Infinite tetration of the imaginary unit

[sheldonison]

Hi Sheldon -

likely a typo in the formula:

case, Then the period "works". For all integers, we can define the superfunction for x=(m mod period), starting from initial value L+z:
\( \text{SuperFunction}_{L}(x) = \exp_B^{o m}(L+z) \). And for

For the parameter of the superfunction I think you want to write m (or for exp_b the iterator/height x ?)
....

Hey Gottfried,

I don't think its a typo, but I tried to make the original post slightly clearer. x=(m mod period), m would be an integer iteration. The "o m" notation is iteration m times. Lower case b works better; here from your previous example, \( b\approx 1.7129i \), \( \text{period}\approx 2.9883 \)

\( \text{SuperFunction}_{L}(0) = (L+z) \)
\( \text{SuperFunction}_{L}(1) = b^{(L+z)} \)
\( \text{SuperFunction}_{L}(2) = \exp_b^{o 2}(L+z) \)
\( \text{SuperFunction}_{L}(0.0117) = \exp_b^{o 3}(L+z) \)
\( \text{SuperFunction}_{L}(1.0117) = \exp_b^{o 4}(L+z) \)
\( \text{SuperFunction}_{L}(2.0117) = \exp_b^{o 5}(L+z) \)
\( \text{SuperFunction}_{L}(0.0234) = \exp_b^{o 6}(L+z) \)
\( \text{SuperFunction}_{L}(x) = \exp_b^{o m}(L+z) \) where \( x=(m \bmod \text{period}) \)

This allows you to use integer iterations to gradually fill in all of the points for the superfunction on the real axis between 0 and the period. Since the function is real periodic, this means we can define the superfunction anywhere on the real axis. Then what I did, was to take the Fourier analysis of the function, using polynomial interpolations from several nearby points on either side to get a fairly exact value for a sampling of evenly spaced points; the results were posted here.

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