Apropos "fix"point: are the fractional iterations from there "fix" as well?

[tommy1729]

So, this is actually a defining property of the standard Schroder iteration. But it's a little difficult to fully flesh out why.  Now, to begin I'll construct an arbitrary iteration which has a constant noodle, and show there are many of them.

If you iterate locally about a fixed point, and your solution satisfies \(f^t(p) = p\), then the iteration is expressible via Shroder iteration (provided that \(|f'(p)| \neq 0,1\)).  For convenience, assume that \(|f'(p)| < 1\).

You can actually prove this pretty fast. Assume that \(f^t(x)\) is a super function in \(t\) about a fixed point \(p\), and \(x\) is in the neighborhood of \(p\). Assume that \(f^t(p) = p\).  Well then:

\[
\Psi(f^{t+1}(x)) = \lambda \Psi(f^t(x))\\
\]

So that:

\[
\theta(t) = \frac{\Psi(f^t(x))}{\lambda^t}\\
\]

And we know that there must be some 1-periodic function \(\theta(t)\), such that:

\[
f^t(x) = \Psi^{-1}\left(\lambda^{t}\theta(t) \Psi(x)\right)\\
\]

In fact, any periodic function will work fine here, and will have a constant noodle, will be a super function, but will not be Schroder iteration.

Now let's add one more constraint, let's say that \(f^{t}(f^{s}(x)) = f^{t+s}(x)\). Well then, we have that \(\theta\) must be constant. By which, we are guaranteed that it's a Schroder iteration...


So to clarify. Any fractional iteration with a constant noodle is a Schroder iteration. But there are plenty of superfunctions of \(f\) which have a constant noodle, but in turn, they aren't fractional iterations then (don't satisfy the semi-group law).

This is not acceptable for me.
Not formal , detailed and general enough.

It is actually quite simple 

even without fixpoints.

\(f^{t}(f^{s}(x)) = f^{t+s}(x)\). 

let theta(v) be a one periodic function.

so do we get 

\(f^{t+theta(t)}(f^{s+theta(s)}(x)) = f^{t+s+theta(s+t)}(x)\). 

well that would require 

theta(v+1) = theta(v)


and

theta(t + s) = theta(t) + theta(s)

so theta must be constant.

keywords :  cauchy functional equation , axiom of choice , linear function , non-constructive , non-periodic.


So we get a (local ?) uniqueness criterion.

This has many consequences.

(for instance probably if you use iterations of another function to get to iterations of yours , it is required that the other function has the  semi-group property.)

So we have uniqueness up to convergeance speed ofcourse.

This should be on page 1 of any dynamics book !


regards

tommy1729