Let a function $f$ be on $R^n$ into $R^n$ and let k be a positive integer. Proposition: There exist a function $g$ on $R^n$ into $R^n$ such that $f$ equals $g$ composed $k$ times with itself: \begin{align} f(x) &= \underbrace{g(g(\cdots g(}_{k}x)\cdots)) \\ &= g^k(x) \end{align} If this is false, then under what conditions does such a function $g$ exist? In either case, if a $g$ exists, then can something be said about the complexity of the function $g$? For example, if $f$ is composed of elementary functions, then $g$ is composed of elementary functions?
I am wondering because when formulating a prediction one can either formulate a 1-step prediction and then derive a k-step prediction, or directly formulate a k-step prediction. If the proposition is false, then there can exist cases where a k-step prediction is better than any compostion of 1-step predictions. Furthermore, if $g$ is in general "more complex" than $f$, then it may in some applications be motivated to first try formulating a k-step prediction.
