  Hamiltonian dynamical systems tend to have infinitely many periodic orbits.
For example, for a broad class of symplectic manifolds almost all levels of a
proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert
conjecture is the existence problem for regular compact energy levels without
periodic orbits.
  Very little is known about how large the set of regular energy values without
periodic orbits can be. For instance, in all known examples of Hamiltonian
flows on linear spaces such energy values form a discrete set, whereas ``almost
existence theorems'' only require this set to have zero measure. We describe
constructions of Hamiltonian flows without periodic orbits on one energy level
and formulate conjectures and open problems.
