(06/08/2022, 03:39 AM)Catullus Wrote: I wonder how the analytical continuation of the schröder tetraion behaves.
It's fairly well studied. It looks essentially like \(f^{\circ s}(z)\) for any transcendental function \(f\) about a repelling fixedpoint \(L\). For any transcendental function, \(f: \mathbb{C} \to \mathbb{C}\) with a repelling fixed point \(L\) has a transcendental inverse Schroder function \(\Psi^{-1}:\mathbb{C} \to \mathbb{C}\). Then:
\[
f(\Psi^{-1}(z)) = \Psi^{-1}(\lambda z)\\
\]
Where, \(\lambda= f'(L)\). Then, so long as \(1\) is in the image of \(f\), there always exists a function:
\[
f^{\circ s}(1) = \Psi^{-1}(\lambda^{s-s_0})\\
\]
It will be entire in \(s\) (transcendental), and will satisfy \(f^{\circ 0}(1) = 1\).
