TITLE: fixed points of quadratic iteration
QUESTION [5 upvotes]: 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.

REPLY [5 votes]: 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 $\tfrac{1}{2n}$.

REPLY [4 votes]: 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).