Functional square root

Summary

A functional square root of a map $f$ is a map $g$ with $g(g(x)) = f(x)$ for all $x$. It is the iterative root of order $2$, and it is often called a half-iterate of $f$. It answers the question of what it means to apply $f$ half a time. Functional square roots of the exponential function and of other simple maps have a long history in iteration theory.