Keith Burns and Howard Weiss
Spheres with positive curvature and nearly dense orbits for 
the geodesic flow
(643K, Postscript)

ABSTRACT.  For any $\ep > 0$, we construct an explicit smooth Riemannian 
metric on the sphere $S^n, n \geq 3$, that is within $\ep$ 
of the round metric and has a geodesic for which the 
corresponding orbit of the geodesic flow is $\ep$-dense in the 
unit tangent bundle. Moreover, for any $\ep > 0$, we construct 
a smooth Riemannian metric on $S^n, n \geq 3$, that is within 
$\ep$ of the round metric and has a geodesic for which the 
complement of the closure of the corresponding orbit of the 
geodesic flow has Liouville measure less than $\ep$.