Mathematics – Symplectic Geometry
Scientific paper
2006-08-31
Mathematics
Symplectic Geometry
12 pages; fixed an error, provided more details, reorganized exposition of proof of Theorem 1.2
Scientific paper
This paper introduces techniques of symplectic topology to the study of homoclinic orbits in Hamiltonian systems. The main result is a strong generalization of homoclinic existence results due to Sere and to Coti-Zelati, Ekeland and Sere, which were obtained by variational methods. Our existence result uses a modification of a construction due to Mohnke (originally in the context of Legendrian chords), and an energy--capacity inequality of Chekanov. In essence, we show the existence of a homoclinic orbit by showing a certain Lagrangian embedding cannot exist. We consider a (possibly time dependent) Hamiltonian system on an exact symplectic manifold (W, \omega = d \lambda) with a hyperbolic rest point. In the case of periodic time dependence, we show the existence of an orbit homoclinic to the rest point if \lambda(X_H) - H is positive and proper, H is positive outside a compact set and proper, and (W, \omega) admits the structure of a Weinstein domain. In the autonomous case, we establish the existence of an orbit homoclinic to the rest point if the critical level is of restricted contact-type, and the critical level has a Hamiltonian displaceable neighbourhood.
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