Is 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.
What is the difference between Julia set and Mandelbrot set?
The Mandelbrot set is the set of all c for which the iteration z → z2 + c, starting from z = 0, does not diverge to infinity. Julia sets are either connected (one piece) or a dust of infinitely many points. The Mandelbrot set is those c for which the Julia set is connected.
Why are Julia sets fractals?
For Julia sets, c is the same complex number for all pixels, and there are many different Julia sets based on different values of c. By smoothly changing c we can transform from one Julia set to another over time, creating animated fractal shapes.
Is Julia set connected?
A Julia set is either connected or disconnected, values of c chosen from within the Mandelbrot set are connected while those from the outside of the Mandelbrot set are disconnected. The disconnected sets are often called dust, they consist of individual points no matter what resolution they are viewed at.
What is 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 self-similar?
Julia sets are all very closely related to self-similar sets – each one can be thought of as the invariant set of something like an iterated function system.
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.
How many Julia sets are there?
There are two types of Julia sets: connected sets (Fatou set) and Cantor sets (Fatou dust).
What is quasi self similarity?
Quasi self-similarity of Julia sets. The fractal quality of Julia sets is formalized by the concept of quasi-self-similarity. Self-similarity occurs when a set is a dilated, isometric copy of one of its proper subsets, see e.g. the Von Koch curve or the Cantor set.