FREDHOLM PROPERTIES OF THE L2 EXPONENTIAL MAP ON THE SYMPLECTOMORPHISM GROUP

Let M be a closed symplectic manifold with compatible symplectic form and Riemannian metric g. Here it is shown that the exponential mapping of the weak L-2 metric on the group of symplectic diffeomorphisms of M is a non-linear Fredholm map of index zero. The result provides an interesting contrast between the L-2 metric and Hofer's metric as well as an intriguing difference between the L-2 geometry of the symplectic diffeomorphism group and the volume-preserving diffeomorphism group.