Is the limit of symplectic diffeomophisms a diffeomorphism? Suppose we have a sequence, { f_n }, of symplectic diffeomorphisms of R^{2n}  converging to a function f.  By f_n converging to f I mean:  f_n converges to f in C^k(B_r) for every r > 0, where B_r is the ball centered at the origin of radius r.  Suppose k > 0.  
Question:  Is f is diffeomorphism?  Certainly, f is symplectic and hence a local diffeomorphism, and so my question is really: is f invertible?
Interestingly enough, this is not true if f_n is not symplectic (e.g. take f_n = x/n, which converges to 0 in C^k(B_r) for every r > 0).
Thanks!
 A: No, $f$ doesn't have to be invertible.  It has to be injective, even in the volume-preserving category rather than the symplectic category.  None of the singular values of $Df$ can go to 0, because some other singular value would go to $\infty$.  This shows that it is locally injective, and then globally it is not possible to make the locally injective pieces overlap.
But even when $n=1$, it does not have to be surjective.  In this case the symplectic condition says just that $f_n$ is area-preserving.  You can reshape larger and larger circles centered at $0$ to non-concentric circles with the same area that lie in the upper half plane.  So the image of $f$ could be the upper half plane.  Actually, I think that the image of $f$ can be any open topological disk with infinite area.
A: To find explicit examples of what Greg Kuperberg suggests, maybe one can try the following. Consider $\mathbb{R}^{2n}$ as the cotangent bundle of $\mathbb{R}^n$. Then diffeomorphisms between opens subsets of $\mathbb{R}^n$ lift to symplectomorphisms of the "cylinders" over these open subsets. So you can find a sequence of diffeomorphisms of $\mathbb{R}^n$ converging to a diffeomorphism of $\mathbb{R}^n$ onto a proper open subset of $\mathbb{R}^n$. When $n =1$ you can take a sequence converging to $f(x) = e^x$ (for instance $f_n(x) = e^x + \frac{x}{n}$). The symplectomorphisms obtained as the lift of these maps should give you an example of what Greg Kuperberg suggested. 
