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

[sheldonison]

Congratulations, you have completed the key work.
All my calculations are testing superroot.gp, I will test it when I have finish the current work.

(New Edit)
So, What happened between fatou_experiment.gp and fatou.gp when bases is no theta mapping?

Thanks for testing, Ember.  The bases with no theta mapping are the bases very close the the Shell Thron region, whose period's have imaginary part of zero.  sexpinit(B) hasn't changed in fatou_experiment.gp.  In fatou_experiment, there is a function bfromp(period).  Let's consider two tetration bases, one near the Shell Thron boundary, and one practically on top of the of the Shell Thron boundary.

Code:

\r fatou_experiment.gp
zb4=1.989+1.193*I;      /* period is 4.000-0.0003i; too close to Shell Thron boundary; no theta mapping */
zb4_005i=1.995+1.195*I; /* period is 4.000+0.005i; has a theta mapping; but its slow */
limitp=20;              /* limit precision so it doesn't run toooo long */
sexpinit(zb4_005i);     /* initializes after 25 iterations, 22 seconds, 20 decimal digits */
matrix_ir(zb4_005i,132,82,8/10,15/16); /* using settings from sexpinit(zb_005i) but faster; 3 seconds */
sexpinit(zb4);          /* no theta mapping; 14 decimal digits after 70 iterations 40 seconds, 888 samples */
matrix_r(zb4,888);      /* no theta mapping, 13.6 decimal digits; 2min 35 seconds */

So that's a Period=4 example.  For period=4, we get some convergence without a theta mapping by taking advantage of Jay's accelerated slog representation, which I use both with and without a theta mapping to improve convergence. 

Near the origin is a much more difficult problem. The period from the upper fixed point for B=exp(-e) is exactly 2.  The accelerated representation doesn't help as much as the Period gets closer to 2, and, as can be seen in the plot for B=0.015 above, the Schroeder function for some of the nearby bases is really badly behaved so it takes a lot of theta points, plus the sample radius has to be closer to 1 to get convergence.  The posted results used matrix_ir(0.15,400,90,14/15,45/46).  What settings to use for matrix_ir is a matter for experimentation.  Relaxing the sample radius, and increasing the number of theta samples seems to give good results too; matrix_ir(0.15,248,160,8/9,45/46).  However, matrix_r(0.15,n) without a theta mapping doesn't not converge, as is shown by the sample chart showing the points need to pair up with points outside of the sampling circle.  The matrix becomes unstable, and the results have negative precision.

[edit1] The theta mapping for these bases greater than exp(-e) seems poorly behaved and requires a larger number of coefficients to converge than one might expect.  It would be nice to better understand the theta mapping and its radius of convergence for base=0.15, and its nearest singularity.  For B=0.1+I*E-30; I needed 250 terms in the theta mapping! matrix_ir(B,400,250,14/15,45/46) to get a bit over 16 decimal digits of precision as compared with 90 terms for B=0.15 matrix_ir(0.15,400,90,14/15,45/46).  Either way, where is the nearest singularity in the theta function?  I also don't know why I needed the small imaginary offset for B=0.1; otherwise the Schroeder function to Abel function conversion gets confused, but I don't know why.

[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.