fixed points of quadratic iteration Consider the well-known iteration $f:z\to z^2 + c,$ and consider the values of $c$ for which $0$ is a periodic point. Experiment shows that most such values of $c$ (about $480$ out of $512$ for period $10$)  lie on the real axis, and those that don't lie near the cusp part of the Mandelbrot set cardioid part, along the boundary (the cardioid itself). I assume all this is well-known, but not to me, so any enlightenment/references welcome.
 A: The principal source is
MR0762431
Douady, A.; Hubbard, J. H.
Étude dynamique des polynômes complexes. Partie I.
Publications Mathématiques d'Orsay, 84-2. Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. 75 pp. 
MR0812271 
Douady, A.; Hubbard, J. H.
Étude dynamique des polynômes complexes. Partie II.
Publications Mathématiques d'Orsay, 85-4. U Orsay, 1985. v+154 pp.
Available on Internet.
These values are called "centers of the hyperbolic components", and there is a combinatorial algorithm which determines for each $n$ how many of them are real.
They did not study statistics but it is unlikely that a majority of them are real. Anyway, the algorithm ("Hubbard trees") permits in principle to reduce this question to combinatorics and number theory. The algorithm encodes all
solutions $c$ by certain trees embedded in the plane, and the question is equivalent to counting trees with certain symmetry property. 
EDIT. On my request, Gena Levin wrote the following:

There is a recursive formula for the number $H(n)$ of "real" hyperbolic components of the Mandelbrot corresponding to cycles of exact period $n$ via the number $P(n)$ of cycles of exact period $n$ for the Tchebyshev polynomial $x^2-2$:
  $H(n)=P(n)/2$, if $n$ is odd, and
  $H(n)=(P(n)+H(n/2))/2$ if $n$ even.
For example, if $n$ is a prime number then $H(n)=(2^{n-1}-1)/n.$

A: The number of real centers of hyperbolic components of exact period $n$ is OEIS A000048. This isn't bad to work out from the combinatorics of kneading sequences which Rivin links to sources for. A number of references linked by OEIS imply this, although I oddly can't find one which explicitly states it and gives a proof. I know this because Sarah Koch, Dylan Thurston and I were computing these numbers earlier this year and I worked out the combinatorics, which I'm willing to write up if you care enough.
As OEIS says, this gives the formula 
$$\frac{1}{2n} \sum_{d|n,\ d \ \mbox{odd}} \mu(d) 2^{n/d} \approx \frac{2^n}{2n}.$$
By contrast, the number of $c$ of exact period $n$ is
$$\sum_{d|n} \mu(d) 2^{n/d} \approx 2^n.$$
So the fraction of real roots is roughly $\tfrac{1}{2n}$.
A: Upon investigation:


*

*Mathematica is screwing up. Using it's own function CountRoots[], which counts real roots (exactly), the number of real roots, while quite far from zero (there are $30$ for $n=9,$ for example) is not the majority of the fixed points.

*The parameter values corresponding to the periodic points are equidistributed on the boundary of $M$ with respect to harmonic measure. References are:


Ahlfors, Lars V., Conformal invariants. Topics in geometric function theory, McGraw-Hill Series in Higher Mathematics. New York etc.: McGraw-Hill Book Company. VII, 157 p. $ 10.95 (1973). ZBL0272.30012.
and 
Brolin, H., Invariant sets under iteration of rational functions, Ark. Mat. 6, 103-144 (1965). ZBL0127.03401.


*Real fixed points correspond to the fixed points of the quadratic iteration on the intervals - a beautiful subject, one of the most beautiful papers on which is


Milnor, John; Thurston, William, On iterated maps of the interval, Dynamical systems, Proc. Spec. Year, College Park/Maryland, Lect. Notes Math. 1342, 465-563 (1988). ZBL0664.58015.
(many thanks to Curt McMullen for the refs).
