2 typos

1

Is the limit of symplecic diffeomophims 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!