Elliptic Superfunctions

[BenStandeven]

From the double-angle formulas for the Jacobi elliptic functions:
\(
cn(2x) = {\frac{cn(x)^2 - sn(x)^2 dn(x)^2} {1 - k^2 sn(x)^4}},
sn(2x) = {{2 sn(x)cn(x)dn(x)}\over{1 - k^2 sn(x)^4}},
dn(2x) = {{dn(x)^2 - k^2 (sn(x)^2 cn(x)^2)}\over{1 - k^2 sn(x)^4}}
\), we can get superfunctions to other rational functions.

For example, \( cn(2^n) \), if my computations are correct, is the superfunction for:
\(
f(z) = {{k - 1 + (2 - 2k) z^2 + k z^4}\over{1 - k^2 + 2 k^2 z^2 - k^2 z^4}}
\), or
\(
f(z) = \frac{\(z^2 - \frac{k+1}{k} + \frac{sqrt(1-k)}{k} \)\(z^2 - \frac{k+1}{k} - \frac{sqrt(1-k)}{k} \)}{-k\(z^2 - \frac{k+1}{k} \)\(z^2 - \frac{k-1}{k} \)}
\).

So for example, picking the modulus to be k=-3, we would get: \( f(z) = \frac{z^2}{3\(z^2 - \frac{2}{3} \)} \). This means that \( nc(2^n, k=-3) \) would be the superfunction for \( f(z) = \(3 - 2 z^2 \) \), which is well outside the Mandelbrot set.

[mike3]

like i said mike , if you read my reply : i think he meant cn instead of nc.

and i pointed out that a periodic function is not a superfunction in the direction of its period.

regards

tommy1729

(Oo, I just deleted that post, I didn't think someone would have gotten to it already... (Just for reference: the post was asking about what "nc" was since I hadn't seen it before, then I looked it up and saw it really does exist and that's why I deleted it))

Yeah, but \( F(z) = \mathrm{cn}(2^z) \) is not periodic in the real axis direction due to the exponential (it does have an imaginary period of \( \frac{2\pi i}{\log(2)} \) but not a real one) It's not straight \( \mathrm{cn} \), but \( \mathrm{cn} \) composed with an exponential.

[bo198214]

From the double-angle formulas for the Jacobi elliptic functions we can get superfunctions to other rational functions.

Actually this topic was already considered in:
Schröder, E. (1871). Ueber iterirte Functionen. (On iterated functions.). Clebsch Ann., 3, 296–322.

Personally interesting for me would be the iterates/superfunctions of rational functions that dont have a real fixed point. Are there some amongst this class obtained from elliptic addition theorems?

In the case of several real fixed points there is still always the question at which fixpoint the obtained elementary iteration/superfunction is the regular iteration/superfunction.