Homoclinic intersections for geodesic flows on convex spheres
  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. 
Zhihong Xia, Pengfei Zhang
math.DS