Which are the n-times differentiable real functions that fit the condition:
$f = f' \circ f'' \circ \ldots \circ f^{(n)}$ ?
I think I have came up with a tentative solution for $n=2$, which may generalize for arbitrary $n$:
Let $f = kx^m$, then $f' = kmx^{m-1}$ and $f'' = km(m-1)x^{m-2}$,
then $$f' \circ f'' = [kmk^{m-1}m^{m-1}(m-1)^{m-1}]x^{(m-1)(m-2)} = f = kx^m.$$
What demands that:
(i) $m = (m-1)(m-2) \Rightarrow m_1 = 2 - \sqrt 2$ or $m_2 = 2 + \sqrt 2$
(ii) with $m = m_1$ or $m = m_2$ fixed, solve for $k$ such that
$[kmk^{m-1}m^{m-1}(m-1)^{m-1}] = k \Rightarrow k^{m}m^{m}(m-1)^{m-1} = k \Rightarrow k^{m-1} = \frac {1}{m^{m}(m-1)^{m-1}}$, since $k \neq 0$ (assuming non-trivial solution), and $m\neq 0, m \neq 1$ (due to (i)) $\Rightarrow k = \frac {1}{m^{m}(m-1)^{m-1}} ^{\frac {1}{m-1}}$.
With $m$ and $k$ as such we can backsubstitute and done.
Are there any other classes of solutions for this?
Extra: Call $f^{*}_n$ the solution of the above problem when the composition goes up to $n$. Does the sequence $\{f^{*}_n\}$ have any interesting property?
Most importantly: is there any associated application or physical interpretation?
Thanks.