Homoclinic Points For Area-Preserving Surface Diffeomorphisms

2006/06/13 by Zhihong Xia, Xia, Zhihong
Mathematics · #Mathematical Dynamics and Fractals #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.math/0606291

Abstract

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, ..., ∞, 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, ..., ∞ 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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