Uniqueness of fractionally iterated functions

[Gottfried]

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