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Infinite tetration of the imaginary unit
(06/24/2011, 12:25 PM)tommy1729 Wrote: 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.

And .... the formal power series of h {the Seigel disc function} is given by

If h is the solution of the functional equation ... , the coefficients of the series must satisfy (formally) the following recursive relation:
=1, for n=1, and for n>=2,

where in the second summation,

"... By the formulas in question, it is possible to determine the coefficients of the formal power series of ; the denominators of these coefficients can be written as products of the form , for n>=2, since 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 , and is an irrational Brjuno number.
- Sheldon

Messages In This Thread
Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 12:09 AM
RE: Infinite tetration of the imaginary unit - by sheldonison - 06/24/2011, 04:36 PM

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