To avoid repetition see What’s a natural candidate for an analytic function that interpolates the tower function?What’s a natural candidate for an analytic function that interpolates the tower function? for my general explanation of tetration.
A distinguishing feature of the exponential function is that it has an infinite number of complex fixed points, for example $0.318132 + 1.33724 i$. The lack of real fixed points results in $f: \mathbb{R} \rightarrow \mathbb{C}$, where $f(f(x))=e^x$.
I recommend at least three books for anyone serious about this subject:
- An Introduction to Chaotic Dynamical Systems; Devaney
- Complex Dynamics; Carleson, Gamelin
- Iterative Functional Equations; Kuczma
In fact, a picture of the exponential map is on the cover of Devaney's book. Both dynamics and functional equations address fractional iteration. Elements of dynamics are covered in functional equations and the reverse is also true, at least when discussing fractional iteration.
One way to validate or invalidate a proposed solution is to see if it is consistent or inconsistent with the theorems of complex dynamics. If a solution doesn't treat a fixed point as a fixed point, then it is defective. If the solution is inconsistent with the linearization theorem in the neighborhood of a fixed point, then the solution is defective. If the solution is inconsistent with the classification of fixed points and violates topological conjugacy the solution is defective.
