The Julia set
Any function defined on the whole set of complex numbers defines a flow on it by iteration: : is then subdivided into regions which are invariant with respect to the flow, i.e. . When is a rational function, i.e. where and are polynomials, there is a finite numer of such regions which are open (these are called the Fatou sets of ) and their union is dense in .
The Julia set of is the smallest closed set containin at least three points and completely invariant with respect to . This applet draws an approximation of the Julia set of the function , with a complex number, exploiting the fact that for almost all points the set is the set of limit points of the full backwards orbit .
You have two alternatives:
- you can use (up to 600) points to build the set, this allows you to change the value of and automatically update but the Julia set thus constructed is very rough, or
- you can trace the set using 15000 points.
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- .