Epsilon-non-squeezing and $C^0$-rigidity of epsilon-symplectic embeddings

An embedding $\varphi \colon (M_1, \omega_1) \to (M_2, \omega_2)$ (of symplectic manifolds of the same dimension) is called $\epsilon$-symplectic if difference $\varphi^* \omega_2 - \omega_1$ $\epsilon$-small with respect to a fixed Riemannian metric on $M_1$. We prove that sequence embeddings converges uniformly (on compact subsets) another embedding, then limit $E$-symplectic, where number $E$ depends only $\epsilon$ and $E (\epsilon) 0$ as $\epsilon 0$. This generalizes $C^0$-rigidity embeddings, answers question in topological quantum computing by Michael Freedman. As case, this rigidity theorem can be deduced from existence properties capacities. preserves capacity up an error, linear maps characterized property they preserve spectrum ellipsoids (centered at origin) error $\epsilon$-small. sketch alternative proof using shape invariant, which gives rise analogous characterization for $\epsilon$-contact embeddings.