Symplectic diffeomorphisms as isometries of Hofer's norm

Abstract

In this paper, we extend the Hofer norm to the group of symplectic diffeomorphisms of a manifold. This group acts by conjugation on the group of Hamiltonian diffeomorphisms, so each symplectic diffeomorphism induces an isometry of the group Ham( M) with respect to the Hofer norm. The C 0-norm of this isometry, once restricted to the ball of radius α of Ham( M) centered at the identity, gives a scale of norms r α on the group of symplectomorphisms. We conjecture that the subgroup of the symplectic diffeomorphisms which are isotopic to the identity and whose norms r α remain bounded when α → ∞ coincide with the group of Hamiltonian diffeomorphisms. We prove this conjecture for products of surfaces.