Is a function of a simplectic matrix still a simplectic matrix?

Question

Given $M$, a simplectic matrix $(2n\times 2n)$, the function $f(M)=\exp(M)$ is still a simplectic matrix? More in general, what kind of properties has to have a function $f(M)$ in order to give a matrix which is simplectic? Thanks.

Answer 1

Consider the case $M = I$, for $n = 1$. Clearly $e^I$ is not symplectic, since $$(eI)' \Omega (eI) = e^2 I'\Omega I = e^2 I.$$ (I've used "prime" for "transpose").