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 rho such that the open ball of radius rho 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 the weak form of the conjecture.