On the large-scale geometry of domains in an exact symplectic 4-manifold

We show that the space of open subsets of any complete and exact symplectic $4$-manifold has infinite dimension with respect to the symplectic Banach-Mazur distance; the quasi-flats we construct take values in the set of dynamically convex domains. In the case of $\mathbb{R}^4$, we therefore obtain the following contrast: the space of convex domains is quasi-isometric to a plane, while the space of dynamically convex ones has infinite dimension. In the case of $T^* S^2$, a variant of our construction resolves a conjecture of Stojisavljevi\'{c} and Zhang, asserting that the space of star-shaped domains in $T^* S^2$ has infinite dimension. Another corollary is that the space of contact forms giving the standard contact structure on $S^3$ has infinite dimension with respect to the contact Banach-Mazur distance.