The sharp C⁰-fragmentation property for Hamiltonian diffeomorphisms and homeomorphisms on surfaces

In this paper, we present a C⁰-fragmentation property for Hamiltonian diffeomorphisms. More precisely, it is known that for a given open covering U of a compact symplectic surface we can write each C⁰-small enough Hamiltonian diffeomorphism as the composition of Hamiltonian diffeomorphisms compactly supported inside the open sets of the covering U. We show that such a decomposition can be done with a Lipschitz estimate on the C⁰-norm of the fragments. We also show the same property for the kernel of θ, the mass-flow homomorphism for homeomorphisms. This answers a question from Buhovsky and Seyfaddini.