new fatou.gp program

[sheldonison]

Hey Sheldon, just want to say how amazed I am how extensively you developed your program, including the findings of Ecalle and the perturbed Fatou coordinates, while I was absent. Perhaps later I will ask some question about the analytic continuation of the base, which still puzzles me. Meanwhile I will read your fatou.gp to finally understand, what you are doing there

Thanks Henryk.  I wanted to prove fatou would converge, but I hit interesting roadblocks when I try.  Here is a base(e) picture showing 60 sample points, and the correspondence.  I don't remember where I posted this before, but it belongs in this thread.
   
You can run sexpinit(exp(1)) which iterates, but you can also solve a linear system of equations, which works equally well, but you need to know how many samples you want for the theta mapping ahead of time, and its generally extra work, but I think more interesting for understanding how the program works.  matrix_ir(1,60,8 ) also solves slog base(e).  With 60 sample points, the precision is good for about 12 digits of precision with 61 terms in the Taylor series ranging from x^1, to x^60.  For the system of equations, or the iterative solution, the constant term is required to be zero.

Anyway, the points in yellow on the middle circle border are paired up with their exponent in yellow in the inner circle.  The points in brown are paired up with their logarithm in the inner circle.  The points in pink in the inner circle (and green for the conjugate) are used to tell the regular slog's Abel function what its theta mapping is.  Here, alpha_u is the upper abel function from the logarithm of the Schroeder equation, and theta_u is the theta mapping.  That is used to define the points in pink (and green) in the middle circle.  Solve the system of equations, or iterate adding sample points as you go, and you are solving this picture, as the number of sample points goes to infinity.  Here, all the points are paired up with data points in a smaller radius circle, so the high frequency components aren't as relevant and the solution is both linear and very stable.  The same is true of the theta mappings.  
For Kneser's slog, there is a complex valued inverse superfunction in the upper half of complex plane, and another in the lower half of the complex plane, generated from the formal Schroder function which my program generates.
\( \alpha_u=\frac{\ln(\psi(z))}{\ln(\lambda)};\;\;\lambda=L;\;\; \) for base_e lambda=L
\( \alpha_r(z)=\alpha_u(z)+\theta_u(\alpha_u(z));\;\; \) real valued Abel function via 1-cyclic theta mapping
\( \alpha_r(z)=\frac{\ln(z-L_1)}{\ln(\lambda_1)} + \frac{\ln(z-L_2)}{\ln(\lambda_2)} + p(z)\;\; \) real valued abel function via Taylor series p(z) centered between fixed points
\( \text{slog}(z)=\alpha_r(z)-\alpha_r(1) \)

I have lots more charts I could post as time permits.  The algorithm works for lots of repelling and attracting complex bases, as long as they're not on the Shell Thron boundary or have a period too close to pseudo period-2, but you can get really close to the boundary and it still works, just a little slower.