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 $\ep$.