continuation of fix A to fix B ?

[tommy1729]

Just a thought.
it can be extended to the family of schroder function by
\[\sigma_f(z) = \int_{\mathbb{R}}{h(k)g(f^k(z))\mathrm{d}k}\] (and thus on measurable set in the way that k has some kinda shift-invariance), where \(h\) is any function that fits \(h(k-1)=s\,h(k)\), \(s\) is the multiplier, and \(g\) is any function.

But you cannot do an integral over something with a f^[k](x) dk term because that requires knowing fractional iteration already.

Since you do not have f^[1/2] this seems like a circular reasoning.

So this is a purely theoretical idea and not a computable one .. at first sight.

And also h(k+1) may be c h(k) , but h(k+1/2) is not sqrt© h(k).
And even h(k+1/2)/h(k) is different for most k.
So i see many objections and issues with that.

As for the g(x) part , this might give additional fixpoints and such to limit the region of convergeance or analyticity.

Also I want to warn extra about using g(x) without care :

the following example illustrates why ;

let M(x) = 1/exp(x) + 1/exp(exp(x)) + 1/exp^[3](x) + ...

this M(x) converges for all real x > 0

but it is nowhere convergeant on the complex away from the real line.

Let alone analytic.

If we lower the base of M(x) to smaller then eta but larger than 1 , the 1/ part was not needed and we would have convergeance locally ( x between the fixpoints ) 

Analytic is a desire and intention here.

I understand why ppl like integral representations over sums.
But I think it is psychological :  It WORKED NICELY for standard functions and laplace and fourier stuff.

But for such exotic cases there are many issues and numerical and theoretical problems ... not even mentioning closed forms lacking and analytic issues ...

There are even more issues but I do not want to give an angry impression

regards

tommy1729