On extension to "other" iteration roots

[tommy1729]

but we in everyday talk and in forum it's default to say something taking singlevalued-ness for granted,

On the other hand, in the complex plane, it is rather the default to consider multivaluedness. E.g. log or sqrt, when you continue from one point around the singularity back to the same point, you arrive at a different value and that's also the secret with the iteration group, it is not so much having multiple separate values, they just come naturally into existence by analytic continuation (and that's why the graph is one line not many separate lines).

Apropos Abelian property: just want to remind you to be cautious with the term, because \(f^{s}\circ f^{t} = f^{t}\circ f^{s}\) does not automatically imply \(f^{s}\circ f^{t} = f^{s+t}\) if the latter is what you actually mean.
I say that because I just encountered that case with the real valued Fibonacci extension
\begin{align}
\phi'_t &:=\frac{\Phi^t+\cos(\pi t)(-\Psi)^t}{\Phi-\Psi}\\
f^{t}(z) &:= \frac{\phi'_t + \phi'_{t-1}z}{\phi'_{t+1} + \phi'_t z}\\
\end{align}
There we have \(f^{s}\circ f^{t} = f^{t}\circ f^{s}\) but we don't have \(f^{s}\circ f^{t} = f^{s+t}\) for most s,t and hence it is not a continous iteration group.
And that would actually be your counterexample to have a real single valued function family, with the Abelian property at a fixed point with negative multiplier

I call the continuous iteration group valid within a jordan curve on the complex plane and being analytic there a " flow " and the related ideas " flow theory ".

which bring me to the natural question

if we have a continuous iteration group for real s and t within real interval [a,b] , meaning s + t is within [a,b] and s and t being positive, does that mean that we have a " flow " near the interval [a,b].

In other words does the property hold for complex s and t near [a,b] and is it analytic there ?

and this is the same as the semi-group homom by simply adding the z in f^[s] ( f^[s](z) ).

If the answer is yes , this implies 2sinh method is analytic.



regards

tommy1729