continuation of fix A to fix B ?

[Leo.W]

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.

...

regards

tommy1729

Well tommy, I wasn't focused on the computability, but on the interrelationships between members of schroder functions' family instead and it just generalizes another way of theta-mapping(analytically continued)
You're right tommy, it's piece of * when it comes to computation, most of the time.
* But, I say maybe, we can use non-analytic \(h\) and \(g\) right? in the summation case, we can recover the integral by \(h(z)= s^{-\lfloor z\rfloor}\) and \(g(z)=z\), in this case it theoretically converge in many cases, and what if we use, say, \(g(z)=e^{-\|z\|^2}\)?
Anyway just a thought