TITLE: When does iterating $z \mapsto z^2 + c$ have an exact solution?
QUESTION [44 upvotes]: If one iterates the map $z \mapsto z^2 + c$ there is obviously a simple formula for the sequence one gets if $c=0$. Less obviously, there is also a simple formula when $c = -2$ (use the identity $2 \cos(2x) = (2\cos(x))^2 - 2)$. Are there any other values of $c$ for which one can solve this recurrence explicitly? (For all initial values of course: there are many trivial explicit solutions for special initial values, such as fixed points.)
Related links:
http://en.wikipedia.org/wiki/Mandelbrot_set (the points $c$ where 0 remains bounded under iteration of this map: this strongly suggests that there is no simple exact solution for general $c$).
http://en.wikipedia.org/wiki/Logistic_map (gives the explicit solutions above, after a change of variable)
Motivation: I once used the map with $c=-2$ in a lecture to show that one could prove limits exist even without a formula for the exact solution. A first year calculus student  pointed out the non-obvious exact solution above, and I don't want to be caught out like this again.

REPLY [4 votes]: Actually, there are many other cases : if $f$ is a surjective map from $C$ to $C$ such that $f(az)=f(z)^2+c$, we get the simple formula $z_n=f(a^nZ0)$, where $f(Z_0)=z_0$. So  the question is : for which values of $a$ and $c$ does such an $f$ exists, with perhaps some more constraints  on $f$ such as smoothness. It turns out that for any $c$ large enough (at least of module greater than 1) there exists exactly one such entire function $f$, with $f'(0)=1$ and $a=2f(0)=1+\sqrt{1-4c}$ (this is shown by an elementary, but tedious calculation of the coefficients of the series by recurrence (done in French here, page 8)), followed by a simple argument of domination of those coefficients, proving convergence for $R>0$ ;  then, analytic continuation shows that the radius of convergence is infinite) ; $f$ is (almost) surjective by Picard theorem. Of course, those $f$ are never "usual" or elementary functions, except for  $c=0$ or $c=-2$, but so what?