Is bugs or features for fatou.gp super-logarithm?

[sheldonison]

[edit2] The Schroeder function for neutral points does not converge if the period is a rational number.  So a period=4 Schroeder function doesn't exist since it doesn't converge.  But if the period is a well behaved irrational number (with imaginary part=0) then the Schroeder function does converge.  For example, a period=pi Schroeder function does converge.  See 
https://en.wikipedia.org/wiki/Brjuno_number, and http://www.scholarpedia.org/article/Sieg...egel_disks.  The Kneser Tetration and its inverse are analytic even if the period is rational.  The experiment is to look at all of the Taylor series coefficients of the slog and see how they vary as the base changes in the complex plane.  There is an analytic function for each of these Taylor series coefficients.  So from the bases around the rational base, one can construct an arbitrarily accurate series even though the fatou.gp algorithm isn't ideal for computation of such neutral rational period bases.

The last edit it's sounds strange. So Will the Kneser Tetration and its inverse can exist if Schroeder function doesn't exist? 
The base = E^-E is a singularity?

B~=1.98933207608102 + 1.19328219946665*I; 
B=bfromp(4); /* using fatou_experiment.gp */
l~=0.540302305868140 + 0.841470984807897*I; 
\( B^{l+x}-l = {i}x + {a_2}x^2 + {a_3}x^3 ... \)
There is no Schroeder function from this neutral fixed point for base B. even though Kneser Tetration is analytic for base B.  For this base, the multiplier at the fixed point is \( \lambda=i=\exp(\frac{2\pi i}{4}) \), which is a period 4 multiplier since \( \lambda^4=1 \)

I know how to use the Riemann mapping theorem showing that Kneser exists for real bases at the real axis.  For complex bases, they start with a kernal around the base with a multipier of 1, where the fixed point bifurcates.  For tetration that is eta=exp(1/e). And then one can use the Measurable Riemann mapping theorem to show the Abel function exists.  But this is a pretty recent result in complex dynamics and I don't understand the details in spite of many attempts to read the paper by Lei Tan and Shishikura, reference here: 
https://math.eretrandre.org/hyperops_wik...oordinates

Just like base B above is not a singularity for Kneser, base exp(-e) with \( \lambda=-1 \) is not a singularity for Kneser either.  I have no algorithm for calculating results for base(exp(-e)), or to even to get approximate results. I am unaware of anyone else who can do calculations for this base either.  The proofs using both the Riemann mapping theorem and the measurable Riemann mapping are proofs of existence that don't give a usable algorithm for calculations.

Adrienne Douady conjectured the merged Abel function can be analytically extended until you hit the neutral boundary a second time (the Shell Thron boundary for tetration).  I posted a tetcomplex.gp program, which doesn't work anymore with current pari-gp, but there are some interesting results at this link: https://math.eretrandre.org/tetrationfor...hp?tid=729