Let $f:[0,\infty)\rightarrow [0,\infty) $ be a smooth and monotone function s.t $f(0)=0$. Let $N\in\mathbb{N}$. Can we find a function $g: [0,\infty) \rightarrow [0,\infty) $ s.t $g\circ\cdots\circ g$ ($g$ composed with itself $N$ times) equals $f$?
Can we say something about $g$‘s monotonicity? Its smoothness? I cannot come up with any basic answers. Thanks in advance to the helpers.
This problem is quite nontrivial even for $N=2$ and specific simplest functions such as the exponential and sine ones.
The function $f(x)=\exp(x)−1$ is a valid candidate.
The fractional iterates have formal powerseries where the coefficients $c_k$ at $x^k$ are not constants but in fact polynomials in iteration-height $h$ (the polynomials having their order equal series-index $k$).
The formal solution of this is for instance in L. Comtet's "advanced combinatorics" around pages 140-144.
For this specific function, unfortunately, I.N.Baker has proved, that the powerseries for the fractional iterates all have convergence radius zero, so if some evaluation can be made, then only if some (strong) summation-procedure for divergent series can be applied.
Beside this, it might be possible to assume the series as asymptotic series and get meaningful approximations to consistent values just by truncation at some finite index of terms.
The formal handling of the problem can be described well in terms of infinite triangular matrices ("Carleman matrices") and their formal diagonalization and fractional powers.
I have explored this in many ways (amateurishly) and could provide links to little workouts of mine on my homepage; a bit more professional is a very simple introduction into the Carleman-matrix method for this function has been given by R. Aldrovandi & L. Freitas. Eri Jabotinsky studied the application and analysis of/with Carleman-matrices in much detail and depth, try to get links to articles via some search engine.
