Homoclinics for geodesic flows of surfaces

TL;DR

This paper proves that for a generic class of Riemannian metrics on closed surfaces, all hyperbolic closed geodesics have associated homoclinic orbits, revealing complex dynamical behavior.

Contribution

It establishes the existence of homoclinic orbits for all hyperbolic closed geodesics in Kupka-Smale metrics on closed surfaces, a new result in dynamical systems.

Findings

Homoclinic orbits exist for all hyperbolic closed geodesics.

The result applies to Kupka-Smale Riemannian metrics on closed surfaces.

It advances understanding of geodesic flow complexity.

Abstract

We prove that the geodesic flow of a Kupka-Smale riemannian metric on a closed surface has homoclinic orbits for all of its hyperbolic closed geodesics.