  In this paper we study the dynamical billiards on a convex 2D sphere. We
investigate some generic properties of the convex billiards on a general convex
sphere. We prove that $C^\infty$ generically, every periodic point is either
hyperbolic or elliptic with irrational rotation number. Moreover, every
hyperbolic periodic point admits some transverse homoclinic intersections. A
new ingredient in our approach is that we use Herman's result on Diophantine
invariant curves to prove the nonlinear stability of elliptic periodic points
for a dense subset of convex billiards.
