The sharp $C^0$-fragmentation property for Hamiltonian diffeomorphisms and homeomorphisms on surfaces

In this paper, we present a $C^0$-fragmentation property for Hamiltonian diffeomorphisms. More precisely, it is known that for a given open covering $\mathcal{U}$ of a compact symplectic surface we can write each $C^0$-small enough Hamiltonian diffeomorphism as the composition of Hamiltonian diffeomorphisms compactly supported inside the open sets of the covering $\mathcal{U}$. We show that such a decomposition can be done with a Lipschitz estimate on the $C^0$-norm of the fragments. We also show the same property for the kernel of $\theta$, the mass-flow homomorphism for homeomorphisms. This answers a question from Buhovsky and Seyfaddini.