Homoclinic intersections for geodesic flows on convex spheres

TL;DR

This paper proves that for generic convex spheres, hyperbolic closed geodesics typically have transversal homoclinic orbits, revealing complex dynamics in geodesic flows.

Contribution

It establishes that generically, hyperbolic closed geodesics on convex spheres possess transversal homoclinic orbits, a new insight into their dynamical structure.

Findings

Hyperbolic closed geodesics have transversal homoclinic orbits.

Results hold for generic $C^r$ convex spheres with $2 \,\le\, r \le \infty$.

Enhances understanding of geodesic flow complexity on convex spheres.

Abstract

In this paper we study some generic properties of the geodesic flows on a convex sphere. We prove that, $C^r$ generically ($2\le r\le\infty$), every hyperbolic closed geodesic admits some transversal homoclinic orbits.