Infinite tetration of the imaginary unit

[sheldonison]

sheldon , we dont know if the period is a brjuno number , which is why we are not certain if its a siegel disk.
....
although you are correct with probability 1 , we know how " tricky and weird " tetration can be.

i would also like to point out that 1.7129i^z is periodic with 2pi i /ln(1.7129 i) = 3.57977339 + 1.22650561 i ...

Hmmm, we don't even know if the period is irrational either, although a random base on the Shell-Thron is going to be irrational, with probability 1.

I wasn't able to follow the other part of your post. 3.57977339 + 1.22650561i would be a period of 1.7129^z, but I'm not sure how that effects the superfunction. The superfunction has a period of 2.9883, and 2.9883 is not a period of 1.7129^z.

Presumably, the coefficients of the series for the superfunction can be calculated with some sort of formula from the series for B^(z-L). The Fourier series coefficients are equivalent to the Taylor Series coefficients of the superfunction function wrapped around the Siegel disc. The algorithm I used seems to works, but it isn't very elegant, and it only works a little bit inside the Siegel disc, where convergence is much better. Update, one Thesis paper I started to read, written by Edgar Arturo Saenz Maldonado on the Brjuno number seems to have the formulas.

\( \lambda=\exp(2\pi i \alpha) \)
\( f(z)=\lambda z + \sum_{n>=2}a_n z^n \).
And .... the formal power series of h {the Seigel disc function} is given by
\( h(z)=\sum_{i>=1}h_i z^n \)
If h is the solution of the functional equation ... \( f(h(z))=h(\lambda z) \), the coefficients of the series must satisfy (formally) the following recursive relation:
\( h_n \)=1, for n=1, and for n>=2,
\( h_n = \frac{1}{\lambda^n-\lambda}\sum_{n=2}^{n}a_m \sum_{n1+...+n_m=n} h_{n1}h_{n2}...h_{n_m} \)
where in the second summation, \( n_i>=1 \)

"... By the formulas in question, it is possible to determine the coefficients of the formal power series of \( h_f \); the denominators of these coefficients can be written as products of the form \( \lambda^n-\lambda \), for n>=2, since \( \alpha \) is an irrational number these products could be very small....", which is where the Brjuno number comes from.

So this would be a closed form equation for the superfunction for bases on the Shell-Thron boundary, where \( \lambda=\exp(2\pi i \alpha) \), and \( \alpha \) is an irrational Brjuno number.
- Sheldon