Fractional iteration of x^2+1 at infinity and fractional iteration of exp

[bo198214]

in thread tid 403 I ,

I consider it as standard on the forum to *link* to posts/threads.
Isnt your reader worth this little extra effort?

Bo , Mike and Ben already discussed moving fixpoints
in particular - as the title says - f(f(x)) = exp(x) + x.
this thread seems similar.

So? You mean I violated copyright law?!
I think there is enough original never here discussed stuff in my post.

however , i consider this slightly different.

exp(x) + x has a " true " fixpoint at oo.

with " true " i mean that it touches the id(x) line.

in this case of x^2 + 1 , i would not call oo a fixpoint , but rather say that Bo uses so-called " linearization " :

In standard literature points z, which satisfy f(z)=z are called fixpoint, so do i here f(oo)=oo. Via the fixpoint exchange 0<->oo you can watch this fixpoint also as the real point 0.

g^[-1](f^[n](g(x))) = [g^[-1](f(g(x))]^[n]

Dunno what that has todo with "fixpoint", nor what it has to do with "linearity".

\( g^t(x)=\lim_{n\to\infty} \beta^{-1}(\beta(x)^{2^t}) \).

you take lim n -> oo ... and then i see no n.

so this needs a correction.

Ya I corrected that.

Again similar to the Schröder case, we have an alternative expression:
\( g^t(x)=\lim_{n\to\infty} g^{-n}\left(g^n(x)^{2^t}\right) \)

If we even roll back our conjugation with \( 1/x \) we get:
\( f^t(x)=\lim_{n\to\infty} f^{-n}\left(f^n(x)^{2^t}\right) \).

i dont get how you arrive at this ... g and f on both sides ?

Sure, we are about to determine the fractional iterates \( f^t \), but we know the function f and its integer iterates. Note, that the Kuczma convention is: If you apply a power to a function, they mean the iterate of the function, e.g. f^n, if however they apply a power to a number then they mean the multiplicative power, e.g. f(x)^n.

This may lead into a new way of computing fractional iterates of exp, because we just approximate exp(x) with polynomials \( \sum_{n=0}^N \frac{x^n}{n!} \) and approximate the half-iterate of exp with the half-iterate of these polynomials.

hmm ... i wonder if (x^2 + 1)^[h] is analytic at x = 0 for small h.

Yes, exactly, thats a topic and may shine also some light on the by you defined iterates of exp with help of the iterates of sinh. It also uses the fixpoint at infinity.

If not analytic at 0 then I guess its at least meromorphic everywhere else on the complex sphere, because the Böttcher iteration yields analytic functions in the vicinity of the fixpoint.

the reason is that lim h-> 0 (x^2 + 1)^[h] = abs(x) and abs(x) is not analytic at 0.

why is that so? Oh you mean you verified numerically, hm interesting observation.

also approximating exp with polynomials might be troublesome ; does the n'th approximation of the half-iterate converge when n -> oo ? do we really get an analytic function at n = oo ?

That needs to be investigated Its also possible to terribly fail! Was just an idea at the end of explaining a new method to iterate polynomials.

polynomials also have zero's and those zero's will need to drift towards oo fast if the sequence of polynomials wants to approximate exp(x) well.

You mean the fixpoint of the odd degree polynomial will need to drift quickly towards -oo?
Ya perhaps, the Böttcher iterate is quite probably not analytic at the fixpoint. But in how far that invalidates the approximation, I have no idea and no numerical experiments made yet.