tag removed - Finding Functional form for a given Scaling Condition

Dear all

While studying the overlap distribution for two random Cantor sets (long story made short), I came across the following problem.

$G(k)$ is a complex valued function, and satisfy the following condition:

$G(kmu) = G(k)^2+ beta$

with $beta,mu$ constant (in my case $beta=frac{2}{9}, mu = frac{4}{3}$)

Is there a way to find the functional form of $G(k)$ which satisfy the condition?

Note that for $beta = 0$, $G(k)=expleft(a k^{log_mu 2}right)$, ($a$ konstant) will satisfy the condition (easily verified), but I have no idea on how to find a solution for non-zero $beta$. I'm a not a math student (I'm studying physics), but I have never seen problems like this before. Is there a way to find analytical expression for $G(k)$? Possible as an expansion?

I can generate a function which has this property on the computer. Writing $G(k)= x(k) + i y(k)$, with $x(k)=x(-k)$ and $y(k)=-y(-k)$ the function should look something like this:

http://dl.dropbox.com/u/483049/xy.pdf

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jon