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Julia sets

What is the difference between the Mandelbrot set and a Julia set?

The Mandelbrot set iterates z^2+c with z starting at 0 and varying c. The Julia set iterates z^2+c for fixed c and varying starting z values. That is, the Mandelbrot set is in parameter space (c-plane) while the Julia set is in dynamical or variable space (z-plane).

What is the connection between the Mandelbrot set and Julia sets?

Each point c in the Mandelbrot set specifies the geometric structure of the corresponding Julia set. If c is in the Mandelbrot set, the Julia set will be connected. If c is not in the Mandelbrot set, the Julia set will be a Cantor dust.

How is a Julia set actually computed?

The Julia set can be computed by iteration similar to the Mandelbrot computation. The only difference is that the c value is fixed and the initial z value varies.

Alternatively, points on the boundary of the Julia set can be computed quickly by using inverse iterations. This technique is particularly useful when the Julia set is a Cantor Set. In inverse iteration, the equation z1 = z0^2+c is reversed to give an equation for z0: z0 = +- sqrt(z1-c). By applying this equation repeatedly, the resulting points quickly converge to the Julia set boundary. (At each step, either the postive or negative root is randomly selected.) This is a nonlinear iterated function system. In pseudocode:

z = 1 (or any value)
loop
   if (random number < .5) then
      z = sqrt(z-c)
   else
      z =-sqrt(z-c)
   endif
   plot z
end loop

What are some Julia set facts?

The Julia set of any rational map of degree greater than one is perfect (hence in particular uncountable and nonempty), completely invariant, equal to the Julia set of any iterate of the function, and also is the boundary of the basin of attraction of every attractor for the map.

Julia set references:

1. A. F. Beardon, _Iteration of Rational Functions : Complex Analytic Dynamical Systems_, Springer-Verlag, New York, 1991.

2. P. Blanchard, Complex Analytic Dynamics on the Riemann Sphere, _Bull. of the Amer. Math. Soc_ 11, 1 (July 1984), pp. 85-141. This article is a detailed discussion of the mathematics of iterated complex functions. It covers most things about Julia sets of rational polynomial functions.