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

[sheldonison]

1.Is the sexpeta() merged upper & low superfunction?
2.Can the cheta/incheta/sexpeta/slogeta normal work when base isn't eta?

1. no sexpeta; and cheta are both generated from the Ecalle assymptotic series solution for \( f(z)=\exp(z)-1 \), which has two sectors.  The Ecalle assymptotic generates two different solutions, depending on whether you approach the fixed point of zero from z>0 by iterating \( f^{-1}(z)=\log(z+1) \) before evaluating the series, which generates the cheta upper superfunction, or whether you approach the fixed point from z<0 by iterating f(z) before evaluating the series which generates the sexpeta lower superfunction.  That two different analytic functions can be generated from the same assymptotic series makes sense when you realize Ecalle's solution is a divergent assymptotic series, and you need to iterate f(z) or f^-1(z) enough times so that |z| of z is small enough to generate an accurate result.

The conjecture is that the limit of Kneser, as the base approaches eta from above would be sexpeta; I don't know if the conjecture has been proven.

2. no, the code for cheta/eta are only for exp(z)-1.   For any given assymptotic Ecalle series, you pretty much need to handcode evaluating that series, taking into account the required error terms and how much precision you want, and how many terms of the series to generate, and which of the 2n sectors of the f^n function you are interested in.  For the superfunction, you need approximations of the inverse of ecalle; the approximations I use are not generic. Also, it might get pretty tricky as the denominator of the derivative at the fixed point grows beyond small single digits ...