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

[sheldonison]

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?

The first half of my answer was only discussing sexp generated via Ecalle for base eta=exp(1/e).  Base exp(1/e) has only one neutral fixed point.  Kneser isn't defined for base exp(1/e).  Kneser is defined for base=e, and if you gradually modify the real valued Kneser base, then you get to a singularity at base exp(1/e) where there is only one fixed point.  Even though Kneser isn't defined at base exp(1/e), the conjecture is that the limit as you approach exp(1/e) from bases a little bit bigger than exp(1/e) is defined, and in the limit Kneser's slog would equal Ecalle's Abel function for base exp(1/e) plus a constant.

\( \lim_{b\to\eta^{+}}\text{slog}_b(z)=\alpha(z)+k\;\; \) where \( \alpha(z) \) is Ecalle's Abel function for base exp(1/e) from the attracting sector, and k is a constant

You can't compare Ecalle's solution for other rational bases on the Shell Thron region to Kneser since Ecalle is the Abel function for \( f^{\circ n}(z) \) where n is the denominator of the rational root of unity that is the derivative of f(z) at the fixed point whereas Kneser's slog could be compared to the Abel function for f(z).  Does that answer your question?