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

[Ember Edison]

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.

1.What base can't merged? Is the all Shell-Thron-region rational base and Singularity base can't merged, or just Singularity base, or just eta?
2.What does "limit" mean? Is the \( {\lim_{\delta \to 0^+}}fatou.gp.sexp_{\eta+I*\delta}(z) \) is merge superfunction? upper? lower?