It is known that an $O( \epsilon)$-close to identity map $x_{1} = x_{0} + \epsilon f(x_{0})$, where $\epsilon$ is a small parameter, is approximated by a time-$\epsilon$ map along a level curve of a certain autonomous flow. If $f \in C^{\infty}$, then it can be approximated to $O(\epsilon^{j})$ for arbitrary large $j$. If $f$ is analytic, then the error can be made exponentially small in $\epsilon$.
I am aware of the fact that the above result is obtained from the averaging theory for odes, however I don't know how to prove it. I'd be grateful if someone could provide a sketch of the proof or some references to published literature where this result is proved.
Also, how would the above accuracy change if $f \in C^{m}$ for some finite $m$? Does it then follow that the map is approximated to $O(\epsilon^{m})$?