complex base tetration program

[sheldonison]

Discussing the (extended) Kneser-method the question of fixpoints is relevant. Here I have produced a picture of the fixpoints of tetration to base(î) , I found 2 simple fixpoint (attracting for exp, attracting for log, both used for the Kneser-method), and three periodic points - making things a bit more complicated. The fixpoints were sought using the Newton-algorithm for the joint threefold exponentiation \( f(z)= \log_i(\log_i(\log_i(z))) \) and the iteration \( z_{k+1} = f(z_k) \) .

This means for example, that for the point in the top-left edge with the blue color having the z-value z_0=-5 - 5i the Newton-algorithm using iteration \( z_{k+1} = f(z_k) \) arrives at the fixpoint 3.0 (having the complex value of about -1.14+0.71I in a moderate number of iterations. The blue point at coordinate z_0=-2.5+1I needs less iterations and the a bit lighter blue points near the periodic point 3.0 need even less iterations.

Perhaps this post should be moved into a discussion of the Kneser-method or of the general problem of fixpoints.

Gottfried

Here is the picture:

Yes, this thread should be moved; it has to do with fatou.gp, and an MSE question for tetration base(i), and the two primary fixed points for Henryk Trapmann's uniqueness sickle.

In the case at hand, the other fixed point, for the lower half of the complex plane for sexp(z), is -1.862-0.411i, which Gottfried has listed strangely as 3-periodic, where as it is a simple repelling fixed point for \( i^z \). The Abel/Slog uniqueness sickle connects the two primary fixed points. For exp(z), both fixed points are repelling for exp(z). But if you move slowly from base(e), to base(i), you see the lower fixed point becomes -1.862-0.411i, which is still repelling, but the upper fixed point becomes attracting; 0.4383+0.3606i. The solution I posted on MSE is based on generated the slog(z) exactly between the two primary fixed points, which is what the fatou.gp program does. My answer on MSE includes the taylor series for p(z), which turns out to have a remarkably mild singularity at the two fixed points; finding that analytic Taylor series is the basis for the fatou.gp program solution, which leads directly to Henryk's uniqueness sickle.

\( \alpha(z)=\frac{\ln(z-l_1)}{\ln(\lambda_1)} + \frac{\ln(z-l_2)}{\ln(\lambda_2)} + p(z)\;\; \) Abel function