Is the Julia set a fractal?

For some functions f(z) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following result on the iterations of a rational function: Theorem. Each of the Fatou domains has the same boundary, which consequently is the Julia set.

Is the Mandelbrot set a fractal?

As it turns out, the Mandelbrot set is not a fractal according to this definition, as its Hausdorff dimension and topological dimension are both 2. However, the boundary of the Mandelbrot set is a fractal, according to this definition. The boundary of a set of topological dimension 2 is, perhaps not surprisingly, 1.

What is the Julia set used for?

In general terms, a Julia set is the boundary between points in the complex number plane or the Riemann sphere (the complex number plane plus the point at infinity) that diverge to infinity and those that remain finite under repeated iteration of some mapping (function). The most famous example is the Mandelbrot set.

Are Julia sets infinite?

In general terms, a Julia set is the boundary between points in the complex number plane or the Riemann sphere (the complex number plane plus the point at infinity) that diverge to infinity and those that remain finite under repeated iteration of some mapping (function).

Is Julia set connected?

is the boundary of the filled-in set (the set of “exceptional points”). There are two types of Julia sets: connected sets (Fatou set) and Cantor sets (Fatou dust).

How many types of fractals are there?

three types
Classification of fractals There are three types of self-similarity found in fractals: Exact self-similarity — This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity.

What are Mandelbrot sets used for?

The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected and not computable.