Holomorphic Approximation of Symplectic Diffeomorphisms for Calogero--Moser Spaces

The real Calogero--Moser space $\mathcal{C}_n^\mathbb{R}$ is a noncompact, totally real submanifold of the complex Calogero--Moser space $\mathcal{C}_n$. We prove that every symplectic diffeomorphism of $\mathcal{C}_n^\mathbb{R}$ smoothly isotopic to the identity can be approximated in the fine Whitney topology -- the strongest in this context -- by holomorphic symplectic automorphisms of $\mathcal{C}_n$ that preserve $\mathcal{C}_n^\mathbb{R}$. A key ingredient in our proof is a refined version of the symplectic density property of $\mathcal{C}_n$.