Infinite tetration of the imaginary unit

[sheldonison]

My conjecture is for bases on the Shell Thron boundary, there is an analytic superfunction with a real period, whose structure depends on what the continued fraction representation of the real period is. As long as the period is a real number (with an infinite continued fraction representation), then I suspect the superfunction is analytic. If the period is a rational number, then I don't think there is an analytic superfunction. For example, this base, with a real period=3, probably doesn't have an analytic superfunction, developed from the neutral fixed point, because starting with a point near L, and iterating the function x=B^x three times, doesn't get you back to the initial starting point.
Base= 0.030953557167612060 + 1.7392241043091316i
L= 0.39294655583435517 + 0.46203078407110528i

Hi Sheldon -

I've inserted your base-parameter and got the following plot for the orbit/for the three partial trajectories in the same style of my previous plots. I seem to have problems to understand your comment correctly. For instance, isn't that fixpoint attracting instead of neutral?

The definition for the Shell-Thron region boundary is \( |\log(L)|=1 \), which is the case. But when the period \( =2\pi i/\log(\log(L))=3 \) is an integer (or a fraction), the equations misbehave. At the Shell-Thron boundary, the period is always a real number, and the fixed point is neither attracting nor repelling. At first, I thought the idea of a superfunction with a real period was nonsense in this post, but then I was able to get it to work, except for the cases when the period was an integer, or a fraction with a small denominator. So that experimentation is where my conjecture came from.

For example, here is another case, on the Shell-Thron boundary, that should work fine because the period is a real number, with a period just a little bit bigger than 3. With a sufficient number of iterations, it generates a very nice plot, that appears to lead to an analytic superfunction. But, in the plot, you can see the influence of the base being just a litle bit bigger than an integer. By the way, these Shell-Thron boundary bases are easy to generate. \( L=\exp(\exp(2 \pi i/\text{period})) \) and \( \text{base}=L^{1/L} \)

base= 0.036314759343852642170871708751 + 1.7435957010705633826865464522i
L= 0.39309905520386861718874315414 + 0.46286165860913191074862913970i
Period= 3.0019951097271885263233102180
- Sheldon