Fundamental Principles of Tetration

[Daniel]

Now, start with the two fixed points, L,L*=0.31813 +/- 1.3372i; each fixed point leads to a Schroeder equation, and its inverse.

Yes, I am familiar with the fixed points of the exponential function \( e^z \)

You are familiar with the Schroeder equation, lets call the Schroeder solution \( S(z) \). Call the Abel equation \( \alpha(z) \), and its inverse, \( \alpha^{-z}(z) \)

\( \alpha^{-1}(z) = S^{-1}(L^z)\;\;\; \) This is the complex valued superfunction, which you are familiar with

Using the Classification of Fixed Points, there are several types of tetration. Hyperbolic tetration given by Schroeder's equation which accounts for almost all values. Parabolic tetration given by Abel's equation is only for \( a=e^{1/e} \), rationally neutral tetration \( a=e^{-e} \), super attracting tetration \( a=1 \). While I have looked for a simple way to move between Abel's and Schroeder's equations, they are not topologically conjugate.