show/hide this revision's text 4 corrected spelling

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 newer 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
show/hide this revision's text 3 corrected spelling

Finding Functional form for given 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:

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

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 newer seen problems like this before. Is there an 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
show/hide this revision's text 2 improved formatting

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:

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

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

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

Note that for $\beta = 0$, $G(k)=\exp\left(a k^{\log_\mu 2}\right)$ 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 newer seen problems like this before. Is there an 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
show/hide this revision's text 1