Consider the function $f_c(z) = z^2 + c$. Applying this function repeatedly, we get the familiar quadratic Julia sets that fractal enthusiasts burn compute cycles plotting.
Infinity is always one attractor of the system. Depending on the choice of $c$, a finite attractor may also exist. Sometimes this is a fixed-point. Sometimes it is a repeating cycle of some finite number of points.
Consider the case of a fixed-point. The actual numerical value of this fixed point depends on $c$. So I set out to investigate a way to compute this number directly.
A fixed point of $f_c$ is simply any $z$ for which $f_c(z) = z$. In other words, we wish to solve $z^2 + c = z$. Rearranging as $z^2 - z + c = 0$, I was easily able to find
$z_1 = \frac{1 \pm \sqrt{1 - 4c}}2$
At this point, something struck me: First, there are obviously two such fixed-points, only one of which is the finite attractor. But, more conspicuously, these two fixed-points always exist. Even when there is no fixed-point attractor, there definitely are two fixed points.
What about a period-2 cycle? That is, we want to solve $f_c(f_c(z)) = z$. Solving $(z^2 + c)^2 + c = z$ is a little more tricky than the last equation - but the formula for $z_1$ gives us two of the solutions, and it's then fortunately easy to discover the other two:
$z_2 = \frac{1 \pm \sqrt{-3-4c}}2$
Again, this cycle always exists.
At this point, I tried to find a period-3 cycle. Clearly $((z^2 + c)^2 + c)^2 + c = z$ has 8 solutions, two of which are $z_1$, which leaves 6 remaining. At this point, I was unable to work out how to solve the equation. The mighty Mathematica™ also refused to give me a closed form. (I suppose it's plausible that none exists.)
It seems clear though that these solutions exist, even if I can't easily compute them. And if there's 6 of them, that's presumably a pair of period-3 cycles. More generally, it seems there is no reason why cycles of any finite length wouldn't exist all the time. So, my actual question is this: Where do all these periodic cycles "live" when they aren't the attractor of the system?