Finding Functional form for a given Scaling Condition

Question

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:

$G(k\mu) = G(k)^2+ \beta$

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 $\beta = 0$, $G(k)=\exp\left(a k^{\log_\mu 2}\right)$, ($a$ 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

-- jon

Answer 1

You do not give any smoothness requirement; I will look for an analytic $G$: $$ G(k)=\sum_{n=0}^\infty a_nk^n.$$ In what follows, I assume also that $\mu=4/3$ and $\beta=2/9$. Expanding in a power series both sides of the equation and equating coefficients, we get that $a_0=1/3$ or $a_0=2/3$. In the first case we obtain the constant solution $G(k)=1/3$. But in the second case, we find a one parameter family of (formal) solutions, parametrized by the value of $a_1$: $$ a_0=\frac23,\quad a_1\in\mathbb{C},\quad a_n=\frac{1}{\mu^n-\mu}\sum_{i=1}^{n-1}a_ia_{n-i},\quad n>2. $$ For $a_1=0$ we obtain the constant solution $G(k)=2/3$. For other values of $a_1$, one should check that the series has a positive radius of convergence.

Another way of obtainig solutions is the following. Choose an arbitrary function $h\colon[1,\mu]\to\mathbb{C}$, and define $G(k)=h(k)$ if $1\le k<\mu$; for $\mu\le k<\mu^2$, let $G(k)=G(k/\mu)^2+\beta$; iterate this procedure to define $G$ on $[1,\infty)$. Now, for $1/\mu\le k<1$, let $G(k)=\pm\sqrt{G(\mu k)^2-\beta}$; iterate the procedure to define $G$ on $(0,1)$. Conditions can be impposed on the arbitrary function $h$ to make $G$ continuous, for instance ($G(\mu)=G(1)^2+\beta$).