  We consider any Finsler metric on a closed, orientable surface of genus
greater than one. H. M. Morse proved that we can associate an asymptotic
direction to minimal rays in the universal cover (in the Poincar\'e disc: a
point on the unit circle). We prove here that, if two minimal rays have a
common asymptotic direction, which is not a fixed point of the group of deck
transformations, then the two rays can intersect at most in a common initial
point. This has strong consequences for the structure of the set of minimal
geodesics, as well as for the set of Busemann functions associated to the
Finsler metric.
