Thread: Fixed points of the Mandelbrot set

1. Fixed points of the Mandelbrot set

Bit of a long read here sorry...

I've just finished writing this huge project on complex dynamics and I have to give a presentation on it on Wednesday.

I'm just going through my project and I realized I'm not quite sure if I have this one part right...

It's about finding the fixed points of the Mandelbrot set (if there is any! It's not exactly a standard set!)

Now the way you normally do this is to set f(z) = z but with the Mandelbrot set, $\displaystyle z_0 = 0$ so what we're really dealing with is a sequence like this...

$\displaystyle z_0 = 0$,
$\displaystyle z_1 = c$,
$\displaystyle z_2 = c^2 + c$,
etc...

So to find fixed points we set the above to be 0 (as 'z' = 0 from $\displaystyle z_0 = 0$).

So we get, by solving the above equations to be equal to 0 we get...

For period 1 (i.e fixed) points
$\displaystyle z_1 = c = 0$

For period 2 points
$\displaystyle z_2 = c^2 + c = 0$
$\displaystyle => c = -1$ or 0 (discard 0 as it is a fixed point) so we get...

$\displaystyle c=-1$ which is the centre of the circular area to the left of the main cardioid.

We can take this further by solving for period 3 points in the same way and we get (ignoring the fixed point)
$\displaystyle -1.754877667$,
$\displaystyle -0.1225611669 - 0.7448617670i$,
$\displaystyle -0.1225611669 + 0.7448617670i$

Which are the centres of the discs at the top and bottom of the main cardioid and also the 'centre' (centre being equivalent to the (0,0) position on the main cardioid) of the mini Mandelbrot set way over to the left.

If you plug these back into the $\displaystyle z^2 + c$ you find that 3 iterations will lead you back to your value so it seems to be true...

For reference see this picture.
Perhaps I should be setting $\displaystyle f(z) = z^2 + c = \mathbf{c}$ and solving like that? Don't think so though as I'll end up with c's and z's so no way to solve for values.