Infinite tetration of the imaginary unit

[sheldonison]

I'm going to read up on what's out there on Fatou sets. Anybody have a suggested reference? Specifically, we're interested in a Periodic Fatou set generated from a complex parabolic fixed point. I'm also trying to formulate a question for math overflow, to verify the existence of real periodic Fatou sets, generated with a superfunction of a function with a periodic parabolic fixed point, similar to what is seen on the Shell-Thron boundary.
- Sheldon

Looks like my intuition is correct. They're called Siegel discs, which are part of the Classification of Fatou components. In particular, Siegel discs occur around "occur around irrationally indifferent periodic points", of exactly the type that we have been describing in this thread. Siegel proved their existence.

I'll do some more reading -- there are requirements on the Brjuno Number of the convergents of the continued fraction expansion of α, which would be 1/period. The requirements don't seem that stringent, which implies that on the Shell-Thron boundary, for many/most cases if the period is irrational, \( \text{period}=2 \pi i/\log(\log(L)) \), then there is a real periodic superfunction. Then the real periodic superfunction can be mapped to a unit disc, where the boundary of the unit disc would be generating the superfunction from an initial starting point of 0, where the fractal behavior caused by \( \log_b(0) \), or \( \exp_b^{-n}(0) \). The interior of the Siegel disc would correspond to where the superfunction is analytic, with a real period, converging to the fixed point as \( \Im(x)\to i\infty \), which would correspond to the center of the disc.
- Sheldon