Any function [math]w=f(z)[/math] defined on the whole set of complex numbers [math]\mathbb{C}[/math] defines a flow on it by iteration: [math]z_{n+1}=f(z_{n})[/math]: [math]\mathbb{C}[/math] is then subdivided into regions [math]R_j\subset\mathbb{C}[/math] which are invariant with respect to the flow, i.e. [math]f(R_j)\subset R_j[/math]. When [math]f[/math] is a rational function, i.e. [math]f(z)=p(z)/q(z)[/math] where [math]p(z)[/math] and [math]q(z)[/math] are polynomials, there is a finite numer of such regions which are open (these are called the [i]Fatou sets[/i] of [math]f[/math]) and their union is dense in [math]\mathbb{C}[/math].[br][br]The [i]Julia set[/i] [math]J(f)[/math] of [math]f[/math] is the smallest closed set containin at least three points and completely invariant with respect to [math]f[/math]. This applet draws an approximation of the Julia set of the function [math]f(z)=z^2+c[/math], with [math]c[/math] a complex number, exploiting the fact that for almost all points [math]z_0[/math] the set [math]J(f)[/math] is the set of limit points of the full backwards orbit [math]\bigcup_{n\in\mathbb{N}}f^{-n}(\{z_0\})[/math].

You have two alternatives:[br][list][br][*] you can use (up to 600) points to build the set, this allows you to change the value of [math]c[/math] and automatically update [math]J(f)[/math] but the Julia set thus constructed is very rough, or[br][*] you can trace the set using 15000 points.[br][/list][br]Select the option you prefer and click on [i]Draw[/i]; to cancel the picture click on [i]Cancel[/i]. We suggest, initially, to use points and change the value of [math]c[/math] in order to select the set you are interested in, and then to trace the set with the chosen value of [math]c[/math].[br][br]Suggested values of [math]c[/math] are:[br][list][br][*] [math]c=0[/math],[br][*] [math]c=0.5[/math],[br][*] [math]c=-1[/math],[br][*] [math]c=-1.5[/math],[br][*] [math]c=0.3-0.4i[/math],[br][*] [math]c=0.36+0.1i[/math],[br][*] [math]c=-0.1+0.8i[/math],[br][*] [math]c=-0.4-0.6i[/math],[br][*] [math]c=-i[/math],[br][*] [math]c=-0.8+0.4i[/math].[br][/list]