[MSE] Shape of orbit of iterations with base b on Shell-Thron-region

[sheldonison]

 Does that answer your question?

Is very helpful. My last question is fatou.gp how to merge upper and lower superfunction.

There is post#15 in the fatou.gp thread which includes some equations and a picture of how I setup a 60x60 system of equations to solve for the base_e slog.  https://math.eretrandre.org/tetrationfor...98#pid8998

You need the two Schroder functions for each fixed point, and then you can get Koenig's Abel function by taking the logarithm of the Schroder function, from each fixed point.  That gives you an upper and a lower Abel complex valued function; each Abel function has a 1-cyclic mapping that goes from the Abel function to Kneser's slog.  The 1-cyclic mapping for the upper Abel function decays to a constant as imag(z) to infinity, which is the uniqueness criteria.  Ditto for the lower Abel function.  So I have three mathematically identical representations for Kneser's slog which work best in different parts of the complex plane.

• slog(z)=\( \sum_{n = 0}^{\infty} a_n (z-\Re(L))^n\;\; \) The Taylor series representation
  
• slog(z)=\( \alpha_u(z)+\theta_u(\alpha_u(z))\;\; \) the upper Abel function plus theta mapping
  
• slog(z)=\( \alpha_l(z)+\theta_l(\alpha_l(z))\;\; \) the lower Abel function plus theta mapping
  

So, the system of equations requires these three functions to be identical over carefully chosen subsets of points in the picture, and this leads to a well behaved linear set of equations that converges very nicely.