Symplectic diffeomorphisms as isometries of Hofer's norm

François Lalonde*, Leonid Polterovich

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 C0-norm of this isometry, once restricted to the ball of radius α of Ham(M) centered at the identity, gives a scale of norms rα on the group of symplectomorphisms. We conjecture that the subgroup of the symplectic diffeomorphisms which are isotopic to the identity and whose norms rα remain bounded when α → ∞ coincide with the group of Hamiltonian diffeomorphisms. We prove this conjecture for products of surfaces.

Original languageEnglish
Pages (from-to)711-727
Number of pages17
Issue number3
StatePublished - May 1997


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