The Julia set

Auteur:a.zampa

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

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.

Select the option you prefer and click on Draw; to cancel the picture click on Cancel. We suggest, initially, to use points and change the value of

in order to select the set you are interested in, and then to trace the set with the chosen value of

The Julia set

Nieuw didactisch materiaal

Ontdek materiaal

Ontdek onderwerpen