53 research outputs found
An almost existence theorem for non-contractible periodic orbits in cotangent bundles
Assume M is a closed connected smooth manifold and H:T^*M->R a smooth proper
function bounded from below. Suppose the sublevel set {H<d} contains the zero
section and \alpha is a non-trivial homotopy class of free loops in M. Then for
almost every s>=d the level set {H=s} carries a periodic orbit z of the
Hamiltonian system (T^*M,\omega_0,H) representing \alpha.
Examples show that the condition that {H<d} contains M is necessary and
almost existence cannot be improved to everywhere existence.Comment: 9 pages, 4 figures. v2: corrected typo
Global surfaces of section for Reeb flows in dimension three and beyond
We survey some recent developments in the quest for global surfaces of
section for Reeb flows in dimension three using methods from Symplectic
Topology. We focus on applications to geometry, including existence of closed
geodesics and sharp systolic inequalities. Applications to topology and
celestial mechanics are also presented.Comment: 33 pages, 3 figures. This is an extended version of a paper written
for Proceedings of the ICM, Rio 2018; in v3 we made minor additional
corrections, updated references, added a reference to work of Lu on the
Conley Conjectur
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