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I'm trying to understand how Julia sets iterations are done and how those iterations differ from the ones that generate the Mandelbrot set.

Both of them use the following function: $f(z) = z^2 + C$

For the Mandelbrot set, one would just start with an initial value of $z=0$. And iterate every single point (in the boundaries of $-2,2$ and $-2i,2i$ because any other bigger point is graphically useless) in the complex plane using the C variable. Depending if the iteration is bounded or goes to infinite, you draw differently the points.

But, what's the difference in Julia sets? I've read that just by giving $C$ a value and $z$ another value (different from zero, I guess) a Julia Set can be generated/calculated. The value of $C$ is the point in the plane, but what is the value of $z$? is it a random value? a fixed value? how exactly the iteration of a Julia set is done?

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In the study of the dynamics of functions of the form $f(z)=z^2+c$, the key difference between the definition of the Julia set and the Mandelbrot set is:

  • For Julia sets, the value of $c$ is fixed and we iterate starting from various initial values of $z_0$.
  • For the Mandelbrot set, the initial starting point of $z_0=0$ is fixed but we perform the iteration for various values of $c$.

To understand why one would make these definitions, you need to understand what the point of the definitions of the sets is in the first place.

The Julia set was defined in an attempt to understand the dynamics of a single function $f:\mathbb C \to \mathbb C$. Thus, we set up a grid (perhaps, bounded by $\pm 2$ along the real axis and $\pm 2i$ along the imaginary axis, as you suggest) and, for each point $z_0$ in that grid, we iterate from that point and see what happens. If the function $f$ happens to have the form $f(z)=z^2+c$, then note that $c$ is fixed because we are dealing with a single function.

The Mandelbrot set, by contrast, was defined in an attempt to understand what types of dynamics might be possible for all functions in the entire family of functions of the form $f(z)=z^2+c$. As it turns out, the orbit of the critical point $z_0=0$ (and, in particular, whether that orbit diverges or stays bounded) determines quite a lot about what can happen. Thus, the Mandelbrot set is defined as the set of all $c$ values such that the critical orbit (the orbit of $z_0=0$ stays bounded). Note that the initial point is now fixed but the value of $c$ varies.

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  • $\begingroup$ What does "the orbit" and "the critical orbit" of $z0=0$ mean in your text? I don't understand that exact term used here (my fault, since I'm not a native English speaker). Thanks $\endgroup$ – Pedro Javier Fernández Aug 21 '17 at 18:33
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    $\begingroup$ When you iterate a function $f$ from a point $z_0$, the sequence that you generate is called an orbit. For example, if $f(z)=z^2$ and $z_0=2$, then the orbit of $z_0$ under the iteration of $f$ is the sequence $(2,4,16,265,\ldots)$. A critical point $c$ is a point where $f'(c)=0$ and a critical orbit is an orbit of a critical point. Using the same example of $f(z)=z^2$, then the only critical point is $0$ and, since $f(0)=0$, the critical orbit is the sequence $(0,0,0,\ldots)$. $\endgroup$ – Mark McClure Aug 22 '17 at 1:05
  • $\begingroup$ Thanks! Now I understand everything. I'm gonna accept the answer. Can I ask a last question? where (or who) was firstly defined or said that the orbit must be called actually orbit? I just want to know it for historical research purposes. Thanks again $\endgroup$ – Pedro Javier Fernández Aug 22 '17 at 17:57

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