INDUCED HAMILTONIAN MAPS ON THE SYMPLECTIC QUOTIENT

Abstract. Let (M,ω) be a symplectic manifold and G a compact Lie group that acts onM. Assume that the action of G onM is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian with compact support on M, with no non-constant closed trajectory in time less than 1 and time-1 map f. If the map fH descends to the symplectic quotient to a map Φ(fH) and the symplectic manifold M is exact and Ham(M,ω) has no short loops, we prove that the Hofer norm of the induced map Φ(fH) is bounded above by the Hofer norm of fH. 1