new results from complex dynamics

[sheldonison]

We did see some results for bases both inside and outside the Shell Thron boundary.

Oh Sheldon, I am really too slow to catch up with all your findings. Indeed now that I plotted some sickles it seems to me the Shell/Thron boundary is not much of an osbtacle anymore. Whether this follows from the propositions of Shishikura is perhaps another matter.

So the general assumption in the moment is that we can continue tetration along bases through the STB?

I have gotten results on the STB itself; although the calculation is indirect, and requires calculating complex tetration for many dozens of complex bases in a circle. See http://math.eretrandre.org/tetrationforu...e=threaded for the irrationally indifferent case and the next entry in that thread for the rationally indifferent case. I think the rationally indifferent case, with a "pseudo" period=5, base=1.96514 + 0.441243i, is a very interesting case, and I was hoping someone had calculated the analogous problem for Mandelbrots inverse perturbed-fatou coordinates on the main cardioid boundary, with a rational Pseudo period.

Dimitrii's also reports that his Cauchy integral algorithm can calculate bases on the Shell Thron boundary directly. I have also extended my indirect results to z anywhere in the complex plane for bases on the STB.

What about \( \eta \) itself, do we conjecture holomorphy there?

What happened anyway with writing your article. I really find that its a novel efficient way to compute tetration, worth that the world should know it!

\( \eta \) has a branch singularity, since starting with a real base \( >\eta \) the result going halfway around the circle clockwise to a real base<eta is different than the result going halfway around the circle counterclockwise. The tetration function is not real valued. Numerically, the branch singularity at eta appears to be very slight, even at \( \eta-0.25 \). See http://math.eretrandre.org/tetrationforu...e=threaded Continuing on past \( \eta \) more than halfway around the circle starting from a real base \( >\eta \), it seems that one encounters a singularity wall when one gets to the STB boundary the second time. Douady saw the same phenomena for \( z^2+\frac{1}{4}+\epsilon \). Problem 6) (Douady) Persistence of the Fatou Coordinate:

I'll try to get to that article ... thanks for the encouragement. I still have much to learn.
- Sheldon