TITLE: $n$th root of $(a,b) \mapsto (\operatorname{gm}, \operatorname{am})$
QUESTION [7 upvotes]: Suppose $0 < a < b$, and let GM and AM be respectively the geometric and arithmetic means of $a$ and $b$.  Does the mapping $(a,b) \mapsto (\operatorname{GM}, \operatorname{AM})$ have a well-behaved compositional $n$th root?  In what senses might such an $n$th root be unique?

REPLY [2 votes]: A simple scheme (a parallel of which can be used to define the Gamma function from the factorial) is as follows: call $T$ your function, then define the functional square root as
$$T^{\frac{1}{2}}=\lim_{n\rightarrow\infty}T^{n+1}ST^{-n}$$
where $S$ is the sensible approximation of $T^{-\frac{1}{2}}$ for $b\gg a$, given by
$$S(a,b)=\Big(\frac{a^\sqrt{2}}{2^{2\sqrt{2}-5/2}b^{\sqrt{2}-1}}, \sqrt{2}b\Big)$$
Simple numerical testing shows $T^{n+1}ST^{-n}$ becoming rapidly more accurate (as $n$ grows) than $TS$ as a functional square root of $T$, and the limit above most likely converges to the only reasonable definition of $T^{\frac{1}{2}}$.
A (tedious) generalization to all fractional functional powers should be straightforward.