Infinite tetration of the imaginary unit

[sheldonison]

The complex point x0=0.0+0.0*I (the "original"/reference) was taken
....
Here I took two computations (and two colors) for the plot because of need of lots of tons of iterates to get a clue of the shape in the sparse regions. The blue points are the first about two-thousand iterates, with an implicte float-accuracy of 400 decimal digits. We see, that there are very sparse regions. After that I switched to 800 and 1200 digits precision and went up to 80000 iterates - no clue, whether the errors accumulate to something horrible. From that 80000 iterates I deleted all in the inner area so I took only the points in the sparse areas into the plot (red dots). It seems, the values are not completely messed, since in principle the blue and red points join reasonably to one curve.

Hmm... what does this tell me?

For instance: is that aequator through x0=0 asymptotically dense? Is it indeed an aequator at all? Is it enclosed in some disk?

For another instance: what does this mean in regard to fractional iteration? Assume an orbit with a starting value x0 inside the limiting aequator (through x0=0), say through x0=0.5+0.5*I which seems to fill a smooth, closed line densely. Are also all fractional iterates on that curve?

Gottfried

Hey Gottfried,

Coincidently, I've been thinking about the same problem -- the behavior of bases on the Shell-Thron boundary. The base was chosen to be on the Shell-Thron Boundary, with a real period~=2.9883. Starting with x0=0 probably generates a fractal.

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

Back to your plot, starting with x0=0 generates a fractal, which represents the lower boundary of the real periodic superfunction. The structure of the fractal appears to depends on the continued fraction of the real period, where \( \text{period} = 2\pi i /\log(\log(L)) \).

Anyway, most of this is pure conjecture, and perhaps others have done more work on these complex bases on the Shell-Thron boundary.
- Sheldon