UnAbel to do that

[sheldonison]

The Abel functional equation and the Schroeder functional equation represent different symmetries. Saying that there can be a mathematical connection between the Abel and Schroeder functional equations doesn't mean that they coexist in the same system. Let's consider complex functions. Generally speaking, we will be looking at a system with an infinite set of hyperbolic fixed points, although complex conjugates can give pairs of parabolic fixed points along with infinite hyperbolic fixed points. The same for parabolically neutral fixed points. So while some folks like the properties of Abel's functional equation, the mathematical existence of these systems are rare.

I'm not that fluent in group theory, which is where symmetries would be studied, but I certainly get that the set of Tetration bases with an attracting fixed point is bound by our familiar Shell-Thron boundary curve.
   

All of the bases inside this curve have an attracting fixed point with a well defined Schroeder equation from which the standard equation can be used to generate the Abel function, and the slog. and this family of slogs is an analytic family of slogs.  And for bases outside this region, the function \( \alpha_b(0) \) is a singularity.  
\( \Psi(b^z)=\lambda_b\Psi(z);\;\;\;\lambda_b \) is the derivative at the attracting fixed point.
\( \alpha_b(z)=\frac{\ln(\Psi_b(z))}{\ln(\lambda_b)};\;\;\;\text{slog}_b(z)=\alpha_b(z)-\alpha_b(1); \)

One interesting question is does this limit exist, and is it equal to the slog generated from the Ecalle's parabolic Abel function at \( \eta=exp(1/e) \)?
\( \lim_{b \to \eta}\text{slog}_b(z)=? \)

And then another interesting question, for the Kneser's family of slogs, is the limit the same as the base approaches eta (from>eta)?  The Kneser family of slogs can be extended to the inside of the Shell-Thron region, and would be unequal to the Schroeder family of slogs everywhere else other than possibly b=eta.