A remark on the Morse Theorem about infinitely many geodesics between   two points
  We show that any two non-conjugate points on a forward or backward complete connected Finsler manifold can be joined by infinitely many geodesics which are not covered by finitely many closed ones, provided that the Betti numbers of the based loop space grow unbounded. 
Erasmo Caponio and Miguel Angel Javaloyes
math.DG