Question

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), 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?

Answer 1

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).