Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles

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The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold. As an immediate consequence, we obtain the existence of contractible periodic orbits on almost all low energy levels for twisted geodesic flows with symplectic magnetic field. We 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 flow of a Hamiltonian with sufficiently large variation has a non-trivial contractible one-periodic orbit, when the Hamiltonian is constant and equal to its maximum near a symplectic submanifold and supported in a neighborhood of the submanifold.