Homoclinic Points For Area-Preserving Surface Diffeomorphisms

TL;DR

This paper proves that for generic area-preserving surface diffeomorphisms and Hamiltonian flows, every hyperbolic periodic point or trajectory has a transversal homoclinic point, indicating complex chaotic dynamics.

Contribution

It establishes a $C^r$ connecting lemma for area-preserving surface diffeomorphisms and Hamiltonian flows, showing the generic existence of transversal homoclinic points.

Findings

Generic hyperbolic points have transversal homoclinic points

Connects hyperbolic periodic points with complex dynamics

Applicable to $C^r$ smooth area-preserving maps and Hamiltonian flows

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 $C^r$, $r=1, 2, ...$, $\infty$, 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.