The ultimate sanity check

[Daniel]

I no longer focus exclusively on tetration, I mostly work with smooth complex iterated functions \( f^n(z) \) to serve my attack on hyperoperations. 
...
I would feel comfortable with these other techniques if they could pass my sanity test.

My deep apologies but what was your own method again ?
I know you did alot of research , such as iterated integer interations and such.
Maybe you use those formula's and replace integer n with real x or complex z.
Im not sure what your analytic method is , what uniqueness it satisfies , what was proven or plotted etc.
Also I sometimes confuse you with Gadilakis and so. sorry.
I see you have a nice website with fractals and such, but i did not see it immediately.

regards

tommy1729

Hey Tommy, thanks for replying. For my work on tetration, use the "Tetration" link in the menu. For iterated functions see "Combinatorics" and "Dynamics". The information has been up for almost twenty years and precedes the Tetration Forum by several years. In my post sanity check I explain that my method for determining \( f^n(z) \) perfectly satisfies \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}({z^8}) \) for almost any smooth iterated function and \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}(z^{30}) \) for Schroeder's functional equation. 

What else does my derivation of \( f^n(z) \) do? Well, it is deeply consistent which Carleson and Gamelin book Complex Dynamics. It lead to the same theorems as the chapter on the classification of fixed points. Since I take the Taylor's series of \( f^n(z) \) it is guaranteed for integer tetrates of positive numbers. As far as what is plotted, that would be the fractals you mentioned.   

Best Wishes,
Daniel