| The main theme of this thesis is the existence of periodic orbits for Hamiltonian dynamical systems.;We prove the relative almost existence theorem asserting that almost all low levels of a function on a symplectic manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold and the symplectic manifold meets some natural requirements. As an immediate consequence, we obtain the existence of contractible periodic orbits on almost all low energy levels for systems describing the motion of a charge in a symplectic magnetic field.;In general, periodic orbits need not exist on all low levels even when the phase space is four dimensional. We construct a proper C 2-smooth function on R4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four. We also give examples of functions with a sequence of regular levels without periodic orbits, converging to an isolated, but very degenerate, minimum.;The proof of the relative almost existence theorem hinges on the notion of the relative Hofer-Zehnder capacity and on showing that this capacity of a small neighborhood of a symplectic submanifold is finite. The latter is carried out by proving that the Floer homology of a test function is non-zero for a suitable interval of actions. |
