Carleman approximation by Symplectomorphisms

We will discuss Carleman approximation in SCV, and in particular we will focus on recent results on Carleman approximation on real co-adjoint orbits in closed complex co-adjoint orbits. A special case is the case of $\mathbb R^{2n}\subset\mathbb C^{2n}$, where the result is that any symplectic automorphism of $\mathbb R^{2n}$ is approximable in the sense of Carleman by holomorphic symplectomorphisms of $\mathbb C^{2n}$ that leaves $\mathbb R^{2n}$ invariant.
This is joint work with Fusheng Deng.