Symplectic capacities of $S^1$-invariant dynamically convex domains in ${\mathbb{R}}^4$

In this paper, we prove that all normalized symplectic capacities agree for dynamically convex domains in ${\mathbb{C}}^2$ that are invariant under any Hamiltonian $S^1$-action isotopic to the Hopf diagonal action. We also give necessary and sufficient conditions for $S^1$-invariant domains to be dynamically convex.