Induced Hamiltonian maps on the symplectic quotient (Q953917)

Let \((M,\omega)\) be an exact symplectic manifold and \(G\) a compact Lie group that acts on \(M\) in a Hamiltonian fashion. Then any \(G\)-equivariant Hamiltonian map on \(M\) induces a map on the symplectic quotient of \(M\) by \(G\). Consider an autonomous Hamiltonian \(H\) with compact support on \(M\), with no non-constant closed trajectory in time less than \(1\) and denote by \(f^H\) the corresponding time-\(1\) map. The main result of the paper states that if the map \(f^H\) descends to the symplectic quotient to a map \(\phi(f^H)\) and if \(\text{Ham}(M,\omega)\) has no short loops, then the Hofer norm of the induced map is bounded above by the Hofer norm of \(f^H\). The proof is based on results in \textit{F. Lalonde} and \textit{D. McDuff} [Invent. Math. 122, No. 1, 1--69 (1995); errata 123, No. 3, 613 (1996; Zbl 0844.58021)].