Julia set

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Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane. Given two complex numbers, c and z₀, we define the following recursion:

z_n+1 = z_n² + c

For a given value of c, the Julia set consists of all values of z₀ for which the modulus of z_n does not tend to infinity ("blow up") over multiple iterations. If the modulus of z_n becomes larger than 2 for some n then z_n will tend to infinity.

Table of contents

1 Relation to Mandelbrot set
2 Generalisation
3 Examples from the Julia set:
4 Reverse Julia sets
5 External links

Relation to Mandelbrot set

Julia sets are closely related to the Mandelbrot set which is the set of all values of c for which z_n = z_n-1² + c does not tend to infinity through application of the recursion with z₀ = 0. Like the Mandelbrot set, the Julia set is often plotted with different colors signifying the number of iterations carried out before the modulus of z becomes larger than 2.

The Mandelbrot set is, in a way, an index of all Julia sets, For any point on the complex plane (which represents a value of c) a corresponding Julia set can be drawn. We can imagine a movie of a point moving about the complex plane with its corresponding Julia set. When the point lies in the Mandelbrot set, the Julia set is connected. Otherwise, the Julia set is a Cantor dust of unconnected points.

Map of 121 Julia sets in position over the Mandelbrot set

If c is on the boundary of the Mandelbrot set, and is not a waist, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. For instance:

At c=1/4, the cusp at the set's mouth, the Julia set outline is a closed curve with cusps all around.
At c=i, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
At c=-2, the tip of the long spiky tail, the Julia set is a straight line segment.
-3/4, 1/4+i/2, and e^2πi/5/2-e^4πi/5/4 (-0.482-0.532i) are waists, The Julia set at c=-3/4 actually does look like the Mandelbrot set there, but the other two do not.

Generalisation

Most books refer to the description above for Julia sets, but the formal mathematical definition is more general and covers other contexts.

The Julia set can be defined for any iterated map of the complex plane to itself, or collection of maps. The Julia set is the smallest closed, completely invariant point set for such a map or collection of maps which contains at least 3 points. It can also be defined (informally) as the set of points for which nearby points do not exhibit similar behaviour under repeated iterations of the map(s).

Julia sets can also be defined for any n-dimensional space, not just the complex plane.

Julia sets typically (though not always) have a fractal structure, and Julia sets can be associated with fractals such as the Sierpinski triangle and the Cantor set.

The complement of the Julia set (i.e. the set of points for which nearby points do exhibit similar behaviour) is sometimes called the Fatou set, although some authors use the term Fatou set or Fatou dust for a disconnected Julia set.

Examples from the Julia set:

       Time escape Julia set from coordinate (phi-2, 0)

       Time escape Julia set from coordinate (phi-2, phi-1)

       Time escape Julia set from coordinate (0.285, 0)

Reverse Julia sets

Reverse Julia set (coordinate phi-2, 0)

Combination sequence, method: <em>z</em>ÃÂ² -> Sierpinski with IFS viewed as overlay

Combination sequence, method: zÃÂ² -> Sierpinski with IFS viewed as overlay

It is also possible to generate Julia sets using a method derived from the IFS "random game" method. Instead of using an "escape time" method to find the points that do not belong to the Julia set, the "random game" method uses the reverse formula to track the convergence of a single point towards the edge of the Julia set.

For example, for the Mandelbrot set, the reverse formula is

At each iteration one of the two roots is selected at random. After a sufficiently large number of iterations the current position of z is plotted. The process is then repeated with a different random sequence of roots.