Hamiltonian loops from the ergodic point of view

The paper provides a link between ergodic theory and symplectic topology. Aclassical notion of ergodic theory is a skew product map associated with a loopin a group of transformations. We study skew products which come from loops inthe group of Hamiltonian diffeomorphisms of a symplectic manifold. Our mainquestion is which homotopy classes of loops can be represented by strictlyergodic skew products. We prove an existence result, and find an obstructionwhich arises from Hofer's geometry on the group of Hamiltonian diffeomorphisms.