new fatou.gp program

[sheldonison]

In addition to finally posting my pari-gp Abel(z)/slog(z) program which I call fatou.gp (see below)

Why is this called "Fatou"? I know this is the name of a mathematician, but other than that, I'm confused. What does "Fatou" mean to you? Is it a reference to a functional equation? If so, then which one?

Pierre Fatou, but the algorithm is named because it works with the perturbed fatou coordinate, from complex dynamics. The work of Shishikura is also relevant. Adrien Douady mentioned Fatou Coordinates in some of his work. They all refer to the Abel function for
\( f(z)=z+\sum_{n=2}^{\infty}a_n\cdot z^n + k\;\; \) where k is the perturbation of the parabolic \( z\mapsto z+z^2 \) case.
There is an explicit asymptotic series/formula for the Abel function for k=0. My program iterates \( f(z)=\exp(z)-1+k \) which matches the perturbation of the parabolic case for exponentiation. We can go back and forth between the "k" perturbation abel formula and the sexp/slog base "b", by using the conversion formula:
\( b=\exp(\exp(k-1)) \)
\( y \mapsto b^y\;\; \) is congruent to working with \( z \mapsto \exp(z)-1+k \)
\( z=y\cdot \exp(k-1)+k-1 \)

So for example, k=0 is the parabolic case, and is iterating \( z \mapsto \exp(z)-1 \) and it is congruent or equivalent to iterating \( y \mapsto \eta^y\;\; \) where \( \eta=\exp(\exp(-1))\;\; z=\frac{y}{e}-1 \)
So k=0, is exactly Jay's base eta; \( \;\eta=\exp(1/e)\;\; \) And there is a formal solution, from Ecalle, for the parabolic case as well. There are posts on Ecalle's formal asymptotic Abel series solution for the parabolic case on mathoverflow by Will Jagy; see http://mathoverflow.net/questions/45608/...onvergence

So my attempt in writing fatou.gp. was to write a program that might help me learn some of the mathematics of perturbed fatou coordinates, and thereby give Tetration a stronger mathematical footing. The fatou.gp program also includes an "x2mode=1;" setting where the program iterates solving the abel/invabel function for \( z\mapsto z^2+z+k\;\; \), whereas in the default x2mode=0, the abel/invabel subroutines are called by the Tetration subroutines for sexp/slog using the conversion formula I just posted because in the default mode, the guts of the program actually works on the mapping \( z \mapsto \exp(z)-1+k\; \) as opposed to the \( y \mapsto b^y\; \) mapping. Also, for the x2mode=1, as far as I know, the numerical solution for solving the Abel function of \( z \mapsto z^2+z+k\;\; \) using both fixed points is also a non-trivial problem also worthy of studying.