there is a box in the upper left corner showing the exact position of the cursor.