Uniqueness of fractionally iterated functions

[JmsNxn]

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.

Daniel - If I got James right the very old thread "Bummer!" should be enlightening, where Henryk  noticed that problem first time.

Gottfried

Ya what is it, the same values up to like 1E-13, but then they disagree.

It's intrinsic to the iteration, if you follow Milnor, he even explains this dilemma. You cannot pass from an iteration in the fatou set and be holomorphic once we hit the Julia set. \(4\) is in the Julia set, and the iteration about \(2\) is in the fatou set. To construct the iteration about \(4\) we use it as in the fatou set of \(\log\), then \(2\) is in the Julia set. It dates back very far. Milnor honestly covers so much in this forum, and so much is rediscovered on this forum.

John Milnor is truly a god among men. His complex dynamics is unmatched, even though it dates to the 70s or what ever.