It is well known that the function $f(z)=2\cos(\sqrt {-z})$ (or more accurately the entire function $f(z)=2\sum\_{n=0}^\infty \frac{z^n}{(2n)!}$) satisfies such a functional equation, i.e. $f(4z)= f(z)^2-2$ ; it is not hard to show that this is the unique solution with $c=-2$ and $f'(0)=-1$. Patient calculations (see for instance  [this note in french](http://denisfeldmann.fr/PDF/mandel.pdf)), or the classical results of Tan Lei, show that this implies that the small cardioids on the left main antenna of the Mandelbrot set (on the real axis near $-2$) are in position $-2+3\pi^2/2^{2n+1}+o(4^{-n})$. It would be easy to generalise this result to other Misiurewicz points, if similar functions were known for other values of $c$, as the small copies of the $M$-set lie similarly near the zeros of these functions (after appropriate rescaling) ; it is not hard to show their existence and unicity, but have they been studied, and what is known on their zeros?



Mandelbrot set and analytic functions such that $f(az)=f(z)^2+c$
Such functions were studied, since Poincare.
They are called Poincare functions, or "global linearizers".
Let $f$ be a rational function, and consider the functional equation
$F(kz)=f(F(z))$. Suppose $F$ is analytic at $0$. Then $a=F(0)$ must be a fixed point of $f$.
If $F'(0)$ not $0$, then $k=f'(0)$ by the chain rule.
If all this is so, and $|k|>1$, one can easily show that F exists and is meromorphic in the whole
plane. When $f$ is a polynomial, $F$ is entire. $F$ is uniquely defined by the condition
$F'(0)=0$, if the fixed point of $f$ is given.


The literature on Poincare functions is quite large.
Specially for quadratic polynomials, see the papers of P. Myrberg.
I have several papers on the general rational case.
(All my papers can be found on my web page).


