Julia set

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 does not tend to infinity ("blow up") over multiple iterations. Tendency to grow is determined by whether or not the modulus of z is larger than 2.

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.

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.

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

The Julia set can be defined for any map of the complex plane to itself, or collection of maps. The julia set is the smallest fixed point set for such a map or collection of maps, not counting the empty set.

For example, the Sierpinski triangle is a fixed point set of three maps, each of which maps the triangle to one of the corners, shrinking by a factor of 1/2.

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

The Cantor set is defined on the line, with two maps. One maps the interval [0,1] to [0, 1/3]. The other maps the interval [0,1] to [2/3, 1]. The cantor set is the julia set of this pair of maps.

It is possible to define the dimension of a fractal. This is usually done with the Hausdorff dimension, but other methods of computing the dimension are also used.\n