| | | | | | | | | | | | | | | | | | | | | | | | | | | | | S | u | b | j | e | c | t | : | | J | u | l | i | a | | s | e | t | s | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|
| Q7a: | What is the difference between the Mandelbrot set and a Julia set? |
| A7a: | 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). |
|
| Q7b: | What is the connection between the Mandelbrot set and Julia sets? |
| A7b: | 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. |
|
| Q7c: | How is a Julia set actually computed? |
| A7c: | 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 = ħsqrt( z1 - c). By |
applying this equation repeatedly, the resulting points quickly converge to |
the Julia set boundary. (At each step, either the positive 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 |
|
| Q7d: | What are some Julia set facts? |
| A7d: | 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. |
|