The Hamiltonian Seifert Conjecture: Examples and Open Problems

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.