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Fractal Frequently Asked Questions and Answers
Section - Julia sets

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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.  
  
You can see an example Julia set on the WWW at  
gopher://life.anu.edu.au:70/I9/.WWW/complex_systems/julia.gif .  
  
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 = +- 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  
  
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.  
  

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Top Document: Fractal Frequently Asked Questions and Answers
Previous Document: The Mandelbrot set
Next Document: Complex arithmetic and quaternion arithmetic

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Last Update March 27 2014 @ 02:11 PM