On extension to "other" iteration roots

[bo198214]

The point is that the iterations are defined in the way \(f^s\circ f^s = f^{s+t}\) when you say that f^s is an iteration, I don't see why would iterations not guarantee \(f^s\circ f^s = f^{s+t}\)...? And if you're not referring to iterations, then I personally think \(f_s\) is a lil better than \(f^s\) as notation.

Then why do you speak about "Abelian property" and "Abelian-ity" at all, why don't you call it just "iteration"?
For me, if someone says "iteration of a function", it means the iteration with natural numbers, perhaps extended to negative integers as iterated inverse function.
And the thing is in this case you can really say "the iteration".
Btw. the notation \(f^t\) is similarly badly chosen because you can not distinguish it from power - particularly if you work in a calculus that does not have the "(x)" appended - i.e. if I write \(f\circ g\) and not \(f(g(x))\), e.g. formal powerseries calculus.
That's why I typically write \(f^{\circ s}\) and I saw that also in literature.
But with non-integer iterations you can not talk about "the" iteration, that's why a notation that does not include the type of iteration is also misleading. I tried to work around that with a notation like \(f^{\mathfrak{R}t}\) for regular iteration, but sure this is a rather clumsy approach and also not used in literature.
In older papers you quite often encounter the simple \(f_s\) or even \(f(s,x)\) with \(f(s+t,x)=f(s,f(t,x))\).