(07/04/2022, 02:12 AM)Daniel Wrote: A troubling question that occured to me is if my derivation of the Taylor's series of \( f^n(z) \) is correct, due to its generality it must contain all solutions that don't have a super-attracting fixed point. Since all functions except the successor function have finite fixed points, if f(z) is smooth then my approach should be valid. So shouldn't all valid methods give the same results? Yeah, lots of different tetrations and all, but shouldn't they all agree on their common areas? Aren't we all studying parts of the same "elephant"?
Your iteration method is the basis for iterations. But no, they don't agree.
The same fallacy seems to be making its way around this forum, and I keep on having to correct it.
If \(f\) is a holomorphic function, and has two fixed points \(x_0, x_1\). Then the iteration \(f^{\circ s}(z)\) for \(z \approx x_0\) is NOT THE SAME FUNCTION, as the iteration \(f^{\circ s}(z)\) for \(z \approx x_1\). You CANNOT make them one function. It's incorrect. If you iterate \(\sqrt{2}^z\) about \(z\approx 2\), it is NOT THE SAME iteration as iterating \(\sqrt{2}^z\) about \(z\approx 4\).
We can also iterate from periodic points too, and that can be even MORE COMPLICATED. They are not the same elephant.
So, as your iteration method works, it works to construct Schroder's iteration about a fixed point. This is commonly referred to as the standard iteration, or the regular iteration.
The higher order problems are to create an iteration which works globally (and hence for no fixed points--like Kneser). Or to construct super functions, which there are uncountably many.
Your iteration method is precisely the local iteration. And any local iteration looks like your iteration method. The trouble lies in extending iterations to larger domains (excluding other fixed points), and dealing with more exotic constructions.