Homoclinic Points For Area-Preserving Surface Diffeomorphisms

Details Homoclinic Points For Area-Preserving Surface Diffeomorphisms Homoclinic Points For Area-Preserving Surface Diffeomorphisms

2006-06-13

arxiv.org/abs/math/0606291v1

Mathematics

Dynamical Systems

Scientific paper

We show a $C^r$ connecting lemma for area-preserving surface diffeomorphisms and for periodic Hamiltonian on surfaces. We prove that for a generic $C^r$, $r=1, 2, ...$, $\infty$, area-preserving diffeomorphism on a compact orientable surface, homotopic to identity, every hyperbolic periodic point has a transversal homoclinic point. We also show that for a $C^r$, $r=1, 2, ...$, $\infty$ generic time periodic Hamiltonian vector field in a compact orientable surface, every hyperbolic periodic trajectory has a transversal homoclinic point. The proof explores the special properties of diffeomorphisms that are generated by Hamiltonian flows.

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