The $n$-th iterative root of a function $x$ refers to a function $f(x)$ such that when composed with itself $n$ times, i.e., $f^{[n]}(x) \equiv \underbrace{f \circ f \circ \cdots f }_{n}(x)$, we get back the original function $x$, that is the solution of functional equation $f^{[n]}(x) = x$.
My questions:
Could you provide some non-trivial examples(not $x$, $a-x$, $1/x$) for different $n$, e.g. $n = 2, 3, 4, \cdots$? Where can I find the list of solutions? Or is there general method to generate all solutions?
One family of solutions uses linear fractional transformations $\frac{ax+b}{cx+d}$, and the nice fact that the composition of linear fractional transformations acts like matrix multiplication: if $f(x) = \frac{ax+b}{cx+d}$ and $g(x) = \frac{jx+k}{mx+n}$, then $f(g(x)) = \frac{sx+t}{ux+v}$ where $\begin{pmatrix} s & t \\ u & v \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} j & k \\ m & n \end{pmatrix}$.
In particular, we can use rotation matrices $R_\theta = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$ of finite order: if $\theta=\frac{2\pi}k$ then $R_\theta^k = I$. For example, the function $$ f(x) = \frac{\cos(\frac{2\pi}7)x+\sin(\frac{2\pi}7)}{-\sin(\frac{2\pi}7)x+\cos(\frac{2\pi}7)} $$ satisfies $f(f(f(f(f(f(f(x)))))))=x$.