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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?

maplemaple
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  • The best known example is $f(f(x)) = e^x $ on the entire real line, H. Kneser in the 1950's https://eudml.org/doc/150158 . When the function has real fixpoints the fractional iterates will be $C^\infty$ but piecewise $C^\omega$ http://mathoverflow.net/questions/45608/formal-power-series-convergence/46765#46765 – Will Jagy Mar 29 '23 at 22:02
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    Usually it is denoted $f^{[n]}(x)$ because $f^{(n)}(x)$ is n-th derivative. – Anixx Mar 30 '23 at 02:08
  • @Anixx Thanks for your correction – maplemaple Mar 30 '23 at 06:07

1 Answers1

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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 $f(f(f(f(f(f(f(x)))))))=x$.

Greg Martin
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