We can define the iterates $f^{n+1}=f\circ f^n$ for a given smooth map $f:X\to X$, where $X$ could be a finite interval, the real line $\mathbb{R}$, or the circle $S^1$, or any general smooth manifold. What about the reverse direction? More precisely, a map $g:X\to X$ is said to be an $n$-th root of $f$ if $g^n=f$. There might be a standard notation for this. 


For convenience, let's say $R\_n(f)=\{g\in C^\infty(X):g^n=f\}$. The iteration map $g\in C^\infty(X)\to g^n$ is continuous. So the set $R\_n(f)$ should be discrete in $C^\infty(X)$.


In the following we may take $n=2$ for certainty.


Question 1. Does there exist a square root for any function $f$? If not, what are the possible obstructions?


Question 2. Suppose there exists one solution. What/when could we say about the uniqueness, finiteness, etc about the solutions?


A trivial example: $X=\mathbb{R}^n$ and $f$ is the identity. Then there are at least two square roots: $g\_1(x)=x$ and $g\_2(x)=-x$. I am not sure if these are the only two solutions. 


**Edit:** see [link](https://mathoverflow.net/questions/162370/solution-of-functional-equation-f-circ-kx-x) for the discussion in this special case.


For question 1 in the case when $X$ has nontrivial homology, we can consider the induced action on various homology groups of $X$. In particular, the action $[f]$ must admit a square root matrix.


**Edit:** See [link](https://mathoverflow.net/questions/12081/does-the-exponential-function-have-a-square-root?rq=1) for discussions about the exponential function and a useful [link](http://reglos.de/lars/ffx.html) provided there. 



Existence of a square root of a functional equation
An answer to a restriction of this problem:


Let $F$ be the space of *strictly increasing* twice-differentiable functions $f:I \mapsto I$ on an open interval $I \subset \Bbb{R}$ such that the first derivative of $f$ is strictly increasing. Then for each $f\in F$ there exists a *unique* function $g \in F$ such that $g^2 = f$. 


This statement holds if you change either (or both) of the words "increasing" to "decreasing.


