Picard's little theorem says that the image of a complex analytic function from $\mathbb{C}$ to itself has as image either a single point, or the entire plane without a single point or the entire complex plane. Inspired by the question For which $n$ can the plane with $n$ points removed be equipped with a Lie group structure? I was wondering what is the analogous statement for real analytic maps from $\mathbb{R}^2$ to $\mathbb{R}^2$?
I was of course pretty happy with the 'topological' answer I gave to the linked question, but I wondered if a different, 'analytical' answer was possible along the following lines: 'suppose that a Lie group structure on $\mathbb{R}^2$ with $n$ points removed existed, then the exponential map to this Lie group from its Lie algebra would be an example of a real analytic map from $\mathbb{R}^2$ to itself whose image is an open subset of $[\mathbb{R}^2$ with $n$ points removed$]$, which is only possible for $n \leq 1$ by the (hypothetical) real analytic version of Picard's theorem.'
Does such a theorem exist? What about real analytic maps from $\mathbb{R}^n$ to $\mathbb{R}^n$? More concretely (forgetting the application to the other MSE question):
What can the image of real analytic function look like?
