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On the injectivity radius in Hofer's geometry

In this note we consider the following conjecture: given any closed symplectic manifold $M$, there is a sufficiently small real positive number $ρ$ such that the open ball of radius $ρ$ in the Hofer metric centered at the identity on the group of Hamiltonian diffeomorphisms of $M$ is contractible, where the retraction takes place in that ball -- this is the strong version of the conjecture -- or inside the ambient group of Hamiltonian diffeomorphisms of $M$ -- this is the weak version of the conjecture. We prove several results that support that weak form of the conjecture.